On finite-index extensions of subgroups of free groups

TL;DR

This paper explores the structure of finite-index extensions of subgroups in free groups, providing algorithms for computing the commensurator and malnormal closure, along with bounds and characterizations.

Contribution

It introduces a combinatorial characterization of the greatest finite-index extension and develops efficient algorithms for key subgroup computations.

Findings

Finite lattice of finite-index extensions is characterized.

A fast algorithm for computing the commensurator is presented.

Bounds on the number of finite-index extensions are established.

Abstract

We study the lattice of finite-index extensions of a given finitely generated subgroup $H$ of a free group $F$. This lattice is finite and we give a combinatorial characterization of its greatest element, which is the commensurator of $H$. This characterization leads to a fast algorithm to compute the commensurator, which is based on a standard algorithm from automata theory. We also give a sub-exponential and super-polynomial upper bound for the number of finite-index extensions of $H$, and we give a language-theoretic characterization of the lattice of finite-index subgroups of $H$. Finally, we give a polynomial time algorithm to compute the malnormal closure of $H$.