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.
Highlights
The present paper is a contribution to the study of large scale geometry of Banach spaces and topological groups and, in particular, to questions of embeddability between these objects
Still our focus will be restricted as we are mainly interested in equivariant maps, that is, cocycles associated to affine isometric actions on Banach spaces
Given a map σ : (X, d) → (Y, ∂) between metric spaces, we define the compression modulus by κσ (t) = inf(∂(σ (a), σ (b)) | d(a, b) t)
Summary
The present paper is a contribution to the study of large scale geometry of Banach spaces and topological groups and, in particular, to questions of embeddability between these objects. Let G be a Følner amenable Polish group admitting a uniformly continuous coarse embedding into a Banach space E. Note that Theorem 9 may be viewed as a generalization of a result of Haagerup and Przybyszewska [30] stating that every locally compact second countable group admits a proper affine isometric action on a reflexive space. A coarsely proper continuous isometric action G X of a topological group G on a metric space X is said to be a geometric Gelfand pair if, for some K and all x, y, z, u ∈ X with d(x, y) d(z, u), there is g ∈ G so that d(g(x), z) K and d(z, g(y)) + d(g(y), u) d(z, u) + K This second condition is typically verified when X is sufficiently geodesic and the action of G is almost doubly transitive. Every continuous affine isometric action of Isom(QU) on a reflexive Banach space or on L1([0, 1]) has a fixed point
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