Around heat decay on forms and relations of nilpotent Lie groups

Michel Rumin

Abstract

One knows that the large time heat decay exponent on a nilpotent group is given by half the growing rate of the volume of its large balls. This work deals with the similar problem of trying to interpret geometrically the heat decay on (one) forms. We will show how it is (partially) related to the depth of the relations required to define the group. The tools used apply in general on Carnot-Carathéodory manifolds.

Contents

1 Introduction 2

2 Some differential geometry on C-C manifolds 3 2.1 Dilations and differential on graded groups . . . . . . . . . . . . . . 3 2.2 Cutting d_{0} out of d . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.2.1 C-C manifolds and graded groups . . . . . . . . . . . . . . . . 4 2.2.2 One homotopy and three complexes . . . . . . . . . . . . . . 5

2.3 A few examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.4 Geometric comments in degree ≤ 2 . . . . . . . . . . . . . . . . . . . 10

3 Near-cohomology 11 3.1 Definition and first applications . . . . . . . . . . . . . . . . . . . . . 11 3.2 From cones to spectral densities . . . . . . . . . . . . . . . . . . . . . 14

3.2.1 Γ-dimension and the rational case . . . . . . . . . . . . . . . 14 3.2.2 Extension to non-rational groups . . . . . . . . . . . . . . . . 15

3.3 The algebraic pinching of heat decay . . . . . . . . . . . . . . . . . . 18

4 Examples 20 4.1 Quadratically presented groups . . . . . . . . . . . . . . . . . . . . . 20 4.2 Some higher rank groups . . . . . . . . . . . . . . . . . . . . . . . . . 23

5 Towards a refined pinching 26 5.1 C-C ellipticity and applications . . . . . . . . . . . . . . . . . . . . . 26 5.2 Last examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

5.2.1 First inaudible relations . . . . . . . . . . . . . . . . . . . . . 31 5.2.2 Around H-groups . . . . . . . . . . . . . . . . . . . . . . . . 32

1 Introduction

Let (M, g) be a compact riemannian manifold with fundamental group Γ = π_{1}(M). We denote by M̃ its universal cover and g̃ the pull back metric. The de Rham differential d : Ω^{p}(M̃) → Ω^{p+1}(M̃) acts between p and p + 1-forms of M̃.

We are interested in the spectrum of the p-Laplacian Δ_{p} = dδ + δd acting on L^{2} p-forms of M̃. More precisely, one would like to know what kind of geometric information about M or Γ is encoded in the near zero spectrum of Δ_{p}.

The first result of this kind deals with the invertibility of Δ_{0}, the Laplacian on functions.

Theorem 1.1. (Brooks [2]) 0 belongs to the spectrum of Δ_{0} iff Γ is amenable.

(That means that Γ may be exhausted by parts Γ_{i} such that \frac{#(∂Γ_{i})}{#(Γ_{i})} → 0.)

Roughly, the argument that Δ_{0} is not invertible if Γ is amenable is based on the idea that one can approach the constant function on M̃ by compactly supported test functions with small differential with respect to their integral norm. These approximative units can also be used to cut off the pull back of any harmonic form of M. This shows therefore that if Γ is amenable and H^{p}(M, ℝ) ≠ 0 then 0 belongs to the spectrum of Δ_{p} on M̃. This is satisfied for any p ≤ dimM in the case we will study of M = G/Γ, with Γ a cocompact group of a nilpotent Lie group G (see [10]).

Note also that in these examples, 0 is never embedded in Sp(Δ_{p}). There is no harmonic L^{2}-form on M̃ = G, otherwise they would have to be invariant through the Killing direction associated to the non-vanishing center of these groups. More precise information about the density of Sp(Δ_{p}) near 0 is obtained with the help of the notions of Γ-dimension, and Γ-trace.

Recall (see [1]) that in the case of a Γ-invariant smoothing operator S, acting on Ω^{∗}(M̃), one can consider the average diagonal trace density of its Schwarz kernel K_{S}, namely

Tr_{Γ}(S) = ∫_{F} Tr(K_{S}(x, x))dvol ,

where F stands for any fundamental domain of the Γ-action. For example S can be the heat operator e^{−tΔ_{p}} or the spectral projection Π(Δ_{p} ≤ λ). Actually we can first take profit of the orthogonal splitting of Δ_{p} in δd and dδ to slightly precise the analysis. So the basic operators in concern will be δd restricted to H = (ker d)^{⊥} and the associated heat e^{−tδd} on H. Since these the second operator is the Laplace transform of the former, the two asymptotics of Tr_{Γ}(e^{−tδd}) when t → +∞ and of F_{δd}(λ) = Tr_{Γ}(Π(δd ≤ λ)) when λ → 0 are related in the following way (see appendix of [14]). One has

Tr_{Γ}(e^{−tΔ_{p}}Π_{H}) ≍ t^{−α_{p}/2} when t → +∞

iff

F_{δd}(λ) ≍ λ^{α_{p}/2} when λ → 0 .

A more general definition of α_{p}, called the pth Novikov-Shubin number of M, is

α_{p} = 2 lim inf_{λ→0}(ln F_{δd}(λ)/ ln λ) ∈ [0, +∞]  (1)

(also it seems there is no known geometric example where this liminf is not an actual limit.)

The first one, α_{0}, describing heat decay on functions is known.

Theorem 1.2. (Varopoulos [27], Gromov) α_{0} < +∞ iff Γ has polynomial growing that is iff Γ is (virtually) nilpotent, and then

α_{0} = growing rate of Γ = lim_{R→+∞} \frac{ln Vol_{Γ}(B(0, R))}{ln R} ∈ ℕ.

It appears then that α_{0} is a rough invariant of M, independent of the metric g, depending only here on the large scale structure of Γ = π_{1}(M). For the other α_{p}, one has

Theorem 1.3. (Gromov-Shubin, Efremov [14]) α_{p} are homotopy invariants of M.

In fact it turns out that α_{p} depends on the (rational) homotopy type of M up to degree p. In particular the next one α_{1}, describing heat on 1-forms, depends only on π_{1}(M), and is thus of particular interest. We will mainly focus on its study here, although some sections will deal with related fields.

The (very partial) results are actually disseminated along the paper. We will illustrate them with (carefully chosen) examples. This work relies on the use of constructions that have already been exposed in [25]. Anyway we recall them first.

2 Some differential geometry on C-C manifolds

2.1 Dilations and differential on graded groups

As time increases, heat spreads in larger and larger domains. It is therefore convenient to re-scale the phenomena. This requires some dilation acting on the space. Typical groups admitting such a structure are graded nilpotent (Lie) groups. These are groups G such that their Lie algebra 𝔤 splits in

𝔤 = ⨁_{i=1}^{r} 𝔤_{i} with [𝔤_{1}, 𝔤_{i}] ⊂ 𝔤_{i+1}.

We call r the rank of G. The multiplication by i on 𝔤_{i} induces a family of dilations h_{ε} on G by h_{ε}(exp X_{i}) = ε^{i}(exp X_{i}) for X_{i} ∈ 𝔤_{i}. The function w = i on 𝔤_{i} is called the weight of vectors, and similarly on covectors. It can be extended to the whole differential algebra of G. Forms of weight p are spanned by

θ^{i_{1}} ∧ θ^{i_{2}} ∧ ⋯ ∧ θ^{i_{k}} with ∑_{j=1}^{k} w(θ^{i_{j}}) = p.

De Rham’s differential d is not homogeneous with respect to the weight. In fact it splits like

d = d_{0} + d_{1} + ⋯ + d_{r}  (2)

with components d_{k} increasing the weight by k. Indeed on functions d_{0}f = 0 since there is no 1-forms of weight 0, whereas d_{k}f for 1 ≤ k ≤ r is just df restricted to vectors of weight k. Now, if α is a (left) invariant form of weight p one has d(fα) = df ∧ α + fdα. Then for k ≥ 1, d_{k}(fα) = d_{k}f ∧ α, while

d_{0}(fα) = fd_{0}α = fdα

is an algebraic (order 0) operator. The assertion that for invariant forms dα = d_{0}α is of the same weight as α is exactly dual to the property that [𝔤_{i}, 𝔤_{j}] ⊂ 𝔤_{i+j}, as is seen starting from invariant 1-forms for which dθ(X, Y) = −θ([X, Y]) on invariant vectors.

Now let G be endowed with an invariant metric such that the 𝔤_{i} are mutually orthogonal. If α is a form of weight p, one has point-wise ||h_{ε}^{∗}α|| = ε^{p}||α||, so that

||d(h_{ε}^{∗}α)|| ≥ ε^{p}||d_{0}α|| = ||d_{0}(h_{ε}^{∗}α)||.

Therefore d_{0}, the zero order part of d, invisible on functions and ℝ^{n}, is an obstruction of decreasing the differential of forms through dilations. In other words, the spectral spaces E(δd ≤ λ) have to contract on ker d_{0} in some sense when λ → 0.

2.2 Cutting d_{0} out of d

One would like to find a way of working directly on ker d_{0} when dealing with small spectrum problems on our groups. The spectral sequence technique achieves this from the algebraic viewpoint. In fact the previous discussion pointed out a natural decreasing filtration of Ω^{∗}G by the spaces F^{p} of forms of weight ≥ p. The decomposition (2) says that d respects it. Thus we have a filtered complex giving rise to a sequence of spaces converging to the (graded part of) de Rham cohomology of G.

We briefly recall this technique, d_{0} is interpreted as d acting on F^{p}/F^{p+1}. Let E_{0} = ker d_{0}/ Im d_{0} denotes its cohomology. This is the bundle which fiber is the Lie algebra cohomology H^{∗}(𝔤, ℝ). We get then d acting on the quotient space E_{0}, whose cohomology is called E_{1}, etc. Note that this process produces successive quotients by image of differential operators, not very easy to handle analytically. The fact that d_{0} is algebraic will allow to perform part of the construction inside the de Rham complex. This will be useful to keep track of the original problem of analyzing the small spectrum of d.

In fact the constructions apply to more general spaces than graded groups called Carnot-Carathéodory manifolds which we introduce now.

2.2.1 C-C manifolds and graded groups

Definition 2.1. An equiregular Carnot-Carathéodory structure on a manifold M consists in an (increasing) filtration of T M by bundles H_{i} such that

[H_{1}, H_{i}] ⊂ H_{i+1} with H_{r} = T M for some r.  (3)

Remarks 2.2.

- Strictly speaking this definition is more general than the usual one given in [13] for instance. Here we don’t require T M to be generated by H_{1} through brackets, that is H_{1} to be an Hörmander distribution and H_{i+1} = [H_{1}, H_{i}]. This will allow us to change the grading if necessary.

- The equiregularity assumption means that we suppose the distributions H_{i} are bundles. Their dimensions are not allowed to jump.

The link with our previous discussion is the well known fact that an equiregular C-C manifold admits a tangent graded nilpotent Lie algebra 𝔤_{x_{0}} at each point. Indeed due to (3) the usual bracket on T M admits a quotient map

[ , ]_{0} : H_{p}/H_{p−1} × H_{q}/H_{q−1} → H_{p+q}/H_{p+q−1}.

which is an algebraic operator (order 0), since [X, fY]_{0} = Π_{H_{p+q}/H_{p+q−1}}(f[X, Y] + (X.f)) = f[X, Y]_{0}. Therefore [ , ]_{0} defines a graded Lie algebra structure on

𝔤_{x_{0}} = ⨁_{i=1}^{r} 𝔤_{𝔦} with 𝔤_{𝔦} = H_{i}/H_{i−1} at x_{0}.  (4)

We recover also a decreasing filtration of Λ^{∗}T^{∗}M. Firstly vectors in H_{p} are called of weight ≤ p, and dually a differential form of degree k is of weight ≥ p iff it vanishes on all sets of k vectors of total weight < p. Let F_{p} be the bundle of forms of weight ≥ p. Again F_{p} is d-stable as come from (3), and we get a filtered complex on Ω^{∗}M. As before d_{0} = d, acting on F_{p}/F_{p−1}, is an algebraic operator. This zero order part of d is identified with the previous d_{0} we introduced on 𝔤_{x_{0}}.

Examples 2.3.

- Let D^{k} be a distribution of k-planes in ℝ^{n}. Generically in the jets of the vector fields generating it, D^{k} will not be integrable and even will be an Hörmander distribution. This gives a lot of (true) C-C structures on ℝ^{n}.

- We just describe some classical cases. An hyperplane distribution D^{2n} in ℝ^{2n+1} is called a contact structure when the Lie bracket [ , ]_{0} is non-degenerate on D. Reducing this bilinear form gives that G_{x_{0}} = exp 𝔤_{x_{0}} is isomorphic to the Heisenberg group. Even more, thanks to Darboux’ theorem the contact structure itself is locally isomorphic to its tangent group G_{x_{0}}.

- A generic D^{2} in ℝ^{4}, called an Engel’s structure, also gives rise to a unique tangent 𝔤_{x_{0}}. This is the 4-dimensional Lie algebra of rank 3 with the following relations [X, Y]_{0} = Z, [X, Z]_{0} = T and 0 elsewhere. Here D = H_{1} = span(X, Y), H_{2} = [D, D] = span(X, Y, Z), and H_{3} = ℝ^{4} = [D, [D, D]] = span(X, Y, Z, T). Again D is an Hörmander distribution, but we can look this structure differently. Consider H′_{1} = span(X), H′_{2} = span(X, Y) = D, H′_{3} = H_{2}, H′_{4} = H_{3} = ℝ^{4}. This C-C structure has the same model group G_{x_{0}} as before, but with a different grading. This will give another way of rescaling the asymptotic problems we consider.

2.2.2 One homotopy and three complexes

We now show how to retract the de Rham complex of a C-C manifold M on its E_{0} part. Recall that E_{0} = ker d_{0}/ Im d_{0} = H^{∗}(𝔤_{x_{0}}), where 𝔤_{x_{0}} is tangent Lie algebra at x_{0}. In order to work on bundles we will assume now that dim(E_{0}) is constant (always satisfied on an open dense set of M).

Definition 2.4. A C-C structure is E_{0}-regular if E_{0} is a bundle.

Choose a metric on M. Let V_{i} be the orthogonal supplement of H_{i} in H_{i−1}. By (4), this fixes an isometry between T_{x_{0}}M = ⨁_{i=1}^{r} V_{i} and 𝔤_{x_{0}} = ⨁_{i=1}^{r} 𝔤_{𝔦} and fixes the weight on T M. This allows to see d_{0} as acting between spaces of (true, not quotiented) forms on M of given pure weight. We identify E_{0} with ker d_{0} ∩ ker δ_{0} = ker d_{0} ∩ (Im d_{0})^{⊥}, where δ_{0} is the metric adjoint of d_{0}. We also get a partial inverse of d_{0} by d_{0}^{−1} = (δ_{0}d_{0})^{−1}δ_{0}. We can use this d_{0}^{−1} as a partial inverse of d.

Define the following retraction on Ω^{∗}M

r = Id −d_{0}^{−1}d − dd_{0}^{−1}.  (5)

This is an homotopical equivalence (preserving d and the cohomology) whose zero order term is r_{0} = Id −d_{0}^{−1}d_{0}−d_{0}d_{0}^{−1} = Π_{E_{0}} a projection, whereas r−r_{0} = −d_{0}^{−1}(d− d_{0})−(d−d_{0})d_{0}^{−1} strictly increases the weight and is therefore nilpotent. This incites to iterate the homotopy r in order to retract the de Rham complex on the smallest possible space. The maps r^{k} actually converge toward a map that certainly have to be both an homotopical equivalence and a projection to a sub-complex along an other. The following lemma is useful to identify quickly the limit spaces and operators.

Lemma 2.5. The map d_{0}^{−1}d induces an isomorphism on Im d_{0}^{−1}, whose inverse is a differential operators P.

Proof. ([25]) On Im d_{0}^{−1}, one can write d_{0}^{−1}d = Id +D where D = d_{0}^{−1}(d − d_{0}) is nilpotent since it strictly increases the weight of forms. One has then

P = (d_{0}^{−1}d)^{−1} = ∑_{k=0}^{d(M)}(−1)^{k}D^{k}

where d(M) = weight(Λ^{max}T^{∗}M).

Define Q = P d_{0}^{−1} with P as in the lemma. One has then.

Theorem 2.6. Let M be a E_{0}-regular C-C space.

