Equivariant Geometry of Banach Spaces and Topological Groups

Abstract

We study uniform and coarse embeddings between Banach spaces and topological groups. A particular focus is put on equivariant embeddings, that is, continuous cocycles associated to continuous affine isometric actions of topological groups on separable Banach spaces with varying geometry.

On Følner sets in topological groups

“…In this section we give an application of Theorem 4.5 concerning coarse geometry of topological groups as studied extensively in [Ros17a, Ros17b]. In fact, we answer a question posed by Rosendal [Ros17b, Problem 1] in the affirmative.…”

“…for all x, y ∈ X. For a detailed discussion of the functions introduced in this paragraph the reader is referred to [Ros15b]. We will also need some additional terminology concerning Banach spaces.…”

“…For a start, let us recall some terminology from [Ros17b]. Let and be pseudo-metric spaces and let .…”

“…Furthermore, the compression modulus of is defined to be with Moreover, we define the exact compression modulus of as with Evidently, for every and for all . For a detailed discussion of the functions introduced in this paragraph, the reader is referred to [Ros17b].…”

“…The focus of the final part of the paper is towards a study of the coarse geometry of amenable topological groups – as recently initiated by Rosendal [Ros17a, Ros17b]. Our theory provides a natural setup in which some classical constructions, that rely on the existence of Følner sets in the realm of locally compact groups, can be carried out without further problems.…”