On the automorphism group of generalized Baumslag–Solitar groups

Levitt, Gilbert¹

¹ Laboratoire de Mathématiques Nicolas Oresme, UMR 6139, BP 5186, Université de Caen, 14032 Caen Cedex, France

Geometry & topology, Tome 11 (2007) no. 1, pp. 473-515.

Résumé

A generalized Baumslag–Solitar group (GBS group) is a finitely generated group $G$ which acts on a tree with all edge and vertex stabilizers infinite cyclic. We show that $Out\left(G\right)$ either contains non-abelian free groups or is virtually nilpotent of class $\le$2. It has torsion only at finitely many primes.

One may decide algorithmically whether $Out\left(G\right)$ is virtually nilpotent or not. If it is, one may decide whether it is virtually abelian, or finitely generated. The isomorphism problem is solvable among GBS groups with Out$\left(G\right)$ virtually nilpotent.

If $G$ is unimodular (virtually $F_n\times \mathbb{Z}$), then $Out\left(G\right)$ is commensurable with a semi-direct product ${\mathbb{Z}}^k⋊Out\left(H\right)$ with $H$ virtually free.

Détail

DOI : 10.2140/gt.2007.11.473

Keywords: Baumslag–Solitar, automorphisms, graphs of groups

Bibliographie

[1] H Bass, R Kulkarni, Uniform tree lattices, J. Amer. Math. Soc. 3 (1990) 843

[2] G Baumslag, D Solitar, Some two-generator one-relator non-Hopfian groups, Bull. Amer. Math. Soc. 68 (1962) 199

[3] M Clay, Deformation spaces of $G$–trees and automorphisms of Baumslag–Solitar groups,

[4] M Clay, Contractibility of deformation spaces of $G$–trees, Algebr. Geom. Topol. 5 (2005) 1481

[5] M Clay, A Fixed Point Theorem for Deformation Spaces of $G$–trees, Comment. Math. Helv. (to appear)

[6] D J Collins, The automorphism towers of some one-relator groups, Proc. London Math. Soc. $(3)$ 36 (1978) 480

[7] D J Collins, F Levin, Automorphisms and Hopficity of certain Baumslag-Solitar groups, Arch. Math. $($Basel$)$ 40 (1983) 385

[8] M Culler, K Vogtmann, Moduli of graphs and automorphisms of free groups, Invent. Math. 84 (1986) 91

[9] A L Fel’Shtyn, The Reidemeister number of any automorphism of a Gromov hyperbolic group is infinite, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 279 (2001) 229, 250

[10] A Fel’Shtyn, D L Goncalves, Twisted conjugacy classes of automorphisms of Baumslag–Solitar groups,

[11] M Forester, Deformation and rigidity of simplicial group actions on trees, Geom. Topol. 6 (2002) 219

[12] M Forester, On uniqueness of JSJ decompositions of finitely generated groups, Comment. Math. Helv. 78 (2003) 740

[13] M Forester, Splittings of generalized Baumslag–Solitar groups, Geom. Dedicata 121 (2006) 43

[14] N D Gilbert, J Howie, V Metaftsis, E Raptis, Tree actions of automorphism groups, J. Group Theory 3 (2000) 213

[15] V Guirardel, A very short proof of Forester's rigidity result, Geom. Topol. 7 (2003) 321

[16] V Guirardel, G Levitt, Deformation spaces of trees, Geometry, Groups, Dynamics (to appear)

[17] V Guirardel, G Levitt, The outer space of a free product, Proc. London Math. Soc. (to appear)

[18] P H Kropholler, Baumslag–Solitar groups and some other groups of cohomological dimension two, Comment. Math. Helv. 65 (1990) 547

[19] P H Kropholler, A note on centrality in $3$–manifold groups, Math. Proc. Cambridge Philos. Soc. 107 (1990) 261

[20] S Krstić, K Vogtmann, Equivariant outer space and automorphisms of free-by-finite groups, Comment. Math. Helv. 68 (1993) 216

[21] G Levitt

[22] G Levitt, Automorphisms of hyperbolic groups and graphs of groups, Geom. Dedicata 114 (2005) 49

[23] G Levitt, Characterizing rigid simplicial actions on trees, from: "Geometric methods in group theory", Contemp. Math. 372, Amer. Math. Soc. (2005) 27

[24] G Levitt, M Lustig, Most automorphisms of a hyperbolic group have very simple dynamics, Ann. Sci. École Norm. Sup. $(4)$ 33 (2000) 507

[25] J Mccool, A class of one-relator groups with centre, Bull. Austral. Math. Soc. 44 (1991) 245

[26] D Mccullough, A Miller, Symmetric automorphisms of free products, Mem. Amer. Math. Soc. 122 (1996)

[27] D I Moldavanskiĭ, On the isomorphisms of Baumslag-Solitar groups, Ukrain. Mat. Zh. 43 (1991) 1684

[28] L Mosher, M Sageev, K Whyte, Quasi-actions on trees. I. Bounded valence, Ann. of Math. $(2)$ 158 (2003) 115

[29] M R Pettet, Virtually free groups with finitely many outer automorphisms, Trans. Amer. Math. Soc. 349 (1997) 4565

[30] M R Pettet, The automorphism group of a graph product of groups, Comm. Algebra 27 (1999) 4691

[31] A Pietrowski, The isomorphism problem for one-relator groups with non-trivial centre, Math. Z. 136 (1974) 95

[32] E Raptis, D Varsos, On the automorphism group of the fundamental group of a graph of polycyclic groups, Algebra Colloq. 4 (1997) 241

[33] A Rhemtulla, D Rolfsen, Local indicability in ordered groups: braids and elementary amenable groups, Proc. Amer. Math. Soc. 130 (2002) 2569

[34] K Whyte, The large scale geometry of the higher Baumslag–Solitar groups, Geom. Funct. Anal. 11 (2001) 1327