  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 $H$.

Group Theory