Generalized Baumslag–Solitar groups: rank and finite index subgroups
Annales de l'Institut Fourier, Volume 65 (2015) no. 2, p. 725-762

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.

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.

DOI : https://doi.org/10.5802/aif.2943
Classification:  20E06,  20E08,  20F05,  20F65
Keywords: Group, Baumslag–Solitar, rank, finite index
@article{AIF_2015__65_2_725_0,
     author = {Levitt, Gilbert},
     title = {Generalized Baumslag--Solitar groups: rank and finite index subgroups},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {65},
     number = {2},
     year = {2015},
     pages = {725-762},
     doi = {10.5802/aif.2943},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2015__65_2_725_0}
}
Levitt, Gilbert. Generalized Baumslag–Solitar groups: rank and finite index subgroups. Annales de l'Institut Fourier, Volume 65 (2015) no. 2, pp. 725-762. doi : 10.5802/aif.2943. http://www.numdam.org/item/AIF_2015__65_2_725_0/

[1] Bass, Hyman Covering theory for graphs of groups, J. Pure Appl. Algebra, Tome 89 (1993) no. 1-2, pp. 3-47 | Article | MR 1239551 | Zbl 0805.57001

[2] Baumslag, G.; Miller, C. F. Iii; Short, H. Unsolvable problems about small cancellation and word hyperbolic groups, Bull. London Math. Soc., Tome 26 (1994) no. 1, pp. 97-101 | Article | MR 1246477 | Zbl 0810.20025

[3] Baumslag, Gilbert; Solitar, Donald Some two-generator one-relator non-Hopfian groups, Bull. Amer. Math. Soc., Tome 68 (1962), pp. 199-201 | Article | MR 142635 | Zbl 0108.02702

[4] Beeker, B. Multiple conjugacy problem in graphs of free abelian groups (http://arxiv.org/abs/1106.3978, to appear in Groups, geometry, dynamics) | MR 3343344

[5] Button, J. O. A formula for the normal subgroup growth of Baumslag-Solitar groups, J. Group Theory, Tome 11 (2008) no. 6, pp. 879-884 | Article | MR 2466914 | Zbl 1153.20024

[6] Clay, Matt Deformation spaces of G-trees and automorphisms of Baumslag-Solitar groups, Groups Geom. Dyn., Tome 3 (2009) no. 1, pp. 39-69 | Article | MR 2466020 | Zbl 1226.20022

[7] Clay, Matt; Forester, Max On the isomorphism problem for generalized Baumslag-Solitar groups, Algebr. Geom. Topol., Tome 8 (2008) no. 4, pp. 2289-2322 | Article | MR 2465742 | Zbl 1191.20021

[8] Clay, Matt; Forester, Max Whitehead moves for G-trees, Bull. Lond. Math. Soc., Tome 41 (2009) no. 2, pp. 205-212 | Article | MR 2496498 | Zbl 1200.20020

[9] Collins, Donald J. Generation and presentation of one-relator groups with centre, Math. Z., Tome 157 (1977) no. 1, pp. 63-77 | Article | MR 466322 | Zbl 0348.20030

[10] Collins, Donald J.; Levin, Frank Automorphisms and Hopficity of certain Baumslag-Solitar groups, Arch. Math. (Basel), Tome 40 (1983) no. 5, pp. 385-400 | Article | MR 707725 | Zbl 0498.20021

[11] Cornulier, Yves; Valette, Alain On equivariant embeddings of generalized Baumslag–Solitar groups, Geom. Dedicata, Tome 175 (2015), pp. 385-401 | Article | MR 3323648

[12] Culler, Marc Finite groups of outer automorphisms of a free group, Contributions to group theory, Amer. Math. Soc., Providence, RI (Contemp. Math.) Tome 33 (1984), pp. 197-207 | Article | MR 767107 | Zbl 0552.20024

[13] Degrijse, D.; Petrosyan, N. Bredon cohomological dimensions for groups acting on CAT(0)-spaces (http://arxiv.org/abs/1208.3884)

[14] Dudkin, F. A. Subgroups of finite index in Baumslag-Solitar groups, Algebra Logika, Tome 49 (2010) no. 3, p. 331-345, 427, 429 | Article | MR 2766392 | Zbl 1215.20026

[15] Dunwoody, M. J. Folding sequences, The Epstein birthday schrift, Geom. Topol. Publ., Coventry (Geom. Topol. Monogr.) Tome 1 (1998), p. 139-158 (electronic) | Article | MR 1668347 | Zbl 0927.20013

