Geometry & Topology

On the automorphism group of generalized Baumslag–Solitar groups

Gilbert Levitt

Full-text: Open access

Abstract

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(G) either contains non-abelian free groups or is virtually nilpotent of class 2. It has torsion only at finitely many primes.

One may decide algorithmically whether Out(G) 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(G) virtually nilpotent.

If G is unimodular (virtually Fn×), then Out(G) is commensurable with a semi-direct product k Out(H) with H virtually free.

Article information

Source
Geom. Topol., Volume 11, Number 1 (2007), 473-515.

Dates
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.gt/1513799838

Digital Object Identifier
doi:10.2140/gt.2007.11.473

Mathematical Reviews number (MathSciNet)
MR2302496

Zentralblatt MATH identifier
1143.20014

Subjects
Primary: 20F65: Geometric group theory [See also 05C25, 20E08, 57Mxx]
Secondary: 20E08: Groups acting on trees [See also 20F65] 20F28: Automorphism groups of groups [See also 20E36]

Keywords
Baumslag–Solitar automorphisms graphs of groups

Citation

Levitt, Gilbert. On the automorphism group of generalized Baumslag–Solitar groups. Geom. Topol. 11 (2007), no. 1, 473--515. doi:10.2140/gt.2007.11.473. https://projecteuclid.org/euclid.gt/1513799838


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References

  • H Bass, R Kulkarni, Uniform tree lattices, J. Amer. Math. Soc. 3 (1990) 843–902
  • G Baumslag, D Solitar, Some two-generator one-relator non-Hopfian groups, Bull. Amer. Math. Soc. 68 (1962) 199–201
  • M Clay, Deformation spaces of $G$–trees and automorphisms of Baumslag–Solitar groups
  • M Clay, Contractibility of deformation spaces of $G$–trees, Algebr. Geom. Topol. 5 (2005) 1481–1503
  • M Clay, A Fixed Point Theorem for Deformation Spaces of $G$–trees, Comment. Math. Helv. (to appear)
  • D J Collins, The automorphism towers of some one-relator groups, Proc. London Math. Soc. $(3)$ 36 (1978) 480–493
  • D J Collins, F Levin, Automorphisms and Hopficity of certain Baumslag-Solitar groups, Arch. Math. $($Basel$)$ 40 (1983) 385–400
  • M Culler, K Vogtmann, Moduli of graphs and automorphisms of free groups, Invent. Math. 84 (1986) 91–119
  • 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) 229–240, 250
  • A Fel'shtyn, D L Goncalves, Twisted conjugacy classes of automorphisms of Baumslag–Solitar groups
  • M Forester, Deformation and rigidity of simplicial group actions on trees, Geom. Topol. 6 (2002) 219–267
  • M Forester, On uniqueness of JSJ decompositions of finitely generated groups, Comment. Math. Helv. 78 (2003) 740–751
  • M Forester, Splittings of generalized Baumslag–Solitar groups, Geom. Dedicata 121 (2006) 43–59
  • N D Gilbert, J Howie, V Metaftsis, E Raptis, Tree actions of automorphism groups, J. Group Theory 3 (2000) 213–223
  • V Guirardel, A very short proof of Forester's rigidity result, Geom. Topol. 7 (2003) 321–328
  • V Guirardel, G Levitt, Deformation spaces of trees, Geometry, Groups, Dynamics (to appear)
  • V Guirardel, G Levitt, The outer space of a free product, Proc. London Math. Soc. (to appear)
  • P H Kropholler, Baumslag–Solitar groups and some other groups of cohomological dimension two, Comment. Math. Helv. 65 (1990) 547–558
  • P H Kropholler, A note on centrality in $3$–manifold groups, Math. Proc. Cambridge Philos. Soc. 107 (1990) 261–266
  • S Krstić, K Vogtmann, Equivariant outer space and automorphisms of free-by-finite groups, Comment. Math. Helv. 68 (1993) 216–262
  • G Levitt in preparation
  • G Levitt, Automorphisms of hyperbolic groups and graphs of groups, Geom. Dedicata 114 (2005) 49–70
  • G Levitt, Characterizing rigid simplicial actions on trees, from: “Geometric methods in group theory”, Contemp. Math. 372, Amer. Math. Soc., Providence, RI (2005) 27–33
  • G Levitt, M Lustig, Most automorphisms of a hyperbolic group have very simple dynamics, Ann. Sci. École Norm. Sup. $(4)$ 33 (2000) 507–517
  • J McCool, A class of one-relator groups with centre, Bull. Austral. Math. Soc. 44 (1991) 245–252
  • D McCullough, A Miller, Symmetric automorphisms of free products, Mem. Amer. Math. Soc. 122 (1996) viii+97
  • D I Moldavanskiĭ, On the isomorphisms of Baumslag-Solitar groups, Ukrain. Mat. Zh. 43 (1991) 1684–1686
  • L Mosher, M Sageev, K Whyte, Quasi-actions on trees. I. Bounded valence, Ann. of Math. $(2)$ 158 (2003) 115–164
  • M R Pettet, Virtually free groups with finitely many outer automorphisms, Trans. Amer. Math. Soc. 349 (1997) 4565–4587
  • M R Pettet, The automorphism group of a graph product of groups, Comm. Algebra 27 (1999) 4691–4708
  • A Pietrowski, The isomorphism problem for one-relator groups with non-trivial centre, Math. Z. 136 (1974) 95–106
  • E Raptis, D Varsos, On the automorphism group of the fundamental group of a graph of polycyclic groups, Algebra Colloq. 4 (1997) 241–248
  • A Rhemtulla, D Rolfsen, Local indicability in ordered groups: braids and elementary amenable groups, Proc. Amer. Math. Soc. 130 (2002) 2569–2577
  • K Whyte, The large scale geometry of the higher Baumslag–Solitar groups, Geom. Funct. Anal. 11 (2001) 1327–1343