Title: Properties of linear groups with restricted unipotent elements,
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.