Regular Sets and Counting in Free Groups

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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.