Rigidity of actions on metric spaces close to three dimensional manifolds

TL;DR

This paper explores a metric variation of the Zimmer program for three-dimensional manifolds, establishing a dichotomy for homomorphisms from higher rank lattices to isometry groups, with implications for geometric orbifolds and Alexandrov spaces.

Contribution

It introduces a new metric approach to the Zimmer program in dimension three, proving a dichotomy for lattice homomorphisms and extending results to orbifolds and Alexandrov spaces.

Findings

Dichotomy between finite image and infinite volume quotients for lattice homomorphisms.

Classification results for finite group actions on three manifolds.

Extension of the Zimmer program to singular spaces like orbifolds and Alexandrov spaces.

Abstract

In this article we propose a metric variation on the C^0-version of the Zimmer program for three manifolds. After a reexamination of the isometry groups of geometric three-manifolds, we consider homomorphisms defined on higher rank lattices to them and establish a dichotomy betweeen finite image or infinite volume of the quotient. Along the way, we enumerate classification results for actions of finite groups on three manifolds where available, and we give an answer to a metric variation on topological versions of the Zimmer program for aspherical three manifolds, as asked by Weinberger and Ye, which are based on the dichotomy stablished in this work and known topological rigidity phenomena for three manifolds. Using results by John Pardon and Galaz-Garc\'ia-Guijarro, the dichotomy for homomorphisms of higher rank lattices to isometry groups of three manifolds implies that a C 0 -isometric version of the Zimmer program is also true for singular geodesic spaces closely related to three dimensional manifolds, namely three dimensional geometric orbifolds and Alexandrov spaces. A topological version of the Zimmer Program is seen to hold in dimension 3 for Alexandrov spaces using Pardon's ideas.