5 Tits Alternative

5.1 Statements

Theorem 5.1 (Tits Alternative) Let G be a finitely generated linear group then G contains a free subgroup or G is virtually solvable.

This alternative implies that a linear group is amenable for a simple reason (it is virtually solvable) or it is not amenable for a simple reason (it contains a free subgroup).

Let us recall that a linear group is a subgroup G of some GLn(k) where n is some integer and k any field. In some case, like k=C, the assumption that G is finitely can be removed. The proof depends on the field and to avoid number theoretical difficulties, we will prove the following. Up to replacing R by another locally compact field, the proof of the Tits alternative reduces (if the group is not virtually solvable) to prove the following.

Theorem 5.2 Let G be a unbounded subgroup of SLn(R) whose Zariski closure acts irreductibly on Rn then G contains a free groups on two generators.

Let us recall that a subgroup GSLn(R) acts irreductibly on Rn if there is non-trivial G-invariant subspace of Rn.

A subgroup H of SLn(R) is algebraic (or Zariski closed) if there is a familly of polynomial maps Pi:Mn(R)R such that gHPi(g)=0 i. The Zariski closure of subgroup G is the intersection of all algebraic subgroups that contains G.

Remark. For a linear space V of dimension n, one can think the notion of algebraic group depends of a choice of an identification VRn, i.e. the choice of a base. Since change of bases corresponds to conjugation for matrices and conjugation are linear maps, this notion of algebraic groups do not depend of the choice of a base.

Example 5.1 The following groups are algebraic:

  • SO(p,q)
  • SO(n)
  • SLn(R)
  • stabilizers of subspaces of Rn.

An algebraic group H is connected if it can’t be written as the union of two proper Zariski closed subspaces. This strange definition is equivalent to connectedness for the Zariski topology.

5.2 Cartan decomposition and contracting projective transformations

Proposition 5.1 (Cartan decomposition) Let gSLn(R). There exist k1,k2SOn(R) and aA+ such that

g=k1ak2

where A+={a=diag(a1,,an), a1a2an>0}.

Moreover, for two such decompositions, the ai are the same and we denote them by ai(g).

Proof. The matrix tgg is symmetric definite positive, so there is kSO(n) such that tktggk is diag(b1,b2,,bn) where the bis are positive and in decreasing order. Let ai=bi and a=diag(a1,a2,,an) then t(a1)tktggka1 is the identity, i.e. gka1SO(n), so, there is k2SO(n) such that g=k2ak1. By setting k1=k1, we have the desired decomposition.

One should observe that the ai’s are the square roots of the eigenvalues of tgg and thus unique.

Observe that a1(g)=||g|| (the operator norm associated to the Euclidean norm on Rn) and thus if ||gm||+ then a1(gm).

Let dn and gSLn(R). The linear map g induces a linear map ¯gSL(dRn) given by ¯g(ei1eid)=gei1geid for i1,,id distincts in {1,,n}.

Lemma 5.1 Let gm be an unbounded sequence of elements of SLn(R) then there some l such the action if ¯gm is the image of gm for the action on lRn then up to extraction α1(¯gm)/α2(¯gm).

On the projective space P(Rn), for two lines l1,l2, we define d(l1,l2)=sin(α) where α is the geometric angle between the two lines. For a metric space (X,d) and a subspace YX, we denote by Yε, the ε-neighborhood of Y.

Definition 5.1 Let g be a projective transformation of P(Rn) and ε>0. This transformation is said to be ϵ-contracting if there a point vP(Rn) and a hyperplane HP(Rn) such that g(P(Rn)Hε)vε.

Lemma 5.2 If g is ε-contracting then for any k1,k2SO(n) k1gk2 is ε-contracting as well.

Proposition 5.2 Let gSL(V). For any ε>0, there is M>0 such that if a1(g)/a2(g)>M then the projective transformation associated to g is ε-contracting and ε-Lipschitz outside Hε .

Lemma 5.3 Let gSL(V) and δ>0 then there is an open subset OP(V) such that g|O is (1+δ)-Lipschitz.

Proposition 5.3 For every ε>0 there is δ>0 such that if gSL(V) is δ-Lipschitz on some open subspace of P(V) then the projective transformation associated to g is ε-contracting.

5.3 Separating points and hyperplanes

Definition 5.2 A subset FSL(V) is (k,r)-separating (where kN, r>0) if for any hyperplanes H1,,Hk in P(V) and v1,,vkP(V) there is fF such that

d(fvi,Hj)>r and d(f1vi,Hj)>r

for all i,j{1,,k}.

Proposition 5.4 Let G be a subgroup of SL(V) with a connected Zariski closure that acts irreductibly on V then for any k, there is r>0 such that G contains a finite (k,r)-separating subset.

Proof. Let H be the Zariski closure of G and let H1,,Hk be hyperplanes in P(V) and v1,,vk points in P(V). We define

V+i,j={hH, hviHj}

and

Vi,j={hH, h1viHj}.

These subsets are algebraic subsets. Define Vi,j=V+i,jVi,j. Since H is irreducible i,jVi,j. Since G is Zariski dense, there is \gamma\in H\setminus \cup_{i,j}V_{i,j}.

