A probabilistic Tits alternative and probabilistic identities

Abstract

We introduce the notion of a probabilistic identity of a residually finite group. We prove that a finitely generated linear group satisfies a probabilistic identity if and only if it is virtually solvable. As an application, we prove a probabilistic variant of the Tits alternative: let G be the profinite completion of a finitely generated linear group. Then either G is virtually solvable, or for any positive integer n, with probability one, n independent, uniformly distributed elements of G freely generate a free subgroup of G of rank n.