Finitely generated subgroups of free groups as formal languages and their cogrowth

TL;DR

This paper explores the language of reduced words representing finitely generated subgroups of free groups, constructing automata to analyze their properties, cogrowth series, and entropy, with applications to computational group theory.

Contribution

It introduces a method to construct minimal automata for these subgroup languages and characterizes when these languages are irreducible, enabling efficient computation of cogrowth and entropy.

Findings

Automata construction for subgroup languages

Characterization of irreducible subgroup languages

Efficient computation of cogrowth series and entropy

Abstract

For finitely generated subgroups $H$ of a free group $F_m$ of finite rank $m$, we study the language $L_H$ of reduced words that represent $H$ which is a regular language. Using the (extended) core of Schreier graph of $H$, we construct the minimal deterministic finite automaton that recognizes $L_H$. Then we characterize the f.g. subgroups $H$ for which $L_H$ is irreducible and for such groups explicitly construct ergodic automaton that recognizes $L_H$. This construction gives us an efficient way to compute the cogrowth series $L_H(z)$ of $H$ and entropy of $L_H$. Several examples illustrate the method and a comparison is made with the method of calculation of $L_H(z)$ based on the use of Nielsen system of generators of $H$.