Amenable groups and measure concentration on spheres

Amenable groups and measure concentration on spheres

It is proved that a discrete group G is amenable if and only if for every unitary representation of G in an infinite-dimensional Hilbert space H the maximal uniform compactification of the unit sphere SH has a G-fixed point, that is, the pair (SH, G) has the concentration property in the sense of Milman. Consequently, the maximal U(H)equivariant compactification of the sphere in a Hilbert space...

Ergodic Theorems on 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 ∈ ...

Ergodic Theorems on Amenable Groups

In 1931, Birkhoff gave a general and rigorous description of the ergodic hypothesis from statistical meachanics. This concept can be generalized by group actions of a large class of amenable groups on σ-finite measure spaces. The expansion of this theory culminated in Lindenstrauss’ celebrated proof of the general pointwise ergodic theorem in 2001. The talk is devoted to the introduction of abs...

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...