A language-theoretic approach to study the density of subsets in free groups

TL;DR

This paper applies language theory to analyze subset densities in free groups, proving new properties, characterizing positive density subsets, and extending classical results with a novel approach.

Contribution

It introduces a language-theoretic framework for studying subset densities in free groups, including an analogue of the Infinite Monkey Theorem and characterizations of rational subsets.

Findings

Proves an analogue of the Infinite Monkey Theorem for freely reduced words.

Characterizes rational subsets with positive density in free groups.

Shows that the automorphic orbit of a nonabelian free group element has density zero.

Abstract

In this paper, we study the density of subsets of nonabelian free groups using relative densities of languages. We start by proving some basic properties about the density of a language $L_1$ relative to another language $L_2$ containing $L_1$. We then focus on the case where $L_2$ is the language of freely reduced words over an alphabet and prove an analogue of the Infinite Monkey Theorem for this language. This result, obtained as a corollary of a broader theorem on irreducible subshifts of finite type, allows for a language-theoretic characterization of rational subsets with positive density. As a consequence, we obtain a language-theoretic proof that the automorphic orbit of an element of a nonabelian free group has natural density zero, which generalizes a result by Burillo and Ventura concerning the density of primitive elements of free groups. We then describe rational subsets of free groups of positive density and explore in depth the case of finitely generated subgroups. We prove that if the rational subset is a subgroup, then it has positive density if and only if it has finite index and characterize those for which there is convergence. In cases where convergence fails due to parity constraints, we show that the density of the subset always exhibits weak convergence and that the average of its supremum and infimum densities converges to the expected value.