Generalized Baumslag–Solitar groups: rank and finite index subgroups

Levitt, Gilbert

doi:10.5802/aif.2943

Generalized Baumslag–Solitar groups: rank and finite index subgroups
[Groupes de Baumslag–Solitar généralisés : rang et sous-groupes d’indice fini]

Levitt, Gilbert ¹

¹ Laboratoire de Mathématiques Nicolas Oresme Université de Caen et CNRS (UMR 6139) BP 5186 F-14032 Caen Cedex, (France)

Annales de l'Institut Fourier, Tome 65 (2015) no. 2, pp. 725-762.

Résumé
Abstract

Un groupe de Baumslag–Solitar généralisé (groupe GBS) est un groupe de type fini agissant sur un arbre avec stabilisateurs de sommets et d’arêtes infinis cycliques. Nous déterminons explicitement le rang (nombre minimal de générateurs) d’un groupe GBS, et en déduisons le rang de la suspension d’un automorphisme d’ordre fini d’un groupe libre $F_{n}$ . Nous montrons aussi que le rang ne peut pas diminuer quand on passe à un sous-groupe d’indice fini d’un groupe GBS. Nous déterminons quels groupes GBS sont larges (un sous-groupe d’indice fini se surjecte sur $F_{2}$ ), et nous résolvons le problème de commensurabilité (décider si deux groupes ont des sous-groupes d’indice fini isomorphes) dans une certaine famille de groupes GBS.

A generalized Baumslag–Solitar (GBS) group is a finitely generated group acting on a tree with infinite cyclic edge and vertex stabilizers. We show how to determine effectively the rank (minimal cardinality of a generating set) of a GBS group; as a consequence, one can compute the rank of the mapping torus of a finite order outer automorphism of a free group $F_{n}$ . We also show that the rank of a finite index subgroup of a GBS group $G$ cannot be smaller than the rank of $G$ . We determine which GBS groups are large (some finite index subgroup maps onto $F_{2}$ ), and we solve the commensurability problem (deciding whether two groups have isomorphic finite index subgroups) in a particular family of GBS groups.

DOI : 10.5802/aif.2943

Classification : 20E06, 20E08, 20F05, 20F65
Keywords: Group, Baumslag–Solitar, rank, finite index
Mot clés : Groupe, Baumslag–Solitar, rang, indice fini

Affiliations des auteurs :

Levitt, Gilbert ¹

¹ Laboratoire de Mathématiques Nicolas Oresme Université de Caen et CNRS (UMR 6139) BP 5186 F-14032 Caen Cedex, (France)

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     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
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Levitt, Gilbert. Generalized Baumslag–Solitar groups: rank and finite index subgroups. Annales de l'Institut Fourier, Tome 65 (2015) no. 2, pp. 725-762. doi : 10.5802/aif.2943. http://archive.numdam.org/articles/10.5802/aif.2943/

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