Measuring cones and other thick subsets in free groups

TL;DR

This paper studies automata over free groups to analyze the structure and measure of regular subsets, providing explicit calculations and improved asymptotic classifications.

Contribution

It introduces special automata for free groups, enabling explicit computation of measures and decomposition of regular subsets, advancing understanding of their asymptotic behavior.

Findings

Decomposition of regular subsets into automata-accepted sets

Explicit formulas for generating functions and measures

Enhanced asymptotic classification of regular subsets

Abstract

In this paper we investigate the special automata over finite rank free groups and estimate asymptotic characteristics of sets they accept. We show how one can decompose an arbitrary regular subset of a finite rank free group into disjoint union of sets accepted by special automata or special monoids. These automata allow us to compute explicitly generating functions, $\lambda-$measures and Cesaro measure of thick monoids. Also we improve the asymptotic classification of regular subsets in free groups.