Aspects of non positive curvature for linear groups with no infinite order unipotents

Abstract

We show that a linear group without unipotent elements of infinite order possesses properties akin to those held by groups of non positive curvature. Moreover in positive characteristic any finitely generated linear group acts properly and semisimply on a CAT(0) space. We present applications, including that the mapping class group of a surface having genus at least 3 has no faithful linear representation which is complex unitary or over any field of positive characteristic.