Chapter

Combinatorial and Geometric Group Theory

Part of the series Trends in Mathematics pp 93-118

Regular Sets and Counting in Free Groups

  • Elizaveta FrenkelAffiliated with
  • , Alexei G. MyasnikovAffiliated withDepartment of Mathematics and Statistics, McGill University
  • , Vladimir N. RemeslennikovAffiliated withOmsk Branch of Mathematical Institute SB RAS

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Abstract

In this paper we study asymptotic behavior of regular subsets in a free group F of finite rank, compare their sizes at infinity, and develop techniques to compute the probabilities of sets relative to distributions on F that come naturally from random walks on the Cayley graph of F. We apply these techniques to study cosets, double cosets, and Schreier representatives of finitely generated subgroups of F with an eye on complexity of algorithmic problems in free products with amalgamation and HNN extensions of groups. Mathematics Subject Classification (2000). 20E05.

Keywords

Geometric group theory regular set measures on free groups Schreier transversals generic and negligible sets