Now, define for \gamma\in G,

X_\gamma=\{(H_1,\dots,H_k,v_1,\dots,v_k),\ \gamma v_i, \gamma^{-1} v_i\notin H_j,\ \forall i,j\}.

Let X=\mathrm{Grass}_{n-1}(V)^k\times P(V)^k. We proved that X=\cup_{\gamma\in G}X_\gamma. Since each X_\gamma is open, by compactness of X, there is F\subset G finite such that X=\cup_{\gamma\in F}X_\gamma. Now let

r=\max_{\gamma\in F}\min_{i,j}\{d(\gamma v_i, H_j),d(\gamma^{-1}v_i, H_j)\}. Now, F is the desired (k,r)-subset of G.

5.4 Completion of the proof

In this subsection we prove Theorem 5.2 and we will need the following definition.

Definition 5.3 Let \varepsilon,r>0. A projective map g\colon P(V)\to P(V) is (\varepsilon,r)-proximal if it \varepsilon-contracting with respect to a pair (v,H) with d(v,H)>r.

Let’s start the proof. So, let G be an unbounded subgroup of SL_n(\mathbf{R}) whose Zariski closure is connected and acts irreducibly on \mathbf{R}^n. We know by Proposition 5.4 that there is some finite set F\subset G that is (2,r)-separating.

Let C be an upper bounds for Lipschitz constants of elements of F^{\pm 1} and \delta\in(0,r). We know there is \gamma that is \delta-contracting outside H_\delta for some hyperplane H. We proved that \gamma^{-1} is 2-Lipschitz on some open subset O\subset P(\mathbf{R}^n). Let v\in O and f\in F such that f(\gamma^{-1}v)\notin H_r and f^{-1}(\gamma^{-1}v)\notin H_r. Now let \gamma_0=\gamma f\gamma^{-1}. It is 2C\delta-Lipschitz and its inverse as well.

Let \varepsilon>0. By choosing \delta small enough, this implies that \gamma_0^{\pm 1} are \varepsilon-contracting with respect to some (v_0^+,H_0^+) and (v_0^-,H_0^-).

By separation, there is f_0 such that

d(f_0v_0^+,H_0^+)>r

and d(f_0^{-1}v_0^-,H_0^-)>r. Since f_0^{-1} is C-Lipschitz, d(v_0^-,f_0H_0^-)>r/C.

For x\notin (H_0^+)_\varepsilon, since f is C-Lipschitz

d(f_0\gamma_0x,f_0v_0^+)\leq Cd(\gamma_0 x,v_0^+)\leq C\varepsilon.

So if we set \gamma_1=f_0\gamma_0, it is (C\varepsilon, r)-proximal with respect to (f_0v_0^+, H_0^+).

Its inverse \gamma^{-1}=\gamma_0^{-1}f_0^{-1} is (C\varepsilon, r/C)-proximal with respect to (v_0^-,f_0H_0^-). Actually, if x\notin (f_OH_0^-)_{C\varepsilon} then

d(\gamma_0^{-1}f_0^{-1}x,v_0^-)<\varepsilon since \gamma_0^{-1} is \varepsilon-contracting with respect to (v_0-, H_0^-) and we have proved that d(v_0^-,f_OH_0^-)>r/C.

Set v_1^+=f_0v_0^+, H_1^+=H_0^+, v_1^-=v_0^- and H_1^-=f_OH_0^-. So \gamma_1 is (C\varepsilon, r)-proximal with respect to (v_1^+, H_1^+) and \gamma^{-1} is (C\varepsilon, r/C)-proximal with respect to (v_1^-,H_1^-).

By separation, find f_1\in F such that

d(f_1v_1^+,H_1^+\cup H_1^-)>r and

d(f_1v_1^-,H_1^+\cup H_1^-)>r.

Set \gamma_2=f_1\gamma_1f_1^{-1}. By similar computations, one can prove that it is (C^2\varepsilon,R/C)-proximal with respect to (f_1v_1^+,f_1H_1^+). Similarly, \gamma_2^{-1} is (C^2\varepsilon,r/C)-proximal with respect to (f_1v_1^-,f_1H_1^-).

Set v_2^\pm and H_2^\pm and we have

d(v_2^\pm,H_1^+\cup H_1^-)>r

and

d(v_1^\pm, H_2^+\cup H_2^-)>r/C.

So for \varepsilon small enough, one can play ping-pong.

5.5 Exercises

Exercise 5.1 Prove that the Zariski closure of a group is also a group.

Exercise 5.2 Prove that any algebraic group is a Lie group.

Exercise 5.3 Prove that a Zariski closed set has finitely many irreducible components. Prove that if G is an algebraic subgroup then the identity is contained in a unique component and this component is open. Prove that this component is a normal finite index subgroup G^0 that coincides with the connected component of the identity for the Zariski topology. This component is called the identity component of G.

What is the identity component (in the algebraic sense) of \mathrm{GL}_n(\mathbf{R}) ? Compare it with the identity component for the usual topology.

Prove that any projective transformation is Lipschitz.