Free by cyclic groups and linear groups with restricted unipotent elements

Abstract We introduce the class of linear groups that do not contain unipotent elements of infinite order, which includes all linear groups in positive characteristic. We show that groups in this class have good closure properties, in addition to having properties akin to non-positive curvature, which were proved in [6]. We give examples of abstract groups lying in this class, but also show that Gersten’s free by cyclic group does not. This implies that it has no faithful linear representation of any dimension over any field of positive characteristic, nor can it be embedded in any complex unitary group.