Invariant random subgroups of the finitary special linear group over a finite field

Abstract

An invariant random subgroup of a discrete group is a conjugation-invariant probability measure on the space of its subgroups. We confirm a conjecture of Simon Thomas and classify the invariant random subgroups of the finitary special and general linear groups over a finite field. These are shown to be stabilizers of random tuples of vectors and functionals. The classification uses the character theory of the finitary special linear group established by Skudlarek and the theory of the invariant random subgroups of the finitary permutation group established by Vershik. In addition, we classify the invariant probability measures for several natural actions of these finitary groups.