On finite-index extensions of subgroups of free groups

Pedro Silva (1) , Pascal Weil (2)

1 CMUP - Centro de Matemática, Faculdade de Ciências
2 LaBRI - Laboratoire Bordelais de Recherche en Informatique

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 $H$.

Mots clés: free groups; subgroups; finite-index extensions

Domaines: Théorie des groupes [math.GR]

Pedro Silva, Pascal Weil. On finite-index extensions of subgroups of free groups. Journal of Group Theory, 2010, 13 (3), pp.365-381. ⟨hal-00311493v2⟩