A probabilistic Tits alternative and probabilistic identities

Michael Larsen, Aner Shalev

Research output: Contribution to journalArticlepeer-review

6 Scopus citations

Abstract

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, ⋯, gn 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.

Original languageAmerican English
Pages (from-to)1359-1371
Number of pages13
JournalAlgebra and Number Theory
Volume10
Issue number6
DOIs
StatePublished - 2016

Bibliographical note

Publisher Copyright:
© 2016 Mathematical Sciences Publishers.

Keywords

  • Probabilistic identity
  • Profinite completion
  • Residually finite
  • Tits alternative
  • Virtually solvable

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