Structure and rigidity in hyperbolic groups. I. (English) Zbl 0818.20042

Rips, E.; Sela, Z.

Geom. Funct. Anal. 4, No. 3, 337-371 (1994).

This paper is on hyperbolic groups in the sense of Gromov. The authors introduce the notion of a Dehn twist automorphism of a group given as an amalgamated product or HNN extension. They call an automorphism obtained by a sequence of Dehn twists and inner automorphisms an internal automorphism. They prove that for a torsion-free hyperbolic group, the group of internal automorphisms is of finite index. They give a proof of a theorem of Gromov which asserts that if \(\Gamma\) is a hyperbolic group and \(\Gamma_ 1\) a finitely presented, torsion-free, freely indecomposable non-cyclic subgroup, then \(\Gamma\) contains at most finitely many conjugacy classes of subgroups isomorphic to \(\Gamma_ 1\). They obtain also the following results: If \(\Gamma\) is a hyperbolic group and \(\Gamma_ 1\) a finitely presented torsion-free, freely indecomposable subgroup which has a finitely generated normalizer in \(\Gamma\), then \(\operatorname{Aut} (\Gamma_ 1)\) is finitely generated. The automorphism group of a torsion-free hyperbolic group is finitely generated.
The authors introduce “weakly rigid groups”, a notion which they say will be important in the second author’s approach to the study of the automorphism group of a free group. The paper has a short appendix which summarizes results (mainly due to the first author and to Bestvina and Feighn) on the theory of group actions on trees which is used in the paper.

Reviewer: A.Papadopoulos (Strasbourg)

MSC:

20F28	Automorphism groups of groups
57M07	Topological methods in group theory
20F65	Geometric group theory
20E06	Free products of groups, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations
20E08	Groups acting on trees
57M60	Group actions on manifolds and cell complexes in low dimensions
57M05	Fundamental group, presentations, free differential calculus
20F05	Generators, relations, and presentations of groups
20E07	Subgroup theorems; subgroup growth

Keywords:

finitely presented torsion-free subgroups; weakly rigid groups; Dehn twist automorphisms; amalgamated products; HNN extensions; Dehn twists; inner automorphisms; torsion-free hyperbolic groups; internal automorphisms; freely indecomposable subgroups; automorphism groups; automorphism groups of free groups; group actions on trees

References:

[1]	[B]M. Bestvina Degeneration of the hyperbolic space, Duke Math. Journal 56 (1988), 143–161. · Zbl 0652.57009 · doi:10.1215/S0012-7094-88-05607-4
[2]	[BF]M. Bestvina, M. Feighn, Stable actions of groups on real trees, preprint. · Zbl 0837.20047
[3]	[CuMo]M. Culler, J. Morgan, Group actions onR-trees, Proc. London Math. Society 55 (1987), 571–604. · Zbl 0658.20021 · doi:10.1112/plms/s3-55.3.571
[4]	[F-R1]D.I. Fouxe-Rabinovich, Uber die Automorphismengruppen der freien Produkte I, Math. Sbornik 8 (1940), 265–276. · JFM 66.0066.04
[5]	[F-R2]D.I. Fouxe-Rabinovich, Uber die Automorphismengruppen der freien Produkte II, Math. Sbornik 9 (1941), 183–220. · Zbl 0025.00904
[6]	[GSh]H. Gillet, P. Shalen, Dendrology of groups in lowQ-ranks, Journal of Differential Geometry 32 (1990), 605–712. · Zbl 0732.20011
[7]	[Gr1]M. Gromov, Hyperbolic groups, Essays in Group Theory (S.M. Gersten ed.), MSRI Publications no. 8 (1987), 75–263.
[8]	[Gr2]M. Gromov, Volume and bounded cohomology, Publ. Math. IHES 56 (1983), 213–307.
[9]	[GrS]M. Gromov, R. Shoen, Harmonic maps into singular spaces andp-adic superrigidity for lattices in groups of rank one, Publ. Math. IHES 76 (1992), 165–246.
[10]	[HT]A. Hatcher, W. Thurston, A presentation for the mapping class group of a closed orientable surface, Topology 19 (1980), 221–237. · Zbl 0