Local Rigidity Of Uniform Lattices

TL;DR

This paper extends classical local rigidity results to uniform lattices in certain non-positively curved spaces, broadening the understanding of their topological properties beyond Lie groups.

Contribution

It generalizes the local rigidity theorem to irreducible uniform lattices in proper CAT(0) spaces without Euclidean factors, beyond hyperbolic spaces.

Findings

Established topological local rigidity for uniform lattices in compactly generated groups.

Extended classical rigidity theorems to CAT(0) spaces with no Euclidean factors.

Derived an analog of Wang's finiteness theorem for specific non-positively curved spaces.

Abstract

We establish topological local rigidity for uniform lattices in compactly generated groups, extending the result of Weil from the realm of Lie groups. We generalize the classical local rigidity theorem of Selberg, Calabi and Weil to irreducible uniform lattices in $\text{Isom}(X)$ where $X$ is a proper $\text{CAT}(0)$ space with no Euclidian factors, not isometric to the hyperbolic plane. We deduce an analog of Wang's finiteness theorem for certain non-positively curved metric spaces.