Ideals of equations for elements in a free group and context-free languages

TL;DR

This paper investigates the structure and complexity of equations in free groups using formal language theory, providing polynomial-time algorithms for key decision problems and analyzing the growth of equations.

Contribution

It introduces new polynomial-time algorithms for determining the triviality and degree of equations in free groups, and characterizes their growth behavior.

Findings

Polynomial-time algorithm for triviality of the ideal of equations

Algorithm to decide if degree-d equations are empty

Sharp upper bound on the minimal degree of non-trivial equations

Abstract

Let $F$ be a finitely generated free group, and let $H\le F$ be a finitely generated subgroup. An equation for an element $g\in F$ with coefficients in $H$ is an element $w(x)\in H*\langle x \rangle$ such that $w(g)=1$ in $F$; the degree of the equation is the number of occurrences of $x$ and $x^{-1}$ in the cyclic reduction of $w(x)$. Given an element $g\in F$, we consider the ideal $\mathfrak{I}_g\subseteq H*\langle x \rangle$ of equations for $g$ with coefficients in $H$; we study the structure of $\mathfrak{I}_g$ using context-free languages. We describe a new algorithm that determines whether $\mathfrak{I}_g$ is trivial or not; the algorithm runs in polynomial time. We also describe a polynomial-time algorithm that, given $d\in\mathbb{N}$, decides whether or not the subset $\mathfrak{I}_{g,d}\subseteq\mathfrak{I}_g$ of all degree-$d$ equations is empty. We provide a polynomial-time algorithm that computes the minimum degree $d_{\min}$ of a non-trivial equation in $\mathfrak{I}_g$. We provide a sharp upper bound on $d_{\min}$. Finally, we study the growth of the number of (cyclically reduced) equations in $\mathfrak{I}_g$ and in $\mathfrak{I}_{g,d}$ as a function of their length. We prove that this growth is either polynomial or exponential, and we provide a polynomial-time algorithm that computes the type of growth (including the degree of the growth if it's polynomial).