1. Ω^{∗}M splits in the direct sum of two sub-complexes

E = ker d_{0}^{−1} ∩ ker(d_{0}^{−1}d) and F = Im d_{0}^{−1} + Im(dd_{0}^{−1}).

The projection Π_{F} on F along E is given by the differential operator Qd+dQ, with Q as above.

2. The homotopies r^{k} converge to Π_{E} the projection on E along F. It is the homotopical equivalence given by the differential operator Id −Π_{F} = Id −Qd − dQ.

3. One has

Π_{E_{0}}Π_{E}Π_{E_{0}} = Π_{E_{0}} and Π_{E}Π_{E_{0}}Π_{E} = Π_{E},  (6)

saying that E and E_{0} are in bijection, and that Π_{E} restricted to E_{0} and Π_{E_{0}} restricted to E are inverse maps of each other. In particular the complex (E, d) is conjugated to another one (E_{0}, d_{c}) with d_{c} = Π_{E_{0}}dΠ_{E}Π_{E_{0}}.

The constructions are summarized in the diagram

\xymatrix{F \ar@{^{(}->}[r]^{i} \ar[d]_{d} & Ω^{∗}M \ar[l]^{Π_{F}} \ar[r]^{Π_{E}} \ar[d]_{d} & E \ar@{^{(}->}[l]^{i} \ar[r]^{Π_{E_{0}}} \ar[d]_{d} & E_{0} \ar[l]^{Π_{E}} \ar[d]^{d_{c}} \\ F \ar@{^{(}->}[r]^{i} & Ω^{∗}M \ar[l]^{Π_{F}} \ar[r]^{Π_{E}} & E \ar@{^{(}->}[l]^{i} \ar[r]^{Π_{E_{0}}} & E_{0} \ar[l]^{Π_{E}}}

A short proof is given in [25](thm 1). We don’t repeat it here, but insist on particular points.

Observe that E is a space of forms satisfying some differential equations (E = ker d_{0}^{−1}∩ker(d_{0}^{−1}d)) that projects bijectively onto the algebraic E_{0} = ker d_{0}∩ker d_{0}^{−1}. Even more, the equation Π_{E_{0}}Π_{E}Π_{E_{0}} = Π_{E_{0}} says that Π_{E} restricted to E_{0} is Id +a (E_{0})^{⊥} part. In other words E is a particular space of liftings (extensions) of E_{0}. For computations, we notice from (5) that the retraction r preserves the space (ker d_{0})^{−1} = ker δ_{0} ⊃ E_{0} on which

r = Id −d_{0}^{−1}(d − d_{0})

with d_{0}^{−1}(d−d_{0}) strictly increasing the weight. Starting from some form α of weight p in E_{0} this gives by iteration the successive extensions in greater weight by

\{ \begin{matrix}(Π_{E}α)_{p} & = α \\ (Π_{E}α)_{p+k+1} & = −d_{0}^{−1}(∑_{l=1}^{r} d_{l}(Π_{E}α)_{p+k+1−l}) \end{matrix}  (7)

where d_{l} is the part of d that increases the weight by l.

Remarks 2.7.

- At this point it is clear that this construction is everything but a new idea, merely an hold one in a fancy dress. Namely this Π_{E} is a realization of the first homotopical equivalence that is predicted by the general fact that the spectral sequence starting on E_{0} will finally compute (the graded part of) the cohomology. The map d_{c} describes the remaining obstructions of extending a form in E_{0} to a true closed one, put all together.

- Formally at least it is tentative to carry on with the retraction process by replacing d_{0} by d_{c,1}, the part of d_{c} that increases the weight by one. Choosing a supplement space to ker d_{c,1} as the L^{2} closure of Im d_{c,1}, leads to a (now not bounded) partial inverse d_{c,1}^{−1} of d_{c,1} and an another retraction r_{1} = Id −d_{c,1}^{−1}d_{c} − d_{c}d_{c,1}^{−1}. It should be iterated and so on. Of course this does not seem realistic analytically. Anyway some analysis (yet a little bit mysterious) related to the spectral sequence structure is working. We will describe it in section 5.

- We didn’t take care of the invariance of d_{c} on the choice of metric, mainly because it is not invariant in general ! This doesn’t matter here. Again this construction is taken as a convenient approximation of the underlying invariant spectral sequence, which is hopefully related to the (even homotopically) invariant problem of studying the asymptotic heat decays.

We conclude this general section with a remark on duality. As observed in [25], the complex (E_{0}, d_{c}) is Hodge ∗-dual. This should look rather surprising since the retraction map r = Id −d_{0}^{−1}d − dd_{0}^{−1} breaks the symmetry between d_{0} and δ_{0}. Anyway, we have

Proposition 2.8. 1. ∗δ_{0} = (−1)^{k+1}d_{0}∗ on E_{0}^{k} and ∗ preserves E_{0}.

2. ∗E is orthogonal to F, equivalently the pairing (α, β) → ∫_{M} α ∧ β vanishes on E × F. One has ∗Π′_{E} = Π_{E}∗, where Π′_{E} is the formal adjoint of Π_{E}.

3. ∗δ_{E} = (−1)^{k+1}d_{E}∗ on k-forms. Similarly ∗δ_{c} = (−1)^{k+1}d_{c}∗ on E_{0}^{k}.

Proof. - The first observation comes from the same (point-wise) duality standing at the Lie algebra level on 𝔤_{x_{0}}.

- By definition F = Im δ_{0} + Im dδ_{0} and therefore

∗(F^{⊥}) = ∗(ker d_{0} ∩ ker d_{0}δ) = ker δ_{0} ∩ ker dδ_{0} = E.

Then,

(α, Π′_{E}β) = (Π_{E}α, β) = ∫_{M} Π_{E}α ∧ ∗β
= ∫_{M} Π_{E}α ∧ Π_{E} ∗ β = ∫_{M} α ∧ Π_{E} ∗ β
= (α, ∗^{−1}Π_{E} ∗ β).

- From d_{E} = dΠ_{E} = Π_{E}d, we find that

∗δ_{E} = ∗Π′_{E}δ = Π_{E} ∗ δ = (−1)^{k+1}Π_{E}d∗ = (−1)^{k+1}d_{E}∗,

and similarly for δ_{c} starting from d_{c} = Π_{E_{0}}d_{E}Π_{E_{0}} and using that Π_{E_{0}} commutes with ∗ and is self-adjoint, being an orthogonal projection.

2.3 A few examples

We briefly describe some specific cases of the previous constructions.

• On ℝ^{n}, or M^{n} with the trivial C-C structure (H_{1} = T M), one has d_{0} = 0, and d_{c} = d.

• Let (M^{2n+1}, H) be a contact structure. Let θ be such that H = ker θ. Ω^{∗}M splits in Ω^{∗}H + θ ∧ Ω^{∗}H and the 0 order part of d(α + θ ∧ β) is readily seen to be dθ ∧ β. From this we get that E_{0}^{k} are primitive horizontal k-forms if k ≤ n, and the vertical co-primitive ones (∈ ker(dθ ∧ Id)) if k ≥ n + 1. These bundles E_{0}^{k} are of pure C-C weight, namely k if k ≤ n and k + 1 if k ≥ n + 1.

This implies that the d_{c} complex consists in first order operators, except an order 2 one in degree n, due to the “jump” of weights between E_{0}^{n} (weight n) and E_{0}^{n+1} (weight n + 2). This is the contact complex.

• Let M = G be the Engel group (see examples 2.3). Recall this is the 4 dimensional Lie group, with relations [X, Y] = Z, [X, Z] = T, and other brackets vanishing. In the dual base of 1-forms, this translates in

d_{0}θ_{X} = d_{0}θ_{Y} = 0, d_{0}θ_{Z} = −θ_{X} ∧ θ_{Y} and d_{0}θ_{T} = −θ_{X} ∧ θ_{Z}.

We find then

E_{0}^{0} = C(G) : functions on G,
E_{0}^{1} = span (θ_{X}, θ_{Y}) : horizontal one forms,
E_{0}^{2} = span (θ_{Y} ∧ θ_{Z}, θ_{X} ∧ θ_{T}) = ∗E_{0}^{2},
E_{0}^{3} = ∗E_{0}^{1} = span (θ_{Y} ∧ θ_{Z} ∧ θ_{T} , θ_{X} ∧ θ_{Z} ∧ θ_{T})
E_{0}^{4} = Ω^{4}G.

We compute d_{c}. As there is nothing to lift on functions Π_{E}f = f, and

d_{c}f = Π_{E_{0}^{1}}df = d_{H_{1}}f = (X.f)θ_{X} + (Y.f)θ_{Y} .

Using (7), we can compute Π_{E}α = α̃ for α = α(X)θ_{X} + α(Y)θ_{Y} ∈ E_{0}^{1}. One gets

−d_{0}(α̃(Z)θ_{Z}) = α̃(Z)θ_{X} ∧ θ_{Y}
= d_{1}α = (X.α(Y) − Y.α(X))θ_{X} ∧ θ_{Y} ,

thus α̃(Z) = X.α(Y) − Y.α(X). Again with (7),

−d_{0}(α̃(T)θ_{T}) = α̃(T)θ_{X} ∧ θ_{Z}
= (X.α̃(Z) − Z.α(X))θ_{X} ∧ θ_{Z},

and α̃(T) = X.α̃(Z) − Z.α(X) = X^{2}.α(Y) − (XY + Z).α(X). Now d_{c}α is the restriction of dα̃ to E_{0}^{2}, that is

d_{c}α = (Y.α̃(Z) − Z.α(Y))θ_{Y} ∧ θ_{Z} + (X.α̃(T) − T.θ(X))θ_{X} ∧ θ_{T} .

The full complex d_{c} may be completed either by ∗-duality or computing Π_{E} in degree 2. The result can be read out from the following diagram, adding all possible travels between points gives the various components of the liftings and d_{c}

\xymatrix{ & (θ_{Y}) \ar[rr]^{−Z} \ar@{-->}[dr]^{X} &  & (θ_{Y ∧Z}) \ar@{-->}[dr]_{−X} \ar@{-->}[drrr]^{T} \\ (f) \ar@{-->}[ur]^{Y} \ar@{-->}[dr]_{X} &  & θ_{Z} \ar@{-->}[ur]_{Y} \ar[dr]^{X} &  & θ_{Y ∧T} \ar@{-->}[dr]_{X} \ar[rr]_{−Z} &  & (θ_{Y ∧Z∧T}) \ar@{-->}[dr]^{X} \\  & (θ_{X}) \ar@{-->}[ur]^{−Y} \ar[rr]^{−Z} \ar@{-->}[drrr]_{−T} &  & θ_{T} \ar[dr]^{X} &  & θ_{Z∧T} \ar@{-->}[ur]_{Y} \ar@{-->}[dr]^{X} &  & (Ω^{4}G) \\  &  &  &  & (θ_{X∧T}) \ar@{-->}[ur]^{−Y} \ar[rr]^{−Z} &  & (θ_{X∧Z∧T}) \ar@{-->}[ur]^{−Y}}  (8)

It should be noted that d_{c} doesn’t depend on the various possible choices of grading on G (see 2.3), if keeping the same basis for E_{0} = ker d_{0}/ Im d_{0}.

Now, what about d_{c} on a general Engel structure D^{2} ⊂ ℝ^{4} (or T M^{4})? Unlike the contact case these are not locally diffeomorphic to their tangent group G. That means that we can’t find a local system of vector fields X, Y generating D^{2} and satisfying exactly the previous bracket relations. Even so, thanks to [ , ]_{0}, they can be satisfied up to vectors of lower C-C weight. The conclusion is that the general d_{c} will be a perturbation of the above d_{c} on G by differential operators of lower C-C weight. This is of course a general feature of the construction.

• We close this series of examples with a glimpse toward the nice case of C-C structures given by a generic 4 dimensional distribution D^{4} in ℝ^{7} (or T M^{7}). Here again we are in an exceptional situation where there are finitely many isomorphic type of possible tangent graded group G_{x_{0}}.

We see this. Its Lie algebra 𝔤_{x_{0}} is generated by 𝔤_{1} = D and 𝔤_{2} = T M/D and determined by its curvature d_{0} : Λ^{1}(T M/D)^{∗} → Λ^{2}D^{4∗} given by d_{0}θ(X, Y) = −θ([X, Y]_{0}). This map is injective iff [ , ]_{0} : Λ^{2}D → T M/D is surjective, that is the distribution D is bracket generating. In that (generic) case L = Im d_{0} is a 3 dimensional subspace in Λ^{2}D^{∗}. This is a famous case where L is determined up to isomorphism by the signature of the quadratic form q(ω)dvol_{D} = ω ∧ ω restricted to it. The distribution D is called elliptic if q is positive definite on L (changing the orientation of D if necessary). This is an open condition, but not dense due to the other open possibility of an hyperbolic (2, 1) signature !

If D is elliptic, our assumption by now, one knows there exists a unique conformal class of metric g on D such that L becomes Λ^{2,+}D^{∗} the space of ∗-self-dual 2 forms of D^{4}. It is convenient to describe L by seeing D as the quaternions ℍ in which case L = span {dθ_{l}}_{1≤l≤3} is such that

d_{0}θ_{l} = g(J_{l}·, ·),

for J_{l} = i, j, k ∈ ℍ. This is an example of an Heisenberg type group or H-group (see [19], [7]), associated here to the quaternion-hyperbolic group Sp(2, 1) .

The cohomology E_{0} of this structure is easily computed.

1. Again E_{0}^{1} = Λ^{1}D^{∗} are the horizontal 1-forms.

2. E_{0}^{2} splits in two weights E_{0}^{2,(2)} = Λ^{2}D^{∗}/L ≃ Λ^{2,−}D^{∗}, the anti self-dual part of Λ^{2}D^{∗}. The weight 3 part E_{0}^{2,(3)} is generated by θ_{1} ∧ J_{1}α − θ_{2} ∧ J_{2}α and θ_{2} ∧ J_{2}α − θ_{3} ∧ J_{3}α for α ∈ Λ^{1}D^{∗}.

3. E_{0}^{3} is seen to be of pure weight 4. It is the 14-dimensional space of 3-forms γ = ∑_{i=1}^{3} θ_{i} ∧ β_{i} with β_{i} ∈ Λ^{2}D^{∗} satisfying d_{0}γ = ∑_{1≤i≤3} (dθ_{i}, β_{i}) dvol_{D} = 0 modulo Im d_{0} generated by d_{0}(θ_{1} ∧ θ_{2}) = −θ_{1} ∧ dθ_{2} + θ_{2} ∧ dθ_{1} and the two others coming from permutation.

4. The missing degrees are obtained by duality.

We summarize the information in the following diagram, with the degree of forms along x and (weight − degree) along y

\xymatrix{ &  &  &  &  & E_{0}^{5,(8)} \ar[r]^{d_{c}} & E_{0}^{6,(9)} \ar[r]^{d_{c}} & Ω^{7}M \\  &  &  &  & E_{0}^{4,(6)} \ar[r]^{d_{c}} \ar@{-->}[ur]^{d_{c}} & E_{0}^{5,(7)} \ar@{-->}[ur]_{d_{c}} \\  &  & E_{0}^{2,(3)} \ar[r]^{d_{c}} & E_{0}^{3,(4)} \ar@{-->}[ur]^{d_{c}} \\ f \ar[r]^{d_{D}} & Λ^{1}D^{∗} \ar[r]^{d^{−}} \ar@{-->}[ur]^{d_{c}} & Λ^{2,−}D^{∗} \ar@{-->}[ur]_{d_{c}}}  (9)

The interesting feature here is that the bottom line looks like an elliptic complex on the 4 dimensional D, except that D is not integrable here. Anyway the ellipticity of this first order part of d_{c} will be helpful in in determining the heat decays of forms on the tangent group.

2.4 Geometric comments in degree ≤ 2

We give some precisions on the structure of d_{c} in degree ≤ 2.

• We start with E_{0}^{1} = H^{1}(𝔤_{x_{0}}) on C-C manifold M. If D is bracket generating, that is an Hörmander distribution in T M, then 𝔤_{x_{0}} will be generated by its vectors of weight 1. (These particular graded groups are often called filtered groups.) This is equivalent to the fact that d_{0} is injective on 1 forms of weight ≥ 1. Indeed d_{0} restricted to one forms is dual to the bracket map [ , ]_{0} : 𝔤_{x_{0}} ∧ 𝔤_{x_{0}} → 𝔤_{x_{0}} and this one is surjective onto vectors of weight ≥ 1 precisely when D is bracket generating.