[16] Edjvet, M.; Pride, Stephen J. The concept of “largeness” in group theory. II, Groups—Korea 1983 (Kyoungju, 1983), Springer, Berlin (Lecture Notes in Math.) Tome 1098 (1984), pp. 29-54 | Article | MR 781355 | Zbl 0566.20014

[17] Forester, Max Deformation and rigidity of simplicial group actions on trees, Geom. Topol., Tome 6 (2002), p. 219-267 (electronic) | Article | MR 1914569 | Zbl 1118.20028

[18] Forester, Max On uniqueness of JSJ decompositions of finitely generated groups, Comment. Math. Helv., Tome 78 (2003) no. 4, pp. 740-751 | Article | MR 2016693 | Zbl 1040.20032

[19] Forester, Max Splittings of generalized Baumslag-Solitar groups, Geom. Dedicata, Tome 121 (2006), pp. 43-59 | Article | MR 2276234 | Zbl 1117.20023

[20] Gelman, Efraim Subgroup growth of Baumslag-Solitar groups, J. Group Theory, Tome 8 (2005) no. 6, pp. 801-806 | Article | MR 2179671 | Zbl 1105.20018

[21] Guirardel, Vincent A very short proof of Forester’s rigidity result, Geom. Topol., Tome 7 (2003), p. 321-328 (electronic) | Article | MR 1988289 | Zbl 1032.20018

[22] Khramtsov, D. G. Finite groups of automorphisms of free groups, Mat. Zametki, Tome 38 (1985) no. 3, p. 386-392, 476 | MR 811572 | Zbl 0595.20036

[23] Kropholler, P. H. Baumslag-Solitar groups and some other groups of cohomological dimension two, Comment. Math. Helv., Tome 65 (1990) no. 4, pp. 547-558 | Article | MR 1078097 | Zbl 0744.20044

[24] Leighton, Frank Thomson Finite common coverings of graphs, J. Combin. Theory Ser. B, Tome 33 (1982) no. 3, pp. 231-238 | Article | MR 693362 | Zbl 0488.05033

[25] Levitt, Gilbert On the automorphism group of generalized Baumslag-Solitar groups, Geom. Topol., Tome 11 (2007), pp. 473-515 | Article | MR 2302496 | Zbl 1143.20014

[26] Levitt, Gilbert Quotients and subgroups of Baumslag-Solitar groups, J. Group Theory, Tome 18 (2015) no. 1, pp. 1-43 | Article | MR 3297728

[27] Mccool, James A class of one-relator groups with centre, Bull. Austral. Math. Soc., Tome 44 (1991) no. 2, pp. 245-252 | Article | MR 1126363 | Zbl 0726.20020

[28] Mecham, Taralee Largeness of graphs of abelian groups, ProQuest LLC, Ann Arbor, MI (2009), pp. 72 (Thesis (Ph.D.)–The University of Oklahoma) | MR 2713429

[29] Neumann, Walter D. On Leighton’s graph covering theorem, Groups Geom. Dyn., Tome 4 (2010) no. 4, pp. 863-872 | Article | MR 2727669 | Zbl 1210.05113

[30] Pietrowski, Alfred The isomorphism problem for one-relator groups with non-trivial centre, Math. Z., Tome 136 (1974), pp. 95-106 | Article | MR 349851 | Zbl 0264.20029

[31] Rips, E. Subgroups of small cancellation groups, Bull. London Math. Soc., Tome 14 (1982) no. 1, pp. 45-47 | Article | MR 642423 | Zbl 0481.20020

[32] Scott, Peter; Wall, Terry Topological methods in group theory, Homological group theory (Proc. Sympos., Durham, 1977), Cambridge Univ. Press, Cambridge-New York (London Math. Soc. Lecture Note Ser.) Tome 36 (1979), pp. 137-203 | MR 564422 | Zbl 0423.20023

[33] Whyte, K. The large scale geometry of the higher Baumslag-Solitar groups, Geom. Funct. Anal., Tome 11 (2001) no. 6, pp. 1327-1343 | Article | MR 1878322 | Zbl 1004.20024

[34] Zimmermann, Bruno Über Homöomorphismen n-dimensionaler Henkelkörper und endliche Erweiterungen von Schottky-Gruppen, Comment. Math. Helv., Tome 56 (1981) no. 3, pp. 474-486 | Article | MR 639363 | Zbl 0475.57015