Recurrence Relations for Cosets in Free Groups

TL;DR

This paper develops a recurrence relation framework for counting elements in cosets of subgroups within free groups, providing an algorithm to compute these relations and analyzing their properties for finite index subgroups.

Contribution

It introduces a method to derive recurrence relations for coset element counts in free groups and offers an algorithm to compute the associated constants.

Findings

Existence of recurrence relations for coset element counts.

Algorithm for calculating recurrence relation constants.

Finite nonzero constants when subgroup has finite index and contains an odd-length element.

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.