As a consequence, E_{0}^{1} = ker d_{0}/ Im d_{0} consists only of forms of weight 1, which can be identified with Λ^{1}D^{∗}, the partial 1-forms on D. Then d_{c}f is simply the restriction of df to vectors in D, a first order operator.

• Assume again that 𝔤_{x_{0}} is filtered, that is D = 𝔤_{1} is bracket generating. We now see how E_{0}^{2} = H^{2}(𝔤_{x_{0}}) is linked to the relations defining G_{x_{0}}. Let G̃ be the free Lie group generated over D. The map Π : G̃ → G_{x_{0}} is surjective since D is bracket generating. Therefore one has G_{x_{0}} = G̃/N where N = Π^{−1}(0) interprets as the normal subgroup of relations of G (with respect to G̃). At the Lie algebra level N = exp 𝔫 with 𝔫 = ker π is the ideal of relations of 𝔤_{x_{0}}.

Since 𝔫 is an ideal, r = 𝔫/[𝔫, 𝔤̃] may be viewed as its space of generators, isomorphic to R = N/(N, G̃) generating the relations N of G_{x_{0}}. Now a (classical) fact is that

Proposition 2.9. Λ^{1}r^{∗} is naturally isomorphic to H^{2}(𝔤_{x_{0}}).

Proof. This is an homological consequence of the short exact sequence 𝔫 → 𝔤̃ → 𝔤_{x_{0}}. We give the principle in our case. Consider d_{0} : Λ^{1}𝔤̃^{∗} → Λ^{2}𝔤̃^{∗} given by d_{0}α(X, Y) = −α([X, Y]).

The space Λ^{1}r^{∗} identifies with 1-forms on 𝔫 vanishing on [𝔫, 𝔤̃]. Let α ∈ Λ^{1}r^{∗}, and \overline{α} be any extension of α to Λ^{1}𝔤̃^{∗}. Then d_{0}\overline{α} vanishes on 𝔫 × 𝔤̃ and is therefore the pull-back of a 2-form β on 𝔤_{x_{0}}. One checks easily that β ∈ ker d_{0,𝔤_{x_{0}}}, and that β → β + d_{0}γ when the extension \overline{α} is changed (by a one form vanishing on 𝔫). We get therefore a map

[d_{0}] : Λ^{1}r^{∗} → H^{2}(𝔤_{x_{0}}) = ker d_{0}/ Im d_{0}
α → [β = d_{0,𝔤̃}\overline{α}].

- Injectivity of [d_{0}]. If [d_{0}]α = 0 then ∃γ ∈ Λ^{1}𝔤_{x_{0}} such that d_{0}(\overline{α} − Π^{∗}γ) = 0. But since D generates 𝔤̃, we knows that \overline{α} − Π^{∗}γ is a form of weight 1. Therefore by restriction on 𝔫, of weight ≥ 2, we get α = 0.

- The surjectivity of [d_{0}] relies on the fact that H^{2}(𝔤̃) = 0.

This can been taken as a definition of 𝔤̃, meaning that there is one construction of 𝔤̃ which is precisely done to achieve this ! (Starting with D, vectors of weight 1, the space of covectors of weight k + 1 in 𝔤̃ is taken to be ker d_{0} among the two forms of weight ≤ k + 1 already constructed.) More geometrically, to β ∈ Λ^{2}𝔤̃^{∗} ∩ ker d_{0}, a two cocycle, corresponds an extension of 𝔤̃ by ℝ, 𝔤′ = 𝔤̃ ⊕ ℝβ with [ , ]′ defined by [(x_{1}, t_{1}), (x_{2}, t_{2})]′ = ([x_{1}, x_{2}], β(x_{1}, x_{2})). By universality of 𝔤̃, this extension has to be trivial, a product, which corresponds to a d_{0}-exact β.

We finish the proof. If β is a d_{0}-closed two form of 𝔤_{x_{0}}, its pull-back Π^{∗}β on 𝔤̃ is still d_{0}-closed and therefore d_{0}-exact. Let α ∈ Λ^{1}𝔤̃^{∗} be such that Π^{∗}β = d_{0}α. One checks easily that the restriction of α on 𝔫 depends only on the cohomology class of β, and vanishes on [𝔫, 𝔤̃]. It is therefore a one form on r = 𝔫/[𝔫, 𝔤̃].

The previous proposition will give a geometric interpretation, in terms of relations of a group G, of the pinching of heat decay on 1-forms that will be first related to the algebraic E_{0}^{2} = H^{2}(𝔤). The use of the free Lie algebra 𝔤̃ is also helpful in understanding d_{c} itself on 1-forms on a fixed filtered group G.

We see this. Let Π : G̃ → G as before and α ∈ E_{0}^{1}(G). One has Π^{∗}α ∈ E_{0}^{1}(G̃) and since E_{0}^{2}(G̃) vanishes there will be no obstruction in extending α on vectors of increasing weights to a closed one β. By injectivity of d_{0} on 𝔤̃_{≥2} this extension is unique. So one must have β = Π_{E}Π^{∗}α as given by the iteration of the retraction map r on G̃ by (5). More concretely β is determined by

β([X_{1}, X_{2}]) = X_{1}.β(X_{2}) − X_{2}.β(X_{1}) and β = Π^{∗}α on 𝔤_{1}.  (10)

We note that in this extension process the components of β stay invariant functions along 𝔫

Y.β(X) = 0  (11)

for Y ∈ 𝔫, as proved from (10) by recurrence on weight of X and using that 𝔫 is an ideal. Observe that the value of β on 𝔫 is determined by its value on the generators of 𝔫 because for any Y ∈ 𝔫

β([X, Y]) = X.β(Y).  (12)

This β is a canonical form, but in order to push its restriction to 𝔫 on E_{0}^{2}(G) with proposition 2.9, we need to choose a supplementary subspace of [𝔫, 𝔤̃] in 𝔫, or equivalently a set of generators of relations of G. Depending on this choice, we finally get d_{c}α.

3 Near-cohomology

So far we have only some formal hint in section 2.1 that the asymptotic spectral problem we consider has something to do with the previous constructions. The notion of near-cohomology, introduced in [14], is the main tool that allows to relate explicitly the problems.

3.1 Definition and first applications

Here is a brief presentation of this notion (see [14, 15] for details). Let M be a complete riemannian manifold. An Hilbert complex over M, consists in a sequence of Hilbert spaces E over M together with d : D(d) ⊂ E → E such that d are closed densely defined operators and d^{2} = 0. On the quotiented Hilbert space F = E/ ker d, one considers the family of closed cones

C_{ε} = {α ∈ F, ∥dα∥_{E} ≤ ε∥α∥_{F} }

depending of ε > 0. These are shrinking (towards {0}) as ε → 0. By definition, the near-cohomology of (E, d) consists in this family of cones up to equivalence induced by dilatational changes of ε → Kε. That means that two Hilbert complexes (E, d) and (E′, d′) have the same near cohomology if for ε small enough there exists a constant K > 0 and bounded maps f : C_{ε} → C′_{Kε} and g : C′_{ε} → C_{Kε} invertible on their images.

One observation of Gromov and Shubin is that bounded homotopical equivalence between two Hilbert complexes induces equivalence of their near-cohomologies.

Definition 3.1. Two Hilbert complexes (E, d), (E′, d′) are homotopy equivalent, if there exists bounded maps f : E → E′ and g : E′ → E such that fd = d′f on D(d), gd′ = dg on D(d′), g ∘ f = Id_{E} +dA + Bd on D(d), resp. f ∘ g = Id_{F} +d′A′ + B′d′ on D(d′) for bounded operators A, A′, B, B′.

Theorem 3.2. (prop 4.1 [14]) Let (E, d), (E′, d′) be two homotopy equivalent Hilbert complexes. Then, for ε small enough, f and g induce maps [f] : C_{ε} → C′_{Kε} and [g] : C′_{ε} → C_{Kε} invertible on their images.

Proof. We recall the proof as it is short. Let α ∈ C_{ε}. Then certainly

∥d′(fα)∥ = ∥fdα∥ ≤ ∥f∥∥dα∥ ≤ ε∥f∥∥α∥.  (13)

We need to control α by \overline{fα}, the projection of fα in F = E/ ker d′ ≃ (ker d′)^{⊥}. One has \overline{fα} = fα + β with β ∈ ker d′, so that

g(\overline{fα}) = g(fα) + gβ = α + dAα + Bdα + gβ,

with dAα + gβ ∈ ker d. Therefore, as α ∈ (ker d)^{⊥},

∥α∥ ≤ ∥α + dAα + gβ∥ ≤ ∥g\overline{fα}∥ + ∥B∥ε∥α∥.

Finally

∥α∥ ≤ \frac{∥g∥}{1 − ε∥B∥}∥\overline{fα}∥,

giving the injectivity of \overline{f} acting on C_{ε}, and the result together with (13).

One can’t apply directly this theorem to the complexes (E, d) and (E_{0}, d_{c}) described in theorem 2.6. This is because one of the relevant map here, Π_{E}, being a differential operator, is not bounded in L^{2}. Yet we observe that its un-boundedness occurs on high energy forms, and we are precisely interested in the bottom of the spectrum. So we are leaded to consider an intermediate cut-off de Rham complex (E(Δ ≤ 1), d), where E(Δ ≤ 1) is the spectral space associated to [0, 1] by the Laplacian Δ.

To use this remark, we first notice that the previous theorem 3.2 applies to the full de Rham complex and the cut-off one’s, because Π_{E(Δ≤1)} = Id −dA − Ad where A = δΔ^{−1}Π_{E(Δ>1)} is bounded. Thus, they have equivalent near-cohomology (observe that although they have exactly the same small spectrum their C_{ε} are distinct). But now by ellipticity of Δ, the spectral projection Π_{E(Δ≤1)} is a smoothing operator and therefore the differential operator Π_{E} = Id −Qd − dQ and Π_{E_{0}}Π_{E} becomes bounded on E(Δ ≤ 1), making theorem 3.2 usable here.

Theorem 3.3. [25] Let M be a complete E_{0}-regular C-C manifold. Then the de Rham complex, (E, d) and (E_{0}, d_{c}) have equivalent near-cohomologies.

Proof. To complete the previous discussion, we make short comments on the closures of (E_{0}, d_{c}) and (E, d).

- Firstly, d_{c}, being a differential operator, its formal adjoint δ_{c} has dense (initial) domain C_{0}^{∞} in L^{2}. Therefore d_{c} is closable, for instance by \overline{d_{c}} = (d_{c})^{∗∗} (see eg [23]).

- About (E, d). Given a closed extension \overline{d} of de Rham’s differential, its restriction to \overline{E} = ker d_{0}^{−1} ∩ ker d_{0}^{−1}\overline{d} may be seen as a closed extension of (E, d).

The boundedness of differential operators on E(Δ ≤ 1) has also a geometric consequence on the shape of the cones C_{ε}(E(Δ ≤ 1), d).

We denote by [Π_{E}] and [Π_{F}] the actions of the retractions Π_{E} and Π_{F} of theorem 2.6 on the quotient space Ω^{∗}M/ ker d. One has still [Π_{F}] = Id −[Π_{E}] with [Π_{E}] a projection, so that they induce a splitting of Ω^{∗}M/ ker d in [E] ⊕ [F] (possibly degenerated) with [E] = Im[Π_{E}] = (E + ker d)/ ker d and [F] = Im[Π_{F}] = (F + ker d)/ ker d.

Proposition 3.4. The cut-off cones C_{ε}(E(Δ ≤ 1), d) ⊂ Ω^{∗}M/ ker d ≃ (ker d)^{⊥} are uniformly shrinking around [E] relatively to [F]. Precisely, ∃C such that

∥Π_{F} α∥_{Ω^{∗}M/ ker d} ≤ Cε∥α∥

for all α ∈ C_{ε}(E(Δ ≤ 1), d).

Proof. Recall that Π_{F} = Qd + dQ, so that for α ∈ C_{ε}(E(Δ ≤ 1), d) one has

∥Π_{F} α∥_{Ω^{∗}M/ ker d} ≤ ∥Qdα∥
≤ C∥dα∥ ≤ Cε∥α∥,

where C is a bound for Q on E(Δ ≤ 1) ⊃ d(E(Δ ≤ 1)).

Remarks 3.5.

- Notice that [Π_{E}] and [Π_{F}] are not orthogonal projections.

- As F = Im d_{0}^{−1} + Im dd_{0}^{−1} is pinched between Im d_{0}^{−1} and Im d_{0}^{−1} + Im d one has [F] = [Im d_{0}^{−1}]. Also E = ker d_{0}^{−1} ∩ ker d_{0}^{−1}d ⊂ ker d_{0}^{−1} so that [E] ⊂ [ker d_{0}^{−1}].

We close this section with an application of theorem 3.3 to d_{c}-harmonic decomposition on compact C-C manifolds.

Proposition 3.6. Let M be a compact E_{0}-regular C-C manifold. Then Im d_{c} and Im δ_{c} are closed in L^{2} and we have an orthogonal Hodge-de Rham splitting

E_{0} = H_{c} ⊕ Im d_{c} ⊕ Im δ_{c},

where H_{c} = ker d_{c} ∩ ker δ_{c}, is the space of d_{c}-harmonic forms, isomorphic to de Rham’s cohomology of M.

Remark 3.7. This d_{c}-harmonic representation of the cohomology is probably more interesting when E_{0} is of pure C-C weight (like on one forms), since in that case H_{c} will be invariant under the natural C-C dilations of the riemannian metric g → λg_{V_{1}} + λ^{2}g_{V_{2}} + ⋯ + λ^{r}g_{V_{r}}, with V_{i} = H_{i} ⊖ H_{i−1}. In that pure weight case, one can also obtain analytic information on the regularizing properties of d_{c}, showing in particular the smoothness of H_{c} (see section 5.1).

Proof. On a compact manifold, 0 is isolated in the (discrete) spectrum of the de Rham’s Laplacian Δ. Therefore, on (ker d)^{⊥}

∥dα∥^{2} = (Δα, α) ≥ λ_{1}∥α∥^{2}

for some λ_{1} > 0. Then C_{ε}(d) = {0} for ε < λ_{1}, and the near-cohomology of M vanishes. By theorem 3.3 the same is true at the level of (E_{0}, d_{c}). This implies the same kind of bound for q_{c} = ∥d_{c}∥^{2}, showing that Im d_{c} is closed, and the same for Im δ_{c} by ∗-duality.

3.2 From cones to spectral densities

An interesting feature of near-cohomology lies in its stability under general bounded maps, whereas the spectral decomposition of an operator (like the Laplacian) is only preserved by isometries. This is the advantage of working with the quadratic form q(α) = ∥dα∥^{2} on Ω^{∗}M/ ker d, up to multiplicative constants, instead of doing the (much more) fine study of the exact spectral decomposition of Δ on forms. Yet, the cones C_{ε} = {α | q(α) ≤ ε^{2}∥α∥^{2}} encoded the “rough” information we are looking for of the asymptotic spectral density of Δ at 0.

We first observe that E(0 < δd ≤ ε^{2}) is a closed linear space in C_{ε} ⊂ (ker d)^{⊥}. Moreover, one has q(α) > ε^{2}∥α∥^{2} on its orthogonal E(δd > ε^{2}) = E(0 < δd ≤ ε^{2})^{⊥} ∩(ker d)^{⊥}. Therefore, C_{ε} ∩E(δd > ε^{2}) = {0}. A consequence is that any linear space L lying in C_{ε} projects injectively into E(0 < δd ≤ ε^{2}), showing in some sense that this space is the “largest” possible one inside C_{ε}.

3.2.1 Γ-dimension and the rational case

In order to translate numerically this observation, one needs a notion of dimension working on some infinite dimensional linear spaces. There is an classical one, called Γ-dimension, in the case a discrete group of isometries Γ is acting co-compactly on the manifold (eg a Galois covering of a compact manifold). A complete exposition of this may be found in Atiyah’s original work [1] (or [21] for a survey of its properties and applications).

