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)

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Summary

Introduction

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

Uniform and coarse structures on topological groups
Uniform versus coarse embeddings between Banach spaces
Cocycles and affine isometric representations
Amenability
Embeddability in Hilbert spaces
Preservation of local structure
A fixed point property for affine isometric actions
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