Characteristic random subgroups of geometric groups and free abelian groups of infinite rank

Lewis Bowen, Rostislav Grigorchuk, Rostyslav Kravchenko

Published 2014-02-15, updated 2015-05-30Version 3

We show that if $G$ is a non-elementary word hyperbolic group, mapping class group of a hyperbolic surface or the outer automorphism group of a nonabelian free group then $G$ has $2^{\aleph_0}$ many continuous ergodic invariant random subgroups. If $G$ is a nonabelian free group then $G$ has $2^{\aleph_0}$ many continuous $G$-ergodic characteristic random subgroups. We also provide a complete classification of characteristic random subgroups of free abelian groups of countably infinite rank and elementary $p$-groups of countably infinite rank.