Briefly, in the case we are concerned in, let M be a compact manifold and M̃ its universal cover. Let P be a positive Γ-invariant Hermitian operator acting on some Γ-invariant Hilbert space H over M̃. We assume H to be a subspace of L^{2}(M̃, V), where V is the pull-back of some finite dimensional vector bundle on M. Then H admits a Γ-invariant Hilbert base (e_{i})_{i∈I} on which Γ-acts freely (H is called a free Γ-module). Choose one vector by Γ-orbit. (For instance one can take an Hilbert base of H restricted to a fundamental domain F of M̃.) The Γ-trace of P is defined by

Tr_{Γ}(P) = ∑_{i∈Γ\I}(P e_{i}, e_{i}),

The definition is shown to be independent of the choices of e_{i} (lemma 2.2 in [21]). If L is a closed Γ-invariant subspace of H, its Γ-dimension is defined by dim_{Γ} L = Tr_{Γ} Π_{L}, where Π_{L} denotes the orthogonal projection onto L.

The basic properties of dim_{Γ} we need here are

• first its invariance under Γ-invariant bounded injection (proved using polar decomposition see eg lemma 2.3 in [21]),

• its link with kernel density in the case of a smoothing projection Π_{L}. For example, if L = E(Δ ≤ λ) then

dim_{Γ} L = Tr_{Γ} Π_{L} = ∫_{F} Tr(K(x, x))dvol,  (14)

where K(x, y) is the smooth Schwartz kernel of Π_{L}, and Tr is the standard finite dimensional trace on End(Λ^{∗}T^{∗}M).

Applying this to a previous remark that any linear subspace in C_{λ} projects injectively into E(0 < δd ≤ λ^{2}), one obtain the following variational principle ([14])

F_{δd}(λ^{2}) = dim_{Γ} E(0 < δd ≤ λ^{2}) = sup_{L∈L_{λ}} dim_{Γ} L,  (15)

where L_{λ} is the set of all Γ-invariant closed linear subspaces L ⊂ C_{λ}(d). Hence the spectral distribution function of δd is encoded in the Γ-linear “thickness” of the near-cohomology cones. Moreover, theorem 3.2 and the invariance of dim_{Γ} trough injective maps, implies Gromov-Shubin’s result that Γ-homotopical equivalent complexes (E, d) and (E′, d′) on M̃ must have equivalent density functions

F_{d}(C^{−1}λ) ≤ F_{d′}(λ) ≤ F_{d}(Cλ),  (16)

for small λ and some C > 0, where F_{d}(λ) is defined in general by

F_{d}(λ) = sup_{L∈L_{λ}} dim_{Γ} L,  (17)

with the same L_{λ} as above. In particular they have the same Novikov-Shubin exponents

α_{d} = lim inf_{λ→0} \frac{ln F_{d}(λ)}{ln λ}  (18)

(but also same limsup, or any other dilatationally invariant limit). This applies to theorem 3.3, if the manifold admits a cocompact discrete action.

Definition 3.8. A graded nilpotent Lie group G is called rational if it admits a cocompact discrete group Γ.

For such a group rationality is equivalent to being able to find a basis of 𝔤 with brackets given by rational coefficients (see [8]). Then a cocompact group Γ is given by the exponential of a corresponding integral lattice in 𝔤.

Theorem 3.9. [25] Let G be a rational graded nilpotent Lie group. Then de Rham’s complex and (E_{0}, d_{c}) have the same Novikov-Shubin exponents.

These exponents do not depend on the choice of Γ here. One can endow G with a (left) invariant metric, in which case all operators and spaces become G-invariant and formulae (15) and (14) reduce to

F_{δd}(λ^{2}) = vol(Γ\G) Tr(K_{Π_{E(0<δd≤λ^{2})}}(e, e)).

Hence lim inf_{λ→0} \frac{ln F_{δd}(λ^{2})}{ln λ} is independent of Γ.

We remark also that theorem 3.9 is true on any fundamental cover of a compact E_{0} regular C-C manifold. We don’t state it in this generality since for applications we will need some dilation respecting the C-C structure, that is more or less to work on a graded group.

3.2.2 Extension to non-rational groups

Although “geometric” nilpotent groups (associated to semi-simple geometries) tend to be rational, they are certainly negligible in the variety of nilpotent Lie groups. Moreover the question of the asymptotic heat decay makes sense on any group, and it seems unlikely this rough invariant should depend so sharply on the existence of a rational structure. (Observe for instance that the value on functions is the growth rate of G, which turns out to be an integer for all nilpotent Lie groups.) Therefore it is natural to extend the previous result to non-rational groups. The point is to use a counterpart to Γ-dimension for G-invariant Hilbert spaces. We describe two different approaches.

• The first one relies on the use of unitary representation theory of nilpotent Lie groups. On such a group G, the Fourier-Plancherel transform gives an isometry between L^{2}(G) with L^{2}(Ĝ) the unitary dual of G (see eg [8] for a complete exposition). We briefly explain how the Plancherel measure on Ĝ gives rise to a relevant measure on G-invariant Hilbert spaces in L^{2}(G).

A bounded (left) G-invariant map P acting on L^{2}(G) has a Schwarz kernel K(x, y) (a distribution on G×G) which is invariant K(gx, gy) = K(x, y). Therefore P acts as a convolution product P f = k ∗ f which “diagonalizes” trough Fourier-Plancherel transform in π(P f) = π(k) ∘ π(f), where π is an irreducible unitary representation of G (in bijection with covectors ξ in 𝔤^{∗} modulo the coadjoint action of 𝔤 by Kirillov’s orbit method). Recall that for f ∈ L^{2}(G), f̂: π → π(f) consists in a measurable field of Hilbert-Schmidt operators acting for each π on the associated Hilbert space H_{π}. For f ∈ L^{1}(G), one has

π(f) = ∫_{G} f(x)π(x^{−1})dx.

Plancherel formula is

∥f∥_{2}^{2} = ∫_{G} |f(x)|^{2}dx = ∥f̂∥_{2}^{2} = ∫Ĝ ∥π(f)∥_{HS}^{2}dμ(π)  (19)

with ∥π(f)∥_{HS}^{2} = Tr(π(f)^{∗}π(f)), while Fourier inversion formula reads

f(e) = ∫Ĝ Tr(π(f))dμ(π)  (20)

for f ∈ S(G). Then to a bounded P is associated a field of bounded π(k) ∈ End(H_{π}). If moreover P is an orthogonal projection onto a G-invariant closed space L, these π(k) have to be orthogonal projections onto spaces L_{π} ⊂ H_{π}. We can now define the G-dimension of L as

dim_{G}L = ∫Ĝ dim(L_{π})dμ(π).

For a smoothing Π_{L} Fourier inversion formula gives

dim_{G}L = k(e) = K_{Π_{L}}(e, e),

to be compared with its Γ-dimension if G is rational. Using the previous discussion one find also that dim_{G} is preserved trough injective bounded G-invariant maps has needed. All this leads to an extension of theorem 3.9 to any graded nilpotent Lie group.

Theorem 3.10. Theorem 3.9 holds without rationality assumption, replacing the Γ by the G-dimension.

• The second approach of the previous result uses more elementary tools, and follows Atiyah’s presentation of the Γ-trace in [1] section 4. Let G be a graded nilpotent group. The main fact we need is that although G has no compact quotient in general, it admits a nice covering. This has been observed in a close form by Helffer and Nourrigat in [16], lemma 4.5.2.

Lemma 3.11. Let G be a graded nilpotent Lie group. There are a relatively compact domain D and a discrete set Z ⊂ G, such that Z^{−1} = Z and

G = ⨆_{z∈Z} zD.

Proof. The Lie algebra 𝔤 splits in 𝔤_{1} ⊕ 𝔤_{2} ⋯ ⊕ 𝔤_{𝔯} with [𝔤_{1}, 𝔤_{𝔦}] ⊂ 𝔤_{𝔦+1}. Choose an integral additive lattice Z_{i} ⊂ 𝔤_{𝔦}, together with a fundamental domain D_{i} of Z_{i} in 𝔤_{𝔦}. Consider D = exp(D_{1} × D_{2} ⋯ × D_{r}) and Z = exp(Z_{1} × Z_{2} ⋯ × Z_{r}). Now at the Lie algebra level, given y = (y_{1}, ⋯, y_{r}) ∈ 𝔤, the equation

(z_{1}, ⋯, z_{r}).(x_{1}, ⋯, x_{r}) = (y_{1}, ⋯, y_{r})

leads to a triangular system whose diagonal terms are

z_{i} + x_{i} = y_{i} + (non linear) function of z_{j}, x_{k} with j, k < i.

Hence it admits a unique solution with z ∈ Z and x ∈ D. (This proof adapts on any nilpotent Lie group.)

This gives a discretization of L^{2}(G) in ℓ^{2}(Z) ⊗ L^{2}(D), which allows to follow the classical construction of Γ-trace as given in [1] section 4 (or [21]). For instance a bounded G-invariant operator P may be called Z-Hilbert Schmidt if its kernel belongs to L^{2}(Z\G × G), which identifies either to L^{2}(D × G) or L^{2}(G × D) by the symmetry Z^{−1} = Z. Its Z-Hilbert Schmidt norm may be defined by

∥P ∥_{ZHS}^{2} = ∫_{G×D} |K(x, y)|^{2}dxdy = ∥P χ_{D}∥_{HS}^{2} = ∑_{e_{i}} ∥(P χ_{D})e_{i}∥^{2}

= ∫_{D×G} |K(x, y)|^{2}dxdy = ∥χ_{D}P ∥_{HS}^{2} = ∑_{e_{i}} ∥(χ_{D}P)e_{i}∥^{2},

for e_{i} an Hilbert base of L^{2}(G) and χ_{D} the characteristic function of D. It is easily seen to be independent on the choice of D and ∗-invariant ∥P ∥_{ZHS} = ∥P^{∗}∥_{ZHS}. We check the first point, if D′ is another “fundamental domain”, one has

∥P ∥_{DHS}^{2} = ∫_{G} (∫_{D} |K(x, y)|^{2}(∑_{z∈Z} χ_{zD′})(y)dy) dx

= ∫_{G} ∑_{z∈Z^{−1}=Z} ∫_{zD} |K(x, y)|^{2}χ_{D′}(y)dydx

= ∫_{G}(∫_{G} |K(x, y)|^{2}χ_{D′}(y)dy)dx = ∥P ∥_{D′HS}^{2}

A positive hermitian P is of Z-trace class if Tr_{Z}(P) = ∑_{i∈I}(P e_{i}, e_{i}) converges where (e_{i}) an Hilbert base of L^{2}(D), equivalently χ_{D}P χ_{D} is of trace class on L^{2}(G). The relation with the Z-HS class being

Tr_{Z}(P^{∗}P) = Tr(χ_{D}P^{∗}P χ_{D}) = ∥P χ_{D}∥_{HS}^{2} = ∥P ∥_{ZHS}^{2}.

Again if Π_{L} is a smoothing G-invariant projection on L, dim_{Z} L = Tr_{Z} Π_{L} reduces to vol(D)K_{Π_{L}}(e, e) as in the case of Γ-dimension in (14). So we get a version of the previous theorem 3.10 using this “Z-dimension” instead.

Observe that in all considered Γ, G, or Z approaches of dim, one can uses only G-invariant closed spaces to determine the cones thickness function F (λ) in formula (17). Indeed the maximal (through injective projection) Hilbert space in C_{λ}(d) is E(0 < δd ≤ λ^{2}) a G-invariant space, not only a Γ or Z-invariant one.

3.3 The algebraic pinching of heat decay

We can now give a first application of these constructions to our initial problem of estimating the heat decay on (1-)forms of graded groups. We define the C-C weight

of G, with respect to its grading, as N(G) = ∑_{k=1}^{r} kdim(𝔤_{k}). This N(G) is the weight

of Λ^{max}T^{∗}G, occurring in the Jacobian of the dilations of G. When the group G is filtered, N(G) is also its Hausdorff dimension, that is the growing rate of the volume of large balls in G. We also consider the splitting of E_{0}^{k} = H^{k}(𝔤), the k-forms of E_{0}, in its various C-C weights p

E_{0}^{k} = ⨁_{p=N_{k}^{min}}^{N_{k}^{max}} E_{0}^{k,p}.

E_{0}^{k} is said of pure weight p if p = N_{k}^{min} = N_{k}^{max}.

Definition 3.12. We will use β_{k} = N(G)/α_{k} as a convenient renormalization of α_{k}, the k-th Novikov-Shubin exponent of G (twice the heat decay on coclosed k-forms of G by (1)).

Theorem 3.13. [25] (Algebraic pinching of β_{k}.) Let G be a graded nilpotent Lie group. If E_{0}^{k} = H^{k}(𝔤) has pure weight N_{k} then,

δN_{k}^{min} = max(N_{k+1}^{min} − N_{k}, 1) ≤ β_{k} ≤ N_{k+1}^{max} − N_{k} = δN_{k}^{max},

giving an estimation of β_{k} = N(G)/α_{k} from the weight gaps δN_{k}.

Remark 3.14. This is of course consistent with the case of functions k = 0 mentioned in theorem 1.2. Namely for a filtered Lie group (bracket generated by D = 𝔤_{1}), one has E_{0}^{1} ≃ Λ^{1}D^{∗} of pure weight 1 (section 2.4). Therefore β_{0} = 1 and we recover that α_{0} = N(G) is the growth of G.

Proof. By theorem 3.10, we can use (E_{0}, d_{c}) to estimate β_{k}. Then the result relies on an homogeneity argument of d_{c} trough the dilations h_{r} of G. Observe that on E_{0}^{k},

d_{c} = d_{c}^{δN_{k}^{min}} + ⋯ + d_{c}^{δN_{k}^{max}},  (21)

where d_{c}^{i} increases the weight by i.

We recall that for a form α of weight N(α), the integral L^{2}-norms transforms as ∥h_{r}^{∗}α∥_{G} = r^{N(α)−N(G)/2}∥α∥_{G}. We now see how the near-cohomology cones C_{λ}(d_{c}) rescale through h_{r}^{∗}. Firstly, by the homogeneity assumption of E_{0}^{k}, the space E_{0}^{k}/ ker d_{c} ≃ (ker d_{c})^{⊥} is preserved. By the previous remarks, one has then for α ∈ E_{0}^{k} ∩ C_{λ}(d_{c}) and r ≥ 1,

∥d_{c}(h_{r}^{∗}α)∥ = ∥h_{r}^{∗}d_{c}α∥ ≤ r^{N_{k+1}^{max}−N(G)/2}∥d_{c}α∥
≤ λr^{N_{k+1}^{max}−N(G)/2}∥α∥
= λr^{δN_{k}^{max}}∥h_{r}^{∗}α∥.

Therefore

h_{r}^{∗}(C_{λ}(d_{c})) ⊂ C_{λr^{δN{max \atop k}}} (d_{c}).

In the opposite direction, for ε ≤ 1 and α ∈ E_{0}^{k} ∩ C_{λ}(d_{c})

∥d_{c}(h_{ε}^{∗}α)∥ = ∥h_{ε}^{∗}d_{c}α∥ ≤ ε^{max(N_{k}^{min},N_{k}+1)−N(G)/2}∥d_{c}α∥
≤ λε^{max(N_{k}^{min},N_{k}+1)−N(G)/2}∥α∥
≤ λε^{δN_{k}^{min}}∥h_{ε}^{∗}α∥,

giving

h_{ε}^{∗}(C_{λ}(d_{c})) ⊂ C_{λε^{δN{min \atop k}}} (d_{c}).

Putting this together we have found that, for r ≥ 1

h_{r}^{∗}(C_{r^{−δN{max \atop k}}} (d_{c})) ⊂ C_{1}(d_{c}) ⊂ h_{r}^{∗}(C_{r^{−δN{min \atop k}}} (d_{c})).  (22)

The next point to check is the effect of dilations on dimension of G-invariant Hilbert spaces L ⊂ E_{0}. We have for n ∈ ℕ,

dim_{Z}(h_{n}^{∗}L) = dim_{h_{n}Z}(L) = n^{N(G)}dim_{Z}(L).  (23)

We use the discrete dilations h_{n} because we will need that h_{n}(Z) ⊂ Z. (We could avoid this working with the less elementary G-dimension). For Γ-dimension this formula is a direct consequence of its behavior through finite covering. The proof is the same here. Since n^{N(G)/2−N_{k}}h_{n}^{∗} is an isometry on L^{2}(E_{0}^{k}), we have

Π_{h_{n}^{∗}L} = h_{n}^{∗}Π_{L}h_{n}^{∗−1}.

