Quasimorphisms, random walks, and transient subsets in countable groups

We study interrelations between the theory of quasimorphisms and theory of random walks on groups, and establish the following criterion of transience for subsets of groups: if a subset of a countable group has bounded images under any three linearly independent homogeneous quasimorphisms on the group, then 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 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 non-elementary hyperbolic group have large stable commutator length, etc.