A probabilistic Tits alternative and probabilistic identities

作者：Larsen Michael^*; Shalev Aner

来源：Algebra and Number Theory, 2016, 10(6): 1359-1371.

DOI：10.2140/ant.2016.10.1359

• Tits alternative
• residually finite
• virtually solvable
• probabilistic identity
• profinite completion

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 >= 1, n random elements g(1.) ..., 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.

• 出版日期2016