Therefore,

dim_{Z}(h_{n}^{∗}L) = Tr_{Z}(Π_{h_{n}^{∗}L}) = Tr(χ_{D}h_{n}^{∗}Π_{L}h_{n}^{∗−1}χ_{D})
= Tr(h_{n}^{∗}χ_{h_{n}D}Π_{L}χ_{h_{n}D}h_{n}^{∗−1}) = Tr(χ_{h_{n}D}Π_{L}χ_{h_{n}D})
= dim_{h_{n}Z}(L),

Now by lemma 3.11, h_{n}D is a “fundamental domain” for the new “integral lattice” h_{n}Z which splits, for n ∈ ℕ, in n^{N(G)} domains D′ for Z, giving (23). This is applied to G-invariant Hilbert linear sub-spaces L ⊂ C_{λ}(d_{c}) and leads with (22) and (17) to the following pinching for the cone thickness function F_{d_{c}}

F_{d_{c}}(n^{−δN_{k}^{max}}) ≤ n^{−N(G)}F_{d_{c}}(1) ≤ F_{d_{c}}(n^{−δN_{k}^{min}}),

for n ∈ ℕ and finally

CF_{d_{c}}(1)λ^{N(G)/δN_{k}^{min}} ≤ F_{d_{c}}(λ) ≤ C′F_{d_{c}}(1)λ^{N(G)/δN_{k}^{max}},  (24)

for λ > 0 (even real) and some positive constants C, C′.

The last observation is that all these numbers are finite and strictly positive. Finiteness comes from the fact that this is true “upstairs” for the homotopically equivalent de Rham complex. Indeed by ellipticity of the de Rham Laplacian, spectral projectors have smooth kernels, and therefore of finite diagonal trace. Nonvanishing amounts to the fact that 0 belongs to the spectrum of Δ on k-forms for any k on nilpotent groups. This can be viewed at the (E_{0}, d_{c}) level. The cones C_{λ}(d_{c}) don’t vanish, because any sufficiently dilated non-closed smooth form of E_{0} lies in it, as d_{c} (unlike d) strictly increases the weight. We note that d_{c} can’t vanish identically in degree k < dimG, because this complex is a resolution of functions of G and has to be of maximal length.

We can give a more geometric meaning of the previous theorem 3.13 in the case of one forms. Namely we have seen in section 2.4 that when the group is filtered,

• firstly E_{0}^{1} identifies with Λ^{1}D^{∗} forms of weight 1,

• E_{0}^{2} = H^{2}(𝔤) may be interpreted as Λ^{1}r^{∗}, the dual to the space R = N/(N, G̃) of generators of the relations N of G, with respect to the free Lie group G̃.

Therefore theorem 3.13 translates into the following result.

Corollary 3.15. Let G be a filtered nilpotent Lie group. Then β_{1} +1 = N(G)/α_{1} + 1 is pinched between the minimal and maximal weight of the relations giving a presentation of G from the free Lie group. In particular 1 ≤ β_{1} ≤ r if G has rank r.

4 Examples

The previous result gives an estimation of heat decay from the knowledge of the weights of the Lie algebra cohomology H^{∗}(𝔤). This pinching relies therefore on finite dimensional linear systems computations (that may of course become incomprehensible as the dimension increases).

4.1 Quadratically presented groups

The first class of nilpotent groups on which the previous results applies sharply are quadratically presented groups. At the Lie algebra level they are groups admitting a presentation using only linear combinations of bracket of first order in the generators D = 𝔤_{1}. By the previous discussion this may also be checked dually by showing that H^{2}(𝔤) contains only 2-forms of weight 2 (horizontal 2-forms in Λ^{2}D^{∗}).

For all these groups one has β_{1} = 1, meaning that α_{1} = N(G) (in the strong sense that on Ω^{1}G, the heat kernel satisfies K_{t}(x, x) ≍ t^{−N(G)/2} when t → +∞, as follows from (24)). Thus heat decay on one forms is the quickest possible and the same as on functions. Fortunately these groups exist.

• The simplest examples are given by the Heisenberg groups H^{2n+1} of dimension 2n + 1 ≥ 5. They may be presented by the relations [X_{i}, X_{j}] = [Y_{i}, Y_{j}] = 0 with 1 ≤ i, j ≤ n, [X_{i}, Y_{j}] = 0 for i ≠ j and (for n > 1), [X_{i}, Y_{i}] = [X_{j}, Y_{j}] (= T the generator of the center). In fact using the Lie algebra computation in section 2.3 and theorem 3.13, we have α_{p} = N(G) for all 0 ≤ p ≤ 2n except α_{n} = N(G)/2. In particular the value α_{1}(H^{3}) = 2 (first computed by Lott [20]) is half the Hausdorff dimension. This corresponds geometrically to the fact that H^{3} has a cubic presentation : [X, [X, Y]] = [Y, [X, Y]] = 0.

• There are in fact many other examples. In [13] section 4.2, Gromov introduced the class of two step graded groups that possess a Ω-regular legendrian two plane P. That means there exists X_{1}, X_{2} in D such that [X_{1}, X_{2}] = 0 and the map

Ω : D → End(P, [D, D])
X → [X, ·] acting on P = span(X_{1}, X_{2}),

is surjective. (Giving T_{1}, T_{2} ∈ 𝔤_{2} = [D, D], there exists X in D such that [X, X_{1}] = T_{1} and [X, X_{2}] = T_{2}.) One can show easily (see [13] 4.2 A”) that this condition is generically satisfied (is a Zariski open dense set) for graded groups associated to distributions D^{k} ⊂ ℝ^{n} as long as n ≤ 3k/2 − 1. (The necessity of this bound is a dimension counting in the previous formula noting that P ⊂ ker Ω.)

The geometric interest of this class of groups is that Gromov has shown they are quadratically fillable in the sense that closed horizontal curves γ can be filled by horizontal disks of area controlled by length(γ)^{2}. (This is a global non-linear hard property relying infinitesimally on the presence of a lot of flexible legendrian planes.) Our observation is simply that

Proposition 4.1. Ω-regular groups are quadratically presented, and thus have β_{1} = 1.

Proof. We have to show that H^{2}(𝔤) doesn’t contain 2-forms of weight 3, that is the map

\begin{matrix}d_{0} : Λ^{2,(3)}𝔤^{∗} = D^{∗} ∧ 𝔤_{2}^{∗} & ⟶ Λ^{3,(3)}𝔤^{∗} = Λ^{3}D^{∗} \\ ∑ α_{i} ∧ θ_{i} & ⟶ − ∑ α_{i} ∧ d_{0}θ_{i} = ∑ α_{i} ∧ θ_{i}([·, ·]) \end{matrix}  (25)

is injective. Let β = ∑ α_{i} ∧ θ_{i} be in ker d_{0}. We first show that α_{i} vanishes on P = span(X_{1}, X_{2}). We fix a base θ_{i} of 𝔤_{2}^{∗} in the previous formula. Given T_{i_{0}} dual to θ_{i_{0}} we can find by Ω-regularity a vector X ∈ D such that [X, X_{1}] = 0 and [X, X_{2}] = T_{i_{0}} (and still [X_{1}, X_{2}] = 0). Hence

0 = d_{0}β(X_{1}, X_{2}, X) = ∑ α_{i}(X_{1}) ∧ θ_{i}([X_{2}, X]) = −α_{i_{0}}(X_{1}),

and similarly on X_{2}. Now taking X as before and Y ∈ D such that [Y, X_{2}] = 0, one gets

0 = d_{0}β(X_{2}, X, Y) = ∑ α_{i}(Y) ∧ θ_{i}([X_{2}, X]) = −α_{i_{0}}(Y),

so that α_{i_{0}} vanishes on ker [ · , X_{2}]. For the last step, choose Y ∈ D such that [Y, X_{1}] = T_{i_{0}} and [Y, X_{2}] = 0 then, for any X ∈ D

0 = d_{0}β(X_{1}, X, Y) = ∑ α_{i}(X) ∧ θ_{i}([Y, X_{1}]) = α_{i_{0}}(X).

This shows that α_{i_{0}} = 0, and finally β = 0.

The previous fact may be seen more geometrically using Gromov’s result that these groups are quadratically fillable. If any horizontal curve can be filled by an horizontal disk then certainly, for any 1-form α on G, the vanishing of dα on Λ^{2}D is sufficient to conclude that α is exact. Namely f(m) = ∫_{γ} α where γ is any horizontal path from 0 to m, is well defined in that case and satisfies df = α along D. Hence θ = α − df is a vertical form such that d_{0}θ = dα on Λ^{2}D vanishes, and finally α = df. This argument means that the quadratic filling property implies that they are no non zero closed two forms of weight > 2. In particular E_{0}^{2} = H^{2}(𝔤) has to be of pure weight 2, because invariant 2-forms on G coming from H^{2}(𝔤) = ker d_{0}/ Im d_{0} are always closed. We recover therefore that G is quadratically presented.

• Gromov’s quadratically fillable groups have “a lot” of integrable legendrian planes P = (X, Y) ⊂ D which lead to simple quadratic relations [X, Y] = 0 and allows a quadratic (in weight) integration of 2-forms. Anyway, the presence of these flat planes is not a necessary condition for a group to have a quadratic presentation. Here is a “sporadic” example with no flat plane at all but which is quadratically presentable anyway. This is a J-group corresponding to a D^{8} ⊂ ℝ^{15}. We have to describe a 7-dimensional subspace L of curvature in Λ^{2}ℝ^{8}, such that

θ ∈ ℝ^{7} → dθ = g_{D}(J(θ) , ) ∈ L

with −J(θ)^{2} = ∥θ∥^{2} Id = J(θ)^{∗}J(θ). This map J can be realized either as a particular Clifford action, or the left multiplication of the imaginary Cayley numbers Im 𝕆 on 𝕆. It can been checked, using direct calculations with the J_{i}, 1 ≤ i ≤ 7 given in tables of Clifford algebras (see eg [18] section 11), or representation theory (see [26]) that the map d_{0} in (25) is injective (even an isomorphism, since both spaces have dimension 56). Therefore this group is quadratically presented, although it contains no integrable horizontal plane since [X, Y] = 0 implies that X⊥J_{i}Y for 1 ≤ i ≤ 7, which gives collinear X and Y.

• The previous example D^{8} ⊂ ℝ^{15} goes beyond the generic bound n ≤ 3k/2 − 1 under which a generic D^{k} ⊂ ℝ^{n} is Ω-regular hence quadratically presented. Unfortunately, due to the eight periodicity of their structure, Clifford modules fail to produce a linearly increasing number of orthogonal complex structures on ℝ^{k} for k > 8 (ℝ^{16k} has only 8 more such complex structures than ℝ^{k}). In fact it doesn’t seem easy to improve the previous bound n ≤ 3k/2 − 1 in general. Here is another (too rough?) method of construction of quadratic groups, independent of Ω-regularity, but which leads to the same bound.

Proposition 4.2. Any increasing family of extensions of D = ℝ^{k} by

L_{p} = span(ω_{1}, ω_{2}, ⋯, ω_{p}) ⊂ Λ^{2}D^{∗},

leads to quadratically presented groups if

1. each ω_{p} is of rank ≥ 2(p + 1),

2. ω_{p} ∉ L_{p−1} + {α ∧ β ∈ Λ^{2}D^{∗}}.

Proof. As in (25), we have to show that ∑_{i=1}^{p} α_{i} ∧ω_{i} = 0 with α_{i} ∈ D^{∗} implies α_{i} = 0.

This is clear for a single term by assumption ω_{1} of rank ≥ 4. Suppose we have a

non trivial relation α_{p} ∧ ω_{p} = ∑_{i=1}^{p−1} α_{i} ∧ ω_{i}, with α_{p} ≠ 0 by recurrence hypothesis.

We will show that all α_{i} are multiple of α_{p}. Firstly ω_{p} ∧ α_{1} ∧ α_{2} ∧ · ∧ α_{p} = 0, and since rk(ω_{p}) ≥ 2(p + 1), this implies α_{1} ∧ α_{2} ∧ · ∧ α_{p} = 0. Therefore some α_{i_{0}} with 1 ≤ i < p is a linear combination of the others. Now ω_{p} ⋀_{i≠i_{0}} α_{i} = 0 implies again ⋀_{i≠i_{0}} α_{i} = 0 and gives another dependent α_{i_{1}}. We repeat this until we obtain

α_{p}∧ω_{p} = ∑_{i=1}^{p−1} c_{i}α_{p}∧ω_{i} for some constants c_{i}. This implies that ω_{p} = ∑_{i=1}^{p−1} c_{i}ω^{i}+α_{p}∧β_{p},

which is contrary to the assumption 2.

Remarks 4.3. - Notice that the first hypothesis implies, and is generically satisfied if 2p + 2 ≤ k, while the second one is generic if p − 1 + 2(k − 2) + 1 < \frac{k(k−1)}{2} = dim(Λ^{2}D^{∗}). This is seen to be implied by the first condition, which amounts finally to n = p + k ≤ 3k/2 − 1 as with Ω-regularity.

- The second condition means that L_{p} doesn’t contain any rank 2 form α ∧ β. This is certainly a necessary condition for quadratic presentation. Otherwise α ∧ θ and β ∧θ, with d_{0}θ = α∧β ∈ L_{p}, will be a d_{0}-closed but not exact 2-forms of weight 3 in the corresponding extension G of D. Another equivalent formulation is that G can’t admit a surjective morphism onto the 3-dimensional Heisenberg group H^{3}.

• So far we have only considered 2-step groups, but quadratically presented groups of arbitrarily high rank do exist. Interest in that field came from its relation with Sullivan’s rational homotopy theory, where these groups are 1-formal 1-minimal. In particular since Kähler manifolds M have formal minimal models, the Malcev’s completion Π_{1}(M) ⊗ ℚ of Π_{1}(M) are examples of such groups, at the condition they are finite dimensional, that is nilpotent. We refer to [12] for details on these problems.

The simplest example of a rank 3 quadratically presented group is given by a generic 3-step distribution D^{5} ⊂ ℝ^{8} studied in [3]. Its structure is given by D = span_{1≤i≤5}(X_{i}), 𝔤_{2} = [D, D] = span(Y_{1}, Y_{2}), and 𝔤_{3} = ℝZ with the dual relations

dy_{1} = x_{1} ∧ x_{3} + x_{2} ∧ x_{4}, dy_{2} = x_{1} ∧ x_{4} + x_{2} ∧ x_{5}, dz = x_{1} ∧ y_{1} + x_{2} ∧ y_{2}.

Observe that the third cubic form x_{1} ∧ y_{1} + x_{2} ∧ y_{2} is closed from the two first one, and thus may be written dz in order to kill it in H^{2}(𝔤). It turns out (see [3]) that there are no other weight ≥ 3 d_{0}-closed 2-forms, and that this group is quadratically presented.

• Lastly, a series of arbitrarily high rank quadratically presented group has been described by Chen in [5]. Given (n, k) ∈ ℕ × ℕ^{∗}, these k + 1-rank groups are generated by D = {X_{1}, ⋯, X_{n}, Y_{α}, |α| = k}, with α = (α_{1}, ⋯, α_{n}) ∈ ℕ^{n} and |α| = α_{1} + ⋯ + α_{n}. The group structure is given by

dx_{i} = 0, dy_{α} = 0 if |α| = k, dy_{α} = ∑ y_{α+e_{i}} ∧ x_{i} if |α| < k,

with e_{i} = (0, ⋯, 1, ⋯, 0). One has 𝔤_{j} = D^{[j]} = span(Y_{α}, |α| = k − j + 1). These examples have still β_{1} = 1, showing that this function may be independent of the rank of the group.

4.2 Some higher rank groups

We now give examples of groups with relations of higher weight.

• Let G^{k,r} be the “free” nilpotent group of rank r with k generators. This is the quotient of the infinite dimensional free Lie group G̃^{k} in k generators by all its elements of weight > r. Is is the largest r-step filtered group, since any other one may be presented as a quotient of it. By definition the relations of G^{k,r} with respect to G̃^{k} are generated by all words of weight r + 1 in G̃^{k}. Therefore corollary 3.15 gives

β_{1}(G^{k,r}) = r,

the upper possible bound for r-step groups, and thus the lowest possible α_{1} = N(G)/r and heat decay N(G)/2r (with respect to the Hausdorff dimension N(G)).

