On finite-index extensions of subgroups of free groups

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.