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$\mathrm {CAT(0)}$ ACTIONS OF UNIPOTENT-FREE LINEAR GROUPS
Let 
$\Gamma $ be a finitely generated group of matrices over 
$\mathbb {C}$. We construct an isometric action of 
$\Gamma $ on a complete 
$\mathrm {CAT}(0)$ space 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.