• These G^{k,r} are helpful in precising the pinching of β_{1} for groups associated to generic distributions D^{k} ⊂ ℝ^{n}. Let n(k, r) = dim(G^{k,r}) (a general formula using Möbius function may be found eg in [13], section 4.1.B).

Proposition 4.4. For a fixed k, one has r − 1 ≤ β_{1}(G) ≤ r for groups G associated to a generic D^{k} ⊂ ℝ^{n}, if r is such that n(k, r − 1) ≤ n < n(k, r).

Proof. In fact these groups have only relations of weight r and r + 1 generically.

Let (X_{i})_{1≤i≤k} be a polynomial germ of k independent vectors fields around 0 in ℝ^{n}. Define D^{k} = span(X_{i}, 1 ≤ i ≤ k), and G the graded Lie group associated to this C-C structure at 0 (by the quotiented Lie bracket [ , ]_{0} see section 2.2.1). Recall that if H_{i} = [H, H_{i−1}], and H_{1} = D^{k} one has 𝔤 = 𝔤_{1} ⊕ ⋯ ⊕ 𝔤_{r}, with 𝔤_{i} = H_{i}/H_{i−1}, at 0. Generically, on a Zariski open dense set in the (j − 1)-jet of (X_{i}) at 0, one has

𝔤^{j} = 𝔤_{1} ⊕ ⋯ ⊕ 𝔤_{j}

of maximal possible dimension, that is min(n(k, j), n). Therefore 𝔤^{j} is isomorphic for j ≤ r − 1 to the Lie algebra of G^{k,j}, and G has no relation of weight < r. For j = r, one has 𝔤^{r} of dim n (meaning D^{k} is an Hörmander r-step distribution in ℝ^{n}), and then 𝔤 = 𝔤^{r} is of rank r. This gives only relations of weight ≤ r + 1.

• In the opposite direction to the previous “generic” situation, nilpotent groups associated to complex semi-simple geometries may have increasing rank but still a presentation of bounded (although not quadratic) depth.

For instance, consider N_{n} ⊂ SL(n, ℂ) the nilpotent group of Id +strictly upper triangular matrices. It is a rank (n − 1) group whose Lie algebra is generated by D = span(X_{i} = E_{i,i+1} ∈ M_{n}(ℂ), 1 ≤ i ≤ n − 1). N_{3} = H^{3} is the 3 dimensional Heisenberg group, already encountered (it is cubically presented and has β_{1} = 2. For all n ≥ 4, N_{n} has a presentation with both quadratic an cubic relations, which are

[X_{i}, X_{j}] = 0 for j − i > 1, and [X_{i}, [X_{i}, X_{i+1}]] = [X_{i+1}, [X_{i}, X_{i+1}]] = 0.

Dually, H^{2}(N_{n}) is generated by θ_{X_{i}} ∧ θ_{X_{j}} for j − i > 1 and θ_{X_{i}} ∧ θ_{Y_{i}}, θ_{X_{i+1}} ∧ θ_{Y_{i}} where θ_{Y_{i}} is the dual form to Y_{i} = E_{i,i+2} = [X_{i}, X_{i+1}]. All this comes from the general description by Kostant of the structure of the cohomology of (maximal) nilpotent Lie algebras in semi-simple Lie algebra (which splits in multiplicity one factors through the adjoint action of the maximal torus) (see [4],[9]).

Therefore, for these groups of increasing rank, we have 1 ≤ β_{1} ≤ 2. We see that β_{1}(N_{4}) = 2.

Proof. For f ∈ C_{0}^{∞}(N_{4}), consider α = fθ_{X_{2}}. We observe that d_{c}α has weight 3. This is because the weight 2 part of E_{0}^{2} = H^{2}(𝔫_{4}) is spanned by θ_{X_{1}} ∧ θ_{X_{3}}, and thus

(d_{c}α)_{2} = dα(X_{1}, X_{3}) = X_{1}.α(X_{3}) − X_{3}.α(X_{1}) = 0.

As a consequence, for a generic f, we have found a non-closed smooth L^{2} form in E_{0}^{1} = Λ^{1}D^{∗} such that d_{c}α has weight 3. We will see in proposition 5.6 this implies β_{1} ≥ 2.

• We close this section with a first example showing that β_{1} = N(G)/α_{1} is not necessarily an integer when N(G) is the Hausdorff dimension of a filtered group G.

We consider the 4-dimensional Engel group G (studied in section 2.3). We observe that with its filtered weight N, one has N(G) = 2 + 2 + 3 = 7, while H^{2}(𝔤) is generated by θ_{Y} ∧ θ_{Z}, of weight 3 and θ_{X} ∧ θ_{T}, of weight 4. Hence, one has the pinching 2 ≤ β_{1}^{N}(G) ≤ 3, leading to

7/3 ≤ α_{1}(G) ≤ 7/2.  (26)

Now, we can change the grading, using instead N′ such that N′(X) = 1, N′(Y) = 2, N′(Z) = 3 and N′(T) = 4. In that case H^{2}(𝔤) becomes homogeneous of pure weight 5, while E_{0}^{3} = H^{3}(𝔤) = span(θ_{X∧Z∧T} , θ_{Y ∧Z∧T}) has now mixed weight 8 and 9. Applying theorem 3.13 at the level of 2 forms, we find that 8−5 = 3 ≤ β_{2}^{N′}(G) ≤ 9 − 5 = 4, giving

10/4 ≤ α_{2}(G) ≤ 10/3  (27)

since N′(G) = 10. This gives a pinching of the spectral density of δd on Ω^{2}(G)/ ker d which is ∗-conjugated to dδ acting on Ω^{2}(G)/ ker δ. This last one is itself conjugated by δ to the spectrum of δd on Ω^{1}(G)/ ker d. We get finally that α_{2}(G) = α_{1}(G) and that the second pinching (27) is strictly sharper than the first one (26). In particular β_{1}^{N}(G) = 7/α_{1}(G) can’t be an integer using the “Hausdorff” scaling N, the relevant one on functions.

• Still on that group, we can use the computations of section 2.3 to understand more precisely some aspect of the asymptotic heat diffusion on its 1-forms. We restrict our study to forms on G whose components along the left invariant vectors fields X, Y, Z, T are invariant functions along Z, T, that is pull-back functions by Π : G → ℝ^{2} = G/(Z, T). We note Ω_{ℝ^{2}}^{∗}G = C(ℝ^{2}) ⊗ Λ^{∗}𝔤^{∗} this class of forms, and look at the asymptotic behavior of the “commutative” heat, or spectrum of Δ_{G} as acting on L_{ℝ^{2}}^{2}(Ω_{ℝ^{2}}^{∗}G). Like in the true L^{2}(G) situation, the de Rham complex on G has the same restricted ℝ^{2}-asymptotics as (E_{0}, d_{c}) (equivalent ℝ^{2}-near cohomology in fact, as given by the same proofs as before).

Now, the action of d_{c} on Ω_{ℝ^{2}}^{1}G may be read from (8), by cutting the Z and T arrows. We see that d_{c} factorizes here in P d_{ℝ^{2}} where

d_{ℝ^{2}}(fθ_{X} + gθ_{Y}) = (X.g − Y.f)θ_{Z},

and

P (fθ_{Z}) = (X^{2}.f)θ_{X∧T} + (Y.f)θ_{Y ∧Z}.

Hence, on E_{0,ℝ^{2}}^{1}∩(ker d_{c})^{⊥}, the operator δ_{c}d_{c} is conjugated by d_{ℝ^{2}} to P^{∗}P = X^{4}+Y^{2} acting on the one dimensional bundle θ_{Z}. Its spectrum can be described by Fourier analysis, in ℝ^{2} here. The Fourier transform of the spectral space E(P^{∗}P ≤ λ^{2}) are the L^{2} functions whose supports are contained in the set

F S(λ) = {(x, y) ∈ ℝ^{2}, x^{4} + y^{2} ≤ λ^{2}}.

When λ → 0, this strange collection of flying saucers are shrinking at speed λ along y and only \sqrt{λ} along x. Therefore the rescaling (X, Y) → (λ^{−1/2}X, λ^{−1}Y) associated to the previous weight N′ appears naturally here. Observe anyway that the F S(λ) have no usable self-similar limit in this rescaling (they converge to the union of two segments along x and y). In fact one finds easily that the asymptotic area of F S(λ) is −λ^{2} ln λ when λ → 0. This measure of F S(λ) is also the density of E(P^{∗}P ≤ λ^{2}), E(δ_{c}d_{c} ≤ λ^{2}), and finally E(δd ≤ λ^{2}) as acting on L^{2}(ℝ^{2}). We get finally by Laplace transform that the asymptotic ℝ^{2}-heat decay on 1-forms is

equivalent to \frac{ln t}{t^{1/2}} as t → +∞ (to be compared to the standard heat decay in 1/t

on ℝ^{2}).

Lastly, one can obtain some information on the anisotropic aspects of large scale heat diffusion here. For ε > 0, consider the orthogonal splitting

L^{2}(ℝ^{2}) = L_{ε}^{2}(ℝ^{2}) ⊕ (L_{ε}^{2}(ℝ^{2}))^{⊥},

with L_{ε}^{2}(ℝ^{2}) the space of L^{2} functions whose Fourier transforms are supported in D_{ε} = {(x, y), |y| ≤ ε|x|}. We note that for each ε, area(F S(λ) ∩ D_{ε}) ≍ −λ ln λ while area(F S(λ) ∩ D_{ε}^{⊥}) ≍ λ = o(−λ ln λ), showing that most the spectral measure of E(P^{∗}P ≤ λ^{2}) is supported on D_{ε} when λ → 0. Recall that the d_{c} Laplacian on horizontal 1-forms is conjugated to P^{∗}P acting on θ_{Z} by the map

δ_{ℝ^{2}}(fθ_{Z}) = (Y.f)θ_{X} − (X.f)θ_{Y} .

Since for functions f ∈ L_{ε}^{2}(ℝ^{2}), one has

∥Y.f∥_{2} = ∥yf̂∥_{2} ≤ ε∥xf̂∥_{2} = ε∥X.f∥_{2},

we see that most the spectral measure of E(δ_{c}d_{c} ≤ λ^{2}), and finally E(δd ≤ λ^{2}) is supported on forms α such that ∥α(X)∥_{2} ≤ ε∥α(Y)∥_{2}, that is forms close to the Y -direction. Topologically this direction corresponds to the largest Massey product we can find in the 1-minimal model of G. Namely, one has θ_{X}, θ_{Y} ∈ H^{1}(𝔤) with

\{ \begin{matrix}−θ_{X} ∧ θ_{Y} & = d_{0}θ_{Z} \\ −θ_{X} ∧ θ_{Z} & = d_{0}θ_{T} \\ −θ_{X} ∧ θ_{T} & ∈ H^{2}(𝔤), \end{matrix}

Analytically, this sequence is recognized in (8) as the X^{3} component of d_{c} from θ_{Y} to θ_{X} ∧θ_{T}, the link being done by taking the elliptic symbol of d_{c} in the X direction. This shows that some information on the multiplicative structure of H^{∗}(𝔤), not only its weight, is analytically encoded in (E_{0}, d_{c}) even at the commutative level. The effect here is that we can “hear” the exceptional direction Y.

5 Towards a refined pinching

The pinching of α_{1} we gave relies on very few analysis, general facts on the Hilbert-Schmidt and trace class operators, and the ellipticity of the de Rham Laplacian. Anyway it is certainly not sufficient to obtain the exact value of α_{1}, β_{1} on groups of non homogeneous presentation. To improve (even partially) the result we have to be more careful on the analytic properties of d_{c} as related to the convergence of the underlying spectral sequence and filtered complex.

5.1 C-C ellipticity and applications

To that purpose we introduce some class of analytic regularity that will fit to the vector valued operators on graded groups we are dealing with. It relies on the notion of (maximal) hypoellipticity as defined for scalar operators.

Let G be a graded group, and P a left invariant differential operator on C^{∞}(G) of order p with respect to natural grading in T G. P is said maximally hypoelliptic if locally ∥P f∥_{2} + ∥f∥_{2} controls all derivatives of f of weight ≤ p. One can show (see [22, 17]) that this is equivalent to the existence for P of a parametrix Q of order −p. That means the kernel of Q is an homogeneous distribution on G of order p − N(G) near the diagonal, and such that Q ∘ P = Id +S for some smoothing operator S. This amounts also to the partial invertibility of P at the convolution level in q ∗ p = δ_{0} + s with s ∈ S(G) (where the convolution product on G is defined by (f ∗ g)(x) = ∫_{G} f(y^{−1}x)g(y)dy). This last criteria makes sense on more general operators than differential ones. On a filtered group the basic example of a second order maximally hypoelliptic operator is given by the horizontal sub-Laplacian defined on functions by

Δ_{H}f = − ∑_{i=1}^{k} X_{i}^{2}f,  (28)

where D = span(X_{i}, 1 ≤ i ≤ k) = 𝔤_{1} generates 𝔤. Starting from this positive selfadjoint operator, one can define |∇| = Δ_{H}^{1/2} which can be shown to be an order one hypoelliptic pseudo-differential operator on G (see [11] section 3) invertible in the previous sense. On a general graded group, not necessarily filtered, such an order one scalar hypoelliptic operator can also be constructed. One can either replace Δ_{H} with the operator P of [16] corollaire 0.2, or use the one of [6] section 6. We denote by |∇|^{−1} its parametrix, in both left and right senses by self-adjointness of |∇|.

We are now ready to define the class of regularity we need. We extend |∇| as acting diagonally on the full exterior algebra Ω^{∗}G. We call N the induced weight function there.

Definition 5.1. An operator P acting on (part) of Ω^{∗}G is called C-C elliptic if P^{∇} = |∇|^{−N}P |∇|^{N} is a maximally hypoelliptic operator in the previous sense.

In the scalar case this is just hypoellipticity. Thus P = Δ_{H} is such an operator, while the full de Rham Laplacian on functions is not (except in ℝ^{n}). Indeed if G is a r-step group, Δ is a 2r-order operator in the graded sense, as it contains X_{r}^{2} terms with X_{r} ∈ 𝔤_{r}, but Δ do not control 2r-derivatives along 𝔤_{1} for r > 1. One interesting feature of this class is that even if de Rham Laplacian is not in it, the de Rham complex itself is, as (E_{0}, d_{c}).

Theorem 5.2. [25] Let G be a graded nilpotent Lie group. Then the de Rham complex and (E_{0}, d_{c}) are C-C elliptic.

That means the Laplacians associated to d^{∇} and d_{c}^{∇}

□_{d} = d^{∇}(d^{∇})^{∗} + (d^{∇})^{∗}d^{∇} and □_{d_{c}} = d_{c}^{∇}(d_{c}^{∇})^{∗} + (d_{c}^{∇})^{∗}d_{c}^{∇}  (29)

(not (Δ)^{∇} and (Δ_{c})^{∇}) are maximally hypoelliptic. Observe that they are all 0 order operators ! Indeed d and d_{c} are homogeneous operators on Ω^{∗}G in the sense the components of d and d_{c} that increase the weight of forms by k are precisely order k operators (the whole construction of section 2 is homogeneous in that sense).

Proof. Thanks to analysts’ work a short proof of thm 5.2 is possible.

It relies on an hypoellipticity criteria (called Rockland’s condition) first proved for differential operators on general graded groups by Helffer and Nourrigat [16], and extended to the pseudo-differential setting by Christ and all in [6]. Given a (pseudo-differential) operator P of order k on G, call P_{k} its homogeneous part of order k. Then the maximal hypoellipticity of P is implied by the injectivity, for all non trivial irreducible unitary representations π of G, of π(P_{k}) on S_{π} ⊂ H_{π} the space of C^{∞} vectors of the representation. (If G = ℝ^{n} this is the well known ellipticity criteria on the injectivity of the principal symbol of P in all directions. In general we recall (see [8] or [17]) that Kirillov has shown that these representations are parameterized by non-zero covectors ξ ∈ 𝔤^{∗} modulo the co-adjoint action of G on 𝔤^{∗}.)

