Low-dimensional representations of matrix groups and group actions on CAT(0) spaces and manifolds

TL;DR

This paper investigates how matrix groups over various rings can be represented in low dimensions by analyzing their actions on CAT(0) spaces, spheres, and acyclic manifolds, revealing insights into their geometric and algebraic properties.

Contribution

It introduces a framework for understanding low-dimensional representations of matrix groups via geometric group actions on non-positively curved spaces.

Findings

Characterization of group actions on CAT(0) spaces

Constraints on low-dimensional representations of matrix groups

Connections between algebraic properties and geometric actions

Abstract

We study low-dimensional representations of matrix groups over general rings, by considering group actions on CAT(0) spaces, spheres and acyclic manifolds.