Equivariant geometry of Banach spaces and topological groups

TL;DR

This paper investigates the properties of equivariant embeddings between Banach spaces and topological groups, emphasizing the role of continuous cocycles and affine isometric actions in understanding their geometric and topological structure.

Contribution

It introduces a detailed analysis of equivariant embeddings and continuous cocycles, advancing the understanding of how topological groups act on Banach spaces with diverse geometries.

Findings

Characterization of equivariant embeddings in terms of continuous cocycles

Insights into the geometry of Banach spaces under group actions

Connections between affine isometric actions and embedding properties

Abstract

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