We will need to know that for 0 ≠ π ∈ Ĝ one can find some X ∈ 𝔤 such that π(X) is the scalar (anti-self adjoint) operator i Id. Indeed, by irreducibility of π the center Z(G) of G acts by scalars on H_{π} (given by a character Z(G) → U(1)). If it is trivial, consider the quotiented action on Z(G/Z(G)) and so on until, by nilpotency of G, some non trivial scalar action is found.

We come to the proof. Let π and X as above. that the 0 order part of π(□_{d}) and π(□_{d_{c}}) are injective on the C^{∞} vectors S_{π} ⊂ H_{π}. These spaces really identify with the standard Schwarz space S(ℝ^{n}) ⊂ L^{2}(ℝ^{n}) with ℝ^{n} ≃ G/H when π is realized as an induced representation from G/H. Invariant differential operators on G become differential operators with polynomial coefficients on ℝ^{n}, while by definition |∇| is such that π(|∇|) is invertible and preserves S_{π} (in the filtered part we have just π(|∇|) = π(Δ_{H})^{1/2}). Hence the injectivity of π(□_{d}) on S_{π} is equivalent to the injectivity of the system π(d^{∇}) + π(d^{∇})^{∗} there. This is certainly implied by

ker π(d^{∇}) ∩ S_{π} = π(d^{∇})(S_{π}).

This is equivalent by dropping the conjugation by |∇|^{N}, which preserves S_{π}, to the vanishing of the smooth cohomology of the complex π(d)

ker π(d) ∩ S_{π} = π(d)(S_{π}).  (30)

Consider L_{X} the Lie derivative along X, with X as above. As a differential operator on Ω^{∗}G it can decomposed in L_{X} = X Id +ad(X) where

ad(X)α = − ∑ α(·, [X, ·], ·) = (L_{X})_{0} = i_{X}d_{0} + d_{0}i_{X}

is the algebraic part of L_{X} when expressed in a left invariant base. Since π(X) = i Id, one has π(L_{X}) = i Id +ad(X) on H_{π} with ad(X) nilpotent. It is therefore invertible of inverse

P_{X} = π(L_{X})^{−1} = ∑_{k≥0} i^{k−1}ad^{k}(X).

From Cartan’s formula L_{X} = i_{X}d + di_{X} we get [L_{X}, d] = [L_{X}, i_{X}] = 0, so that [P_{X}, π(d)] = [P_{X}, i_{X}] = 0, and finally

Id = (P_{X}i_{X})π(d) + π(d)(P_{X}i_{X}),

which gives (30). The proof for (E_{0}, d_{c}) is a formal consequence of this one. We recall that by theorem 2.6, d_{c} = Π_{E_{0}}Π_{E}dΠ_{E}Π_{E_{0}}, with Π_{E_{0}}Π_{E}Π_{E_{0}} = Id on E_{0}. Composing (31) with π(Π_{E_{0}}Π_{E}) on the left and π(Π_{E}Π_{E_{0}}) on the right we get on E_{0}

Id = Q_{X}π(d_{c}) + π(d_{c})Q_{X},

with Q(X) = π(Π_{E_{0}}Π_{E})P_{X}i_{X}π(Π_{E}Π_{E_{0}}). As before this implies the injectivity of π(□_{d_{c}}).

Remark 5.3. The role of the conjugation by |∇|^{N} in this proof appears very formal. Anyway it is crucial to put everything in a Sobolev scale and work with an homogeneous operator.

The twisted d_{c}^{∇} has a rather strange harmonic theory, due to the fact that (d_{c}^{∇})^{∗} = |∇|^{N}δ_{c}|∇|^{−N} is not δ_{c}^{∇} = |∇|^{−N}δ_{c}|∇|^{N}. On the other hand we observed in remark 3.7 that the d_{c}-harmonic theory is not so attractive, because not scale invariant, except in the case we have E_{0}^{k} = H^{k}(𝔤) of pure weight N_{k}. This is precisely the assumption we did in the previous pinching results. In that case the C-C regularity d_{c} is actually useful for the d_{c} harmonic theory itself. We remark that C-C ellipticity is preserved by any conjugation by an invertible fixed |∇|^{N_{0}}. So one can use any shifted weight function N − N_{0} instead of N. In the case E_{0}^{k} is of pure weight N_{k} the interesting one is N′ = N − N_{k}, since then on E_{0}^{k}

d_{c}^{∇} + (d_{c}^{∇})^{∗} = |∇|^{−N′}d_{c} + |∇|^{N′}δ_{c} = P (d_{c} + δ_{c}),  (31)

with P of strictly negative order ≤ −1, because d_{c} increases the weight by 1 at least.

We now show some applications of these techniques. Let again G be a graded group G and some k with E_{0}^{k} = H^{k}(𝔤) of pure weight N_{k}. The C-C ellipticity of (E_{0}, d_{c}) allows to define some number r_{k} describing the analytic order of convergence of the spectral sequence associated to d on k-forms. Starting form the splitting (21) of d_{c} in d_{c}^{δN_{k}^{min}} + ⋯ + d_{c}^{δN_{k}^{max}}, we consider for r ∈ [δN_{k}^{min}, δN_{k}^{max}] its cut-off

d_{c}^{[r]} = d_{c}^{δN_{k}^{min}} + ⋯ + d_{c}^{r}.

By theorem 5.2, we know that the complex

\xymatrix{E_{0}^{k−1} \ar[r]^{d_{c}} & E_{0}^{k} \ar[r]^{d_{c}^{[r]}} & E_{0}^{k+1}}

is C-C elliptic on E_{0}^{k} if r = δN_{k}^{max}.

Definition 5.4. Define r_{k} as the smallest integer r such that the previous complex is C-C elliptic on E_{0}^{k}.

In view of the proof of theorem 5.2 and (30) this r_{k} is the smallest number r such that ker π(d_{c}^{[r]}) ∩ S_{π} = π(d_{c})(S_{π}) for all non-trivial π ∈ Ĝ. From a purely algebraic viewpoint, this r_{k} is actually the order of convergence, at the level of representation, of the underlying spectral sequence associated to the natural filtered complex here (see section 2). In other words, r_{k} is also the smallest integer such that for all π, any smooth k-form α ∈ S_{π} of weight p which satisfies π(d)α of weight ≥ p + r is necessarily π(d)-closed (and finally exact) up to a form of weight p + 1.

Theorem 5.5. [25] Let G, E_{0}^{k} as in theorem 3.13. Then for r_{k} as above

δN_{k}^{min} ≤ β_{k} ≤ r_{k} ≤ δN_{k}^{max}.

We will give examples of groups with r_{k} < δN_{k}^{max} in the next section.

Proof. We have to show that β_{k} ≤ r_{k}. We use the same near-cohomology approach as in the proof of theorem 3.13. Firstly since for α ∈ E_{0}^{k}

∥d_{c}^{[r_{k}]}α∥^{2} = ∥d_{c}^{δN_{k}^{min}}α∥^{2} + ⋯ + ∥d_{c}^{r_{k}}α∥^{2} ≤ ∥d_{c}α∥^{2},

we certainly have that the near-cohomology cones thickness functions of d_{c} and d_{c}^{[r_{k}]} satisfy

F_{d_{c}}(λ) ≤ F_{d_{c}^{[r_{k}]}}(λ).

Hence the same homogeneity argument as in the proof of thm 3.13 gives the result if we show that F_{d_{c}^{[r_{k}]}}(λ) is finite. This is a consequence of the C-C ellipticity assumption. Namely the same proof as theorem 5.2 together with (31) shows that |∇|^{−N′}d_{c}^{[r_{k}]} + |∇|^{N′}δ_{c} = P (d_{c}^{[r_{k}]} + δ_{c}) is an order 0 maximally hypoelliptic pseudodifferential operator. That means (see [6]) it is bounded and invertible in L^{2} up to a smoothing term

QP (d_{c}^{[r_{k}]} + δ_{c}) = Id +S,

for some bounded (order 0) Q and smoothing S. We have already observed that P is of order ≤ −1, hence B = |∇|QP is bounded and satisfies

B(d_{c}^{[r_{k}]} + δ_{c}) = |∇| + S.  (32)

Therefore there exists some C such that

∥|∇|α∥ ≤ C(∥d_{c}^{[r_{k}]}α∥ + ∥δ_{c}α∥ + ∥α∥),

giving on ker δ_{c} the control of F_{d_{c}^{[r_{k}]}}(λ) by F_{|∇|}(Cλ + C). But this last function is finite because |∇| is a first order maximally hypoelliptic operator with smoothing spectral projectors (for any p, |∇|^{−k} has a C^{p} kernel for k large enough by the Sobolev embedding theorems).

In the opposite direction to the previous result, one can sometimes improve the bound from below of β_{k}, if d_{c}^{[r]} is sufficiently degenerated.

Proposition 5.6. Let G, E_{0}^{k} as above. Suppose that for some r, there exists nonclosed forms in D(d_{c}) ∩ ker d_{c}^{[r]}, then β_{k} ≥ r + 1.

Proof. Consider the closed quadratic form q(α) = ∥d_{c}α∥^{2} on (ker d_{c})^{⊥}. Its restriction to D(d_{c}) ∩ ker d_{c}^{[r]} is still closed, positive, and therefore associated to a self-adjoint Δ_{r} (see [23] thm VIII.15). By hypothesis Δ_{r} ≠ 0. Then for some λ, L = E(0 < Δ_{r} ≤ λ^{2}) is a non zero G-invariant linear space in the near-cohomology cone C_{λ}(d_{c}). Since d_{c} is homogeneous of order ≥ r + 1 on D(d_{c}) ∩ ker d_{c}^{[r]} we have, as in the proof of theorem 3.13, that h_{ε}^{∗}(L) ⊂ C_{λε^{r+1}}(d_{c}). This gives the bound from below for β_{k}.

Remark 5.7. We have already used this for the triangular group N_{4} in section 4.2.

We can interpret the assumption in proposition 5.6 in terms of representation theory. Namely by the Fourier-Plancherel isomorphism between G and Ĝ, the existence of some non-zero L^{2} form α ∈ D(d_{c}) ∩ ker d_{c}^{[r]} implies that π(α) is a field of Hilbert-Schmidt operators in ker π(d_{c}^{[r]}) with π(α) ≠ 0 on a set of representations of strictly positive Plancherel measure. Then the cohomology ker π(d_{c}^{[r]})/ Im π(d_{c}) is non-vanishing for “a lot” of representations of maximal dimensions (called generic). This is opposite (but not complementary) to the case of definition 5.4 and theorem 5.5, where we required this cohomology to vanish on all representations.

We close this section with an application the d_{c} Hodge-de Rham decomposition on compact E_{0} regular C-C manifolds M. We have seen in proposition 3.6 that the we have a closed splitting in L^{2}(Ω^{∗}M)

E_{0} = H_{c} ⊕ Im d_{c} ⊕ Im δ_{c},

where H_{c} = ker d_{c} ∩ ker δ_{c}. Again we restrict to the case of E_{0}^{k} = H^{k}(𝔤_{x_{0}}) is of pure weight N_{k} (see also remark 3.7). We assume also that C-C structure is given by a bracket generating distribution D (the filtered case).

Proposition 5.8. Let M, E_{0}^{k} as above. Then H_{c} = ker d_{c}∩ker δ_{c} consists of smooth forms.

To see this we define a differential operator of the form □_{c} = P (d_{c} + δ_{c}), with P chosen such that □_{c} becomes homogeneous. Fix any horizontal connection ∇_{D} on Ω^{∗}M. let K be the maximal order of the components of d_{c} + δ_{c} on E_{0}^{k}. Then we consider

□_{c} = ∑_{p} ∇_{D}^{K−p}(d_{c}^{p} + δ_{c}^{p}),

where d_{c}^{p} (resp. δ_{c}^{p}) is the components of d_{c} that increases (resp. decreases) the C-C weight by p. This □_{d} is a differential operator of order K. We show that

Lemma 5.9. □_{c} is maximally hypoelliptic on M (in the bracket generating case).

We recall this means that □_{c}α controls the full the K-horizontal jet of α (in L^{2} norms). One consequence of this is that weak solutions to □_{c}α = 0 are smooth, giving proposition 5.8.

Proof. Some characteristic features of maximal hypoellipticity of differential operators are its stability under perturbations of lower order and the fact, in the filtered case, it can be checked on the model group G_{x_{0}} with the freezed operator □_{c,x_{0}} (see [17]). We have already observed that d_{c} at x_{0} may be viewed as a perturbation of d_{c} on G_{x_{0}} (by Cartan’s formula dα(X_{i}) = ∑ X_{i}α(·) − α(·, [X_{i}, ·], ·) and the fact that [ , ]_{0} is the lower part of [ , ]). Hence we just have to check the result on the tangent group. There □_{c} looks like an algebraic version of d_{c}^{∇} + (d_{c}^{∇})^{∗} in (31) with the shift weight N′ = N − N_{k}, except the scalar pseudo-differential |∇| is replaced by the full first order horizontal jet ∇_{D}. The same proof as in theorem 5.2 actually gives the injectivity of π(□_{c}) since on S_{π}, ker π(□_{c}) = ker π(d_{c}) ∩ ker π(δ_{c}) and we have already seen that ker π(d_{c}) ∩ S_{π} = π(d_{c})(S_{π}).

Remark 5.10. The last part of this proof applies on any graded group, not only the filtered ones. It can therefore be taken as an alternative approach to C-C ellipticity using only differential operators, in this case of E_{0}^{k} of homogeneous weight.

5.2 Last examples

We start with a series of groups illustrating the “analytic” pinching of β_{1} obtained in theorem 5.5. We show that some groups may have (arbitrarily) high order relations and still r_{1} and thus β_{1} = 1.

5.2.1 First inaudible relations

Consider some quadratically presented group G of rank r > 3. The examples of Carlson-Toledo and Chen have been described in section 4.1. Let I be any ideal in 𝔤 generated by elements of weight ≥ 3, N = exp I and consider H = G/N. The relations of H as referred to the free Lie group, are generated by the quadratic relations of G and the generators of weight ≥ 3 of N. It may therefore have high order relations, but anyway they are “inaudible” in the heat decay.

Proposition 5.11. For such groups H, r_{1} = 1 = β_{1}(H), and finally α_{1}(H) = N(H). In particular these values are the same as on quadratic groups.

Proof. We use theorem 5.5. Let π_{H} be a non-trivial irreducible unitary representation of H. Suppose α is a form in 𝔥_{1}^{∗} ∩ S_{π} such that π_{H}(d_{c}^{[1]})α = 0. This means that α has an extension α̃ (given by π_{H}(Π_{E})α) such that π_{H}(d)α̃ is of weight ≥ 3. Consider P : G → H. Then P^{∗}(π_{H}) = π_{G} is an irreducible unitary representation of G and π_{G}(d)(P^{∗}α̃) = P^{∗}(π_{H}(d)α̃) is still of weight ≥ 3. But since G is quadratically presented this implies that π_{G}(d_{c}^{G})(P^{∗}α̃) = 0, and finally P^{∗}α̃ can be written π_{G}(d_{c}^{G})f for some f ∈ S_{π}. Restricting this to 𝔤_{1}, gives α = π_{F} (d_{c})f and the result.

These examples show that the spectral sequence may converge in the L^{2} sense quicker than in the algebraic one. More precisely H^{2}(𝔥) always contain forms of weight ≥ 3 here, dual to the relations we added by proposition 2.9. View as left invariant forms in H they are closed, therefore locally exact. Thus we obtain one forms α such that dα ∈ H^{2}(𝔥) are non-zero forms of weight ≥ 3. In other words α_{D} = Π_{E_{0}}α, the restriction of α to D = 𝔥_{1} satisfies d_{c}α_{D} of weight ≥ 3. This shows, as claimed, that the convergence rank of the spectral sequence as a local tool is actually greater than in L^{2}.

We can explain (partially) more geometrically why these high order relations do not have an L^{2} trace. We have to study

E_{2} = {α ∈ E_{0}^{1}(H) = Ω^{1}D | d_{c}α is of weight ≥ 3}.  (33)

Proposition 5.12. Up to closed forms, E_{2} contains only forms whose components are polynomial functions (of bounded degree) in the coordinates of G.

Thus certainly E_{2} do not contain any non-closed L^{2} form. We can’t make an L^{2} “wave packet” of them.

Proof. We first show than the components of d_{c}α are polynomial functions. Then lifting α to Ω^{1}H (with π_{E}) and integrating dα̃ with Poincaré’s lemma along polynomial vector fields gives the result.

