A probabilistic Tits alternative and probabilistic identities

Abstract

We introduce the notion of a probabilistic identity of a residually finite group $\Gamma$. By this we mean a nontrivial word $w$ such that the probabilities that $w=1$ in the finite quotients of $\Gamma$ are bounded away from zero.

We prove that a finitely generated linear group satisfies a probabilistic identity if and only if it is virtually solvable.

A main application of this result is a probabilistic variant of the Tits alternative: Let $\Gamma$ be a finitely generated linear group over any field and let $G$ be its profinite completion. Then either $\Gamma$ is virtually solvable, or, for any $n\ge 1$, $n$ random elements $g_1,\dots ,g_n$ of $G$ freely generate a free (abstract) subgroup of $G$ with probability $1$.

We also prove other related results and discuss open problems and applications.