Recurrence Relations for Cosets in Free Groups

TL;DR

Let $F_2$ be the free group on two generators and $H\le F_2$. The paper proves that for any coset $yH$, the number of length-$n$ elements in $yH$, $|yH\cap S_n|$, satisfies a linear recurrence $|yH\cap S_n|=\sum_{i=1}^{n-1}\sum_{xH\in F_2/H} a_{i,xH}\cdot |xH\cap S_{n-i}|$ with computable coefficients $a_{i,xH}$. It introduces the Directed Coset Graph $F_2/H$ and an explicit edge-highlighting algorithm to determine the coefficients, and it proves the recurrence exists for all $H\le F_2$; furthermore, if $[F_2:H]<\infty$ and $H$ contains an element of odd length, the recurrence stabilizes after finitely many steps. The termination proof combines combinatorial constructions via vital walks in the coset graph with an ergodic-limit argument. The work includes explicit examples, discusses the sharpness of the odd-length condition, and raises open questions about termination bounds and potential non-termination for certain finite-index subgroups.

Abstract

Let $F_2$ be the free group on two generators and let $H$ be a subgroup of $F_2$. We investigate a method for calculating the number of elements in a coset of $H$ that have a given length when written in reduced form. More specifically, taking $S_n\subseteq F_2$ to be the set of elements of length $n$, we show that for any coset $yH$ there always exists a recurrence relation of the form \[ |yH\cap S_n| = \sum_{i=1}^{n-1}\sum_{xH\in F_2/H}a_{i,xH}\cdot |xH\cap S_{n-i}| \] for some constants $(a_{i,xH})_{i\in \mathbb{N}, xH\in F_2/H}$, and we give an algorithm that calculates these constants. Further, we show that when $H$ has finite index and contains an element of odd length, only finitely many of the constants $a_{i,xH}$ are nonzero.

Theorems & Definitions (24)

• Theorem 1.1
• Theorem 1.2
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Lemma 3.3
• proof
• proof
• Lemma 4.1