Amenable groups with very poor compression into Lebesgue spaces

A finitely-generated amenable group with very poor compression into Lebesgue spaces

We construct an example of a finitely-generated amenable group that does not admit any coarse 1-Lipschitz embedding with positive compression exponent into Lp for any 1 ≤ p < ∞, answering positively a question of Arzhantseva, Guba and Sapir.

Finite Metric Spaces and Their Embedding into Lebesgue Spaces

The properties of the metric topology on infinite and finite sets are analyzed. We answer whether finite metric spaces hold interest in algebraic topology, and how this result is generalized to pseudometric spaces through the Kolmogorov quotient. Embedding into Lebesgue spaces is analyzed, with special attention for Hilbert spaces, `p, and EN .

Ramsey–milman Phenomenon, Urysohn Metric Spaces, and Extremely Amenable Groups

In this paper we further study links between concentration of measure in topological transformation groups, existence of fixed points, and Ramsey-type theorems for metric spaces. We prove that whenever the group Iso (U) of isometries of Urysohn’s universal complete separable metric space U, equipped with the compact-open topology, acts upon an arbitrary compact space, it has a fixed point. The ...

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 ∈ ...

Decomposition of Lebesgue Spaces