Introduction

The construction of the Lebesgue measure in 1901 raised many natural questions. Is it the unique measure (up to scaling) invariant the group of isometries of \(\mathbf{R}^n\) ? Can one measure any subset of \(\mathbf{R}^n\). What happens if you remove the \(\sigma\)-additivity of the measure (equivalently remove the monotone convergence property)?

Some of these questions have been answered negatively by showing existence of non-measurable subsets of \(\mathbf{R}\) or by constructing paradoxes such as the Banach-Tarski paradox (using ideas due to Hausdorff). One can cut a unit ball of \(\mathbf{R}^3\) into finitely many pieces, move them with isometries and move them to build to new balls that are isometric to the original one! Such a paradox does not exist in \(\mathbf{R}\) or \(\mathbf{R}^2\).

Analyzing these paradoxes, von Neumann understood that the difference between dimension \(n\leq 2\) and dimension \(n\geq 3\) comes from the group \(\mathrm{Isom}(\mathbf{R}^n)\).

In the Banach-Tarski, the pieces cannot be measurable, otherwise the volume of the unit would be twice itself. Roughly speaking a mean is a finitely additive probability measure on a set \(X\) such that every subset of \(X\) is measurable. The Banach-Tarski paradox shows that such mean on the unit ball and invariant under the action of the isometry group of \(\mathbf{R}^n\) does not exist for \(n\geq3\) while it exists for \(n\leq2\).

The difference comes from the fact that the group \(\mathrm{Isom}(\mathbf{R}^n)\) is solvable for \(n\leq2\) and for \(n\geq3\), \(\mathrm{Isom}(\mathbf{R}^n)\) contains a free subgroup on 2 generators.

Von Neumann defined amenable groups as groups that admit an invariant mean for the action on themselves by left multiplications. One leading direction for us we will be to determine whether a given group is amenable or not. We will see several equivalent definitions and in particular a fixed point property that can be extended for topological groups. So, we will see locally compact and non-locally compact groups that are or not amenable.

We will see some strong consequences for amenable groups acting on spaces of non-positive curvature (Adams-Ballman theorem). This will be an important tool for us to determine what is the amenable linear Lie groups, i.e., amenable connected closed subgroups of \(\mathrm{GL}_n(\mathbf{R})\).

We will also see a very strong form of the dichotomy amenable versus non-amenable for linear subgroups (subgroups of \(\mathrm{GL}_n(k)\) for some field \(k\)) which is called the Tits alternative. A finitely generated subgroup of \(\mathrm{GL}_n(k)\) contains a free subgroup on two generators or it is virtually solvable. So it is amenable for a simple reason (it is virtually solvable) or it is non-amenable for a simple reason (it contains a free subgroup).