A probabilistic Tits alternative and probabilistic identities

TL;DR

This paper introduces probabilistic identities in residually finite groups and proves a probabilistic version of the Tits alternative, showing that finitely generated linear groups are either virtually solvable or generate free subgroups with probability one.

Contribution

It establishes a new concept of probabilistic identities and proves a probabilistic Tits alternative for finitely generated linear groups.

Findings

Finitely generated linear groups satisfy a probabilistic identity iff they are virtually solvable.

Proves a probabilistic Tits alternative for the profinite completion of such groups.

With probability one, independent elements generate free subgroups of any rank.

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.