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

Abstract

Let $Γ$ be a finitely generated group of matrices over $\mathbb{C}$. We construct an isometric action of $Γ$ on a complete CAT(0) space $X$ such that the restriction of this action to any subgroup of $Γ$ 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 $π_1(M)$ maps a nontrivial element of $π_1(M)$ to a unipotent matrix. In particular, the fundamental groups of such 3-manifolds do not admit any faithful finite-dimensional unitary representations.