Properties of linear groups with restricted unipotent elements

TL;DR

This paper studies linear groups without infinite order unipotent elements, revealing they share properties with non-positively curved groups, and applies these findings to show certain surface mapping class groups lack specific linear representations.

Contribution

It establishes new structural properties of linear groups restricted from containing infinite order unipotent elements, with implications for their subgroup structure and representation theory.

Findings

Finitely generated such groups have virtually splitting centralisers and undistorted abelian subgroups.

Virtually torsion-free groups in this class have subgroups either containing free groups or being virtually abelian.

Surface mapping class groups of genus ≥ 3 do not admit faithful complex unitary or positive characteristic linear representations.

Abstract

We consider linear groups which do not contain unipotent elements of infinite order, which includes all linear groups in positive characteristic, and show that this class of groups has good properties which resemble those held by groups of non positive curvature and which do not hold for arbitrary characteristic zero linear groups. In particular if such a linear group is finitely generated then centralisers virtually split and all finitely generated abelian subgroups are undistorted. If further the group is virtually torsion free (which always holds in characteristic zero) then we have a strong property on small subgroups: any subgroup either contains a non abelian free group or is finitely generated and virtually abelian, hence also undistorted. 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.