Day's fixed point theorem, Group cohomology and Quasi-isometric rigidity

TL;DR

This paper demonstrates how Day's fixed point theorem can be applied to conjugate groups of biLipschitz maps into similarity groups, leading to new proofs of quasi-isometric rigidity and related theorems.

Contribution

It introduces a novel application of Day's fixed point theorem to establish Tukia-type theorems and quasi-isometric rigidity results.

Findings

Groups of biLipschitz maps can be conjugated into similarity groups using Day's theorem

New proofs of quasi-isometric rigidity are provided

The approach simplifies existing proofs of related theorems

Abstract

In this note we explain how Day's fixed point theorem can be used to conjugate certain groups of biLipschitz maps of a metric space into special subgroups like similarity groups. In particular, we use Day's theorem to establish Tukia-type theorems and to give new proofs of quasi-isometric rigidity results.