Proper CAT(0) actions of unipotent-free linear groups

TL;DR

This paper constructs specific CAT(0) space actions for linear groups over complex numbers, revealing constraints on their representations and implications for 3-manifold groups without nonpositively curved metrics.

Contribution

It introduces a method to produce CAT(0) actions for linear groups that are well-behaved on subgroups without unipotent elements, linking geometric actions to algebraic properties.

Findings

Groups with no nontrivial unipotent elements have well-behaved CAT(0) actions.

Fundamental groups of certain 3-manifolds cannot have faithful finite-dimensional unitary representations.

Abstract

Let $\Gamma$ be a finitely generated group of matrices over $\mathbb{C}$. We construct an isometric action of $\Gamma$ on a complete CAT(0) space $X$ such that the restriction of this action to any subgroup of $\Gamma$ containing no nontrivial unipotent elements is well behaved. As an application, we show that if $M$ is a graph manifold that does not admit a nonpositively curved Riemannian metric, then any finite-dimensional $\mathbb{C}$-linear representation of $\pi_1(M)$ maps a nontrivial element of $\pi_1(M)$ to a unipotent matrix. In particular, the fundamental groups of such 3-manifolds do not admit any faithful finite-dimensional unitary representations.