Regular sets and counting in free groups

TL;DR

This paper investigates the asymptotic properties of regular subsets in free groups, introduces techniques for probability computation based on random walks, and applies these to subgroup structures and coset analysis.

Contribution

It develops new methods for analyzing the size and probability of regular sets in free groups, including applications to subgroup cosets and Schreier representatives.

Findings

Regular subsets exhibit specific asymptotic behaviors at infinity.

Techniques for probability computation using no-return random walks are introduced.

Applications include size analysis of cosets, double cosets, and Schreier representatives.

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 no-return 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 and also to analyze relative sizes of regular prefixed-closed subsets in F.