A probabilistic Tits alternative and probabilistic identities

We introduce the notion of a probabilistic identity of a residually finite group Γ. By this we mean a nontrivial word w such that the probabilities that w D 1 in the finite quotients of Γ 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 Γ be a finitely generated linear group over any field and let G be its profinite completion. Then either Γ is virtually solvable, or, for any n ≥ 1, n random elements g₁, ⋯, 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.