Quasimorphisms, random walks, and transient subsets in countable groups

Abstract

We study the interrelations between the theory of quasimorphisms and the theory of random walks on groups, and establish the following transience criterion for subsets of groups: if a subset of a countable group has bounded images under any three linearly independent homogeneous quasimorphisms on the group, the this subset is transient for all nondegenerate random walks on the group. From this it follows, by results of M. Bestvina, K. Fujiwara, J. Birman, W. Menasco, and others, that, in a certain sense, generic elements in the mapping class groups of surfaces are pseudo-Anosov, generic braids in Artin’s braid groups represent prime links and knots, generic elements in the commutant of every nonelementary hyperbolic group have large stable commutator length, etc. Bibliography: 20 titles.