Growth rates of amenable groups

Amenable Groups

Throughout we let Γ be a discrete group. For f : Γ → C and each s ∈ Γ we define the left translation action by (s.f)(t) = f(s−1t). Definition 1.1. A group Γ is amenable is there exists a state μ on l∞(Γ) which is invariant under the left translation action: for all s ∈ Γ and f ∈ l∞(Γ), μ(s.f) = μ(f). Example 1.2. Finite groups are amenable: take the state which sends χ{s} to 1 |Γ| for each s ∈ ...

Structural Properties of Inner Amenable Discrete Groups

On growth of homology torsion in amenable groups

Suppose an amenable group G is acting freely on a simply connected simplicial complex X̃ with compact quotient X. Fix n ≥ 1, assume Hn(X̃,Z) = 0 and let (Hi) be a Farber chain in G. We prove that the torsion of the integral homology in dimension n of X̃/Hi grows subexponentially in [G : Hi]. This fails if X is not compact. We provide the first examples of amenable groups for which torsion in homol...

Some extremely amenable groups

A topological group G is extremely amenable if every continuous action of G on a compact space has a fixed point. Using the concentration of measure techniques developed by Gromov and Milman, we prove that the group of automorphisms of a Lebesgue space with a non-atomic measure is extremely amenable with the weak topology but not with the uniform one. Strengthening a de la Harpe’s result, we sh...

Are Unitarizable Groups Amenable?

We give a new formulation of some of our recent results on the following problem: if all uniformly bounded representations on a discrete group G are similar to unitary ones, is the group amenable? In §5, we give a new proof of Haagerup's theorem that, on non-commutative free groups, there are Herz-Schur multipliers that are not coefficients of uniformly bounded representations. We actually prov...