Let Π : G → H and α ∈ E_{2}. Again Π^{∗}α is in E_{2}(G) which are d_{c}^{G}-closed because G is quadratically presented. That means Π^{∗}α has a true closed extension β on G. Recall that by section 2.4, d_{c}α interprets as the components of β along the generators of I the ideal of relations of H with respect to G. So we have to show these components are polynomial. This is consequence of (12). Namely, suppose G is of rank r. Applying (12) to Y ∈ I of weight r, gives X.β(Y) = 0 because [X, Y] = 0 for any X. Therefore β(Y) is constant. Applying now (12) to Y ∈ I of weight r − 1 gives then that X.β(Y) are constants functions, since [X, Y] is in Y and of weight r. And so on, all components of β along I are polynomials.

5.2.2 Around H-groups

The previous groups give us first example of “inaudible” relations. We give another illustration of the fact that “audible” relations have to be sufficiently flexible. We consider again H-type groups (see 4.1). A complete study may be found in [19] or [7]. Let D = ℝ^{n}, and ℝ^{k} be endowed with scalar products. A Clifford module structure on D is a linear map

J : ℝ^{k} → GL(n)

such that J(θ)^{2} = −∥θ∥^{2} Id and J(θ)^{∗} = −J(θ). Consider then

L : ℝ^{k} → Λ^{2}D^{∗}
θ → g_{D}(J(θ)·, ·).

The H-type group associated to J is the two step group G determined by the extension of D by Im L. That means 𝔤 = D ⊕ T with the Lie bracket structure D × D → T = ℝ^{k∗} given by θ([X, Y]) = −L(θ)(X, Y) = −⟨J(θ)X, Y ⟩. When k = 1, 3, 7 such groups appears as the maximal nilpotent (Iwasawa) subgroups of the rank 1 semi-simple groups SU(n, 1), Sp(n, 1) and F_{4}^{−20} (acting on the Cayley plane).

Proposition 5.13. Let G be a H-type group, then either β_{1}(G) = 1 or β_{1}(G) = 2 and G is the 3 dimensional Heisenberg group, or the 7 dimensional quaternionic group (associated to an elliptic D^{4} ⊂ ℝ^{7} see section 4.1).

Proof. Observe that D is necessarily of even dimension n = 2p, and that the p = 1 case corresponds to to the Heisenberg group H^{3} we have already met (β_{1} = 2 by cubic presentation). So we restrict now to p ≥ 2. We first show how the use of theorem 5.5 reduces things to a problem on the Heisenberg group H^{2p+1}. We study r_{1}. By the orbit method non-trivial irreducible representation of G are of two types, up to the coadjoint action.

- The first ones are trivial on T and come from some character π : D → U(1) with π(V) = e^{i⟨ξ,V ⟩} and non zero ξ ∈ D. We show that ker(π(d_{c}^{[1]})) ⊂ Im π(d_{c}) on Λ^{1}D^{∗}.

Namely π(d_{c}^{[1]})α = 0 means that iξ ∧ α ∈ Im L (because d_{c}^{[1]} is the quotient of the horizontal differential by Im L). But Im L contains only non-degenerate forms of top rank 2p ≥ 4 here. Therefore ξ ∧ α = 0 and α = iCξ = π(d_{c})C.

- The other (generic) ones are induced by a vertical θ, and factors trough a representation π of the Heisenberg group H^{2p+1} with the contact form θ and dθ = g_{D}(J(θ)·, ·). We orthogonally split Im L = ℝdθ ⊕ L′, and observe that by the Clifford structure L′ ⊂ Λ^{2,0}D^{∗} ⊕ Λ^{0,2}D^{∗} with respect to J(θ) (while dθ ∈ Λ^{1,1}D^{∗}). Again the equation π(d_{c}^{[1]})α = 0 means π(d_{D})α ∈ Im L, and our system can finally be interpreted on H^{2p+1}, as

π(d_{c}^{H})α ∈ L′ ⊂ Λ^{(2,0)+(0,2)}D^{∗} and π(δ_{c}^{H})α = 0,  (34)

with α ∈ S_{π} a smooth vector of the representation. We arrived at the turning.

Lemma 5.14. Equation (34) has a non trivial solution iff there exists a non zero α ∈ D^{∗} such that

α^{1,0} ∧ Λ^{1,0}D^{∗} ⊂ L^{′(2,0)}.

In real notations α ∧ β − Jα ∧ Jβ ∈ L′ for all β ∈ D^{∗}.

Note that in this case the forms α ∧ β − Jα ∧ Jβ ∈ L′ ⊂ Im L are of rank 4. Since here Im L contains only top rank forms one has necessarily D of dimension 4. Moreover counting dimensions shows that dimD − 2 ≤ dim(Im L) − 1 = k − 1, ie k ≥ 3, and finally k = 3 since this is the maximal number of complex structures on D = ℝ^{4}. Then certainly G has to be the seven dimensional quaternionic group Q_{7}. If not (34) has no solution and theorem 5.5 gives that r_{1} = 1 and thus β_{1} = 1.

Reversely Q_{7} satisfies the condition of lemma 5.14, because L^{′(2,0)} = Λ^{(2,0)}D^{∗} = ℂdZ_{1} ∧ dZ_{2}. It remains to see that β_{1}(Q_{7}) = 2. By proposition 5.6 it is sufficient to find some non d_{c}-closed α ∈ Ω^{1}D ∩ D(d_{c}) such that d_{c}^{[1]}α = 0. This is achieved by making a wave packet of solutions α_{π} of (34). Namely fix some compact set K of positive Plancherel measure in the generic representations of Q_{7}. Let P_{α_{π}} be the orthogonal projection on α_{π} ⊂ S_{π}. Fourier inversion formula (20) leads to consider for g ∈ Q_{7}

α(g) = ∫_{K⊂Ĝ} Tr(π(g)P_{α_{π}})dμ(π) = ∫_{K}⟨π(g)α_{π}, α_{π}⟩_{H_{π}}dμ(π).

α_{π} being smooth vectors, g → α(g) is a smooth form. Moreover

∥α_{π}∥^{2} = ∫_{K} ∥P_{α_{π}}∥_{HS}^{2}dμ(π) = μ(K) ∈]0, +∞[,

and for any derivative ∥X^{I}α∥^{2} = ∫_{K} ∥π(X^{I})α_{π}∥^{2}dμ(π) < ∞. Of course d_{c}^{[1]}α = ∫_{K}⟨π(d_{c}^{[1]})α_{π}, α_{π}⟩dμ(π) = 0 while d_{c}α ≠ 0 because π(d_{c})α_{π} ≠ 0 by C-C ellipticity of d_{c}.

We are left with the proof of lemma 5.14

Proof. We drop the π and H to lighten notations and work on H^{2p+1} (for p ≥ 2) with the contact complex d_{c}. A know fact is that the Laplacian Δ_{c} = (p − k)δ_{c}d_{c} + (p − k + 1)d_{c}δ_{c} preserves the bigrading of Λ^{k}D^{∗} (for k < p). Applying this to Δ_{c}α gives here

(Δ_{c}α, α) = (p − 1)∥d_{c}^{2,0}α^{1,0}∥^{2} + (p − 1)∥d_{c}^{0,2}α^{0,1}∥^{2} by (34)
= (Δ_{c}α^{1,0}, α^{1,0}) + (Δ_{c}α^{0,1}, α^{0,1})
= p∥δ_{c}α^{1,0}∥^{2} + (p − 1)∥d_{c}^{2,0}α^{1,0}∥^{2} + (p − 1)∥d_{c}^{1,1}α^{1,0}∥^{2}
+ p∥δ_{c}α^{0,1}∥^{2} + (p − 1)∥d_{c}^{0,2}α^{0,1}∥^{2} + (p − 1)∥d_{c}^{1,1}α^{0,1}∥^{2},

leading to the vanishing of δ_{c}α^{1,0}, d_{c}^{1,1}α^{1,0}, δ_{c}α^{0,1} and d_{c}^{1,1}α^{0,1}. By [24], this is equivalent to the holomorphy of each component of α^{1,0}, meaning \overline{Z_{i}}.α(Z_{j}) = 0 for Z_{i,j} ∈ D^{1,0} (resp. anti-holomorphy of α^{0,1}). At the representation level holomorphic functions are generated by f = e^{− ∑_{i=1}^{p} x_{i}^{2}/2} ∈ S(ℝ^{p}) = S_{π}, the vacuum state of the harmonic oscillator. Hence there exists a fixed β^{1,0} such that α^{1,0}(x) = f(x)β^{1,0}. Differentiating gives

π(d_{c})α^{1,0} = π(d_{c}^{2,0})α^{1,0} = ∑(π(Z_{i})f)θ_{i} ∧ β^{1,0} ∈ L^{′(2,0)},

but the functions π(Z_{i})f = (\frac{∂}{∂x_{i}} − x_{i})f = −2x_{i}f are independent giving that each θ_{i} ∧ β^{1,0} belongs to L^{′(2,0)}.

We observe that this result gives rise to a group in the β_{1} = 1 class not quadratically presented, and yet of a different type than in the previous section. Consider the 6 dimensional G_{6}, quotient of Q_{7} by a direction of its center, say T_{3} associated to θ_{3}, J_{3}. This G_{6} is the tangent group to a generic D^{4} ⊂ ℝ^{6}. It is a two dimensional extension of D = ℝ^{4} with 2 orthogonal complex structures J_{1} and J_{2}.

Consider again E_{2} as in (33). This E_{2} has a lot of non-closed sections. For instance in view of the proof of lemma 5.14, it contains any form α ∈ Λ^{1}D^{∗} whose (1, 0) components with respect to J_{3} are J_{3} holomorphic functions on D invariant along T_{1}, T_{2}. Here E_{2} contains many other forms than polynomials of bounded degree like in proposition 5.12. Thus the cubic relations of G_{6} (dual to the forms θ_{1} ∧ J_{1}α − θ_{2} ∧ J_{2}α by section 2.3) won’t be solved (without adding others) in any finite dimensional extension of G_{6}. They are inaudible in the heat decay anyway. The fact that E_{2} has no L^{2} section while plenty of local ones interprets here as a vanishing theorem similar for instance of the vanishing of L^{2} holomorphic functions on ℂ. Observe these J_{3} holomorphic forms were not controlled in the representation associated to the T_{3} direction on Q_{7}, but can be here since we have removed it!

Lastly we complete the study of the other Novikov-Shubin exponents α_{p} of Q_{7}. In view of (9) the algebraic pinching theorem 3.13 gives β_{3}(Q_{7}) = 2. By duality (see discussion around (27)) we are left with the study of β_{2}(Q_{7}) = β_{4}(Q_{7}). We show that theorem 5.5 gives β_{4}(Q_{7}) = r_{4} = 1, that is the system d_{c}^{[1]} + δ_{c} is C-C elliptic on E_{0}^{4,(6)}.

Proof. If α ∈ S_{π} belongs to ker π(d_{c}^{[1]} +δ_{c}), then by C-C ellipticity of d_{c}, β = π(d_{c})α is non trivial if α is. Moreover β ∈ E_{0}^{5,(8)} ∩ ker π(d_{c}), and this space is Hodge-∗ conjugated to ker π(d^{−∗}) from Λ^{2,−}D^{∗} to Λ^{1}D^{∗}. The elliptic symbol (on degenerated representations) of this map is well known to be injective. At the level of generic representations, one computes easily that π(d^{−}d^{−∗}) is an inversible Folland-Stein (or Tanaka) operator, giving the result.

Of course these C-C ellipticity results transplant, like in lemma 5.9, on any elliptic distribution D^{4} ⊂ T M^{7} with M^{7} compact, without integrability condition on the structure.

References

[1] M.F. Atiyah. Elliptic operators, discrete groups and von Neumann algebras. Astérisque, 32-33 (1976), 43–72.

[2] R. Brooks. The fundamental group and the spectrum of the Laplacian. Comment. Math. Helv., 56 (1981), 581–598.

[3] James A. Carlson and Domingo Toledo. Quadratic presentations and nilpotent Kähler groups. J. Geom. Anal., 5(3):359–377, 1995.

[4] William Casselman and M. Scott Osborne. The 𝔫-cohomology of representations with an infinitesimal character. Compositio Math., 31(2):219–227, 1975.

[5] Shirnping Chen. Examples of n-step nilpotent 1-formal 1-minimal models. C. R. Acad. Sci. Paris Sér. I Math., 321(2):223–228, 1995.

[6] Michael Christ, Daryl Geller, Paweł Głowacki, and Larry Polin. Pseudodifferential operators on groups with dilations. Duke Math. J., 68(1):31–65, 1992.

[7] Michael Cowling, Anthony H. Dooley, Adam Korányi, and Fulvio Ricci. H-type groups and Iwasawa decompositions. Adv. Math., 87(1):1–41, 1991.

[8] L. Corwin and F. P. Greenleaf. Representations of nilpotent Lie groups and their applications. Part I. Cambridge University Press, Cambridge, 1990.

[9] Ch. Deninger and W. Singhof. On the cohomology of nilpotent Lie algebras. Bull. Soc. Math. France, 116(1):3–14, 1988.

[10] J. Dixmier. Cohomologie des algèbres de Lie nilpotentes. Acta Sci. Math., 16:246–250, 1955.

[11] G. B. Folland. Subelliptic estimates and function spaces on nilpotent Lie groups. Ark. Mat., 13(2):161–207, 1975.

[12] Phillip A. Griffiths and John W. Morgan. Rational homotopy theory and differential forms. Birkhäuser Boston, Mass., 1981.

[13] M. Gromov. Carnot-Carathéodory spaces seen from within. In A. Bellaïche and J.-J. Risler, editors, Sub-Riemannian Geometry, volume 144 of Progress in Math, pages 79–323. Birkhäuser, 1996.

[14] M. Gromov and M.A. Shubin. Von Neumann spectra near zero. Geometric and Functional Analysis, 1(4) (1991), 375–404.

[15] M. Gromov and M.A. Shubin. Near-cohomology of Hilbert complexes and topology of non-simply connected manifolds. In Méthodes semi-classiques. Astérisque, 210 (1992), 283–294.

[16] B. Helffer and J. Nourrigat. Caractérisation des opérateurs hypoelliptiques homogènes invariants à gauche sur un groupe de Lie nilpotent gradué. Comm. Partial Differential Equations, 4(8):899–958, 1979.

[17] Bernard Helffer and Jean Nourrigat. Hypoellipticité maximale pour des opérateurs polynômes de champs de vecteurs. Birkhäuser Boston Inc., Boston, MA, 1985.

[18] Dale Husemoller. Fibre bundles. Springer-Verlag, New York, third edition, 1994.

[19] Aroldo Kaplan. On the geometry of groups of Heisenberg type. Bull. London Math. Soc., 15(1):35–42, 1983.

[20] John Lott. Heat kernels on covering spaces and topological invariants. J. Differential Geom., 35(2):471–510, 1992.

[21] P. Pansu. Introduction to L^{2} Betti numbers. In Riemannian geometry (Waterloo, ON, 1993), pages 53–86. Amer. Math. Soc., Providence, RI, 1996.

[22] Linda Preiss Rothschild and E. M. Stein. Hypoelliptic differential operators and nilpotent groups. Acta Math., 137(3-4):247–320, 1976.

[23] M. Reed and B. Simon. Methods of Modern Mathematical Physics, volumes 1 and 2 : Functional Analysis. Academic Press, 1972.

[24] M. Rumin. Un complexe de formes différentielles sur les variétés de contact. C. R. Acad. Sci. Paris Sér. I Math., 310(6):401–404, 1990.

[25] M. Rumin. Differential geometry on C-C spaces and application to the Novikov-Shubin numbers of nilpotent Lie groups. C. R. Acad. Sci. Paris Sér. I Math., 329(11):985–990, 1999.

[26] Simon Salamon. Riemannian geometry and holonomy groups. Longman Scientific & Technical, Harlow, 1989.

[27] N. Th. Varopoulos, L. Saloff-Coste, and T. Coulhon. Analysis and geometry on groups. Cambridge University Press, Cambridge, 1992.

———————————— Mathématique, Bât. 425, Université de Paris-Sud, 91405 Orsay, France

e-mail : michel.rumin@math.u-psud.fr
http://topo.math.u-psud.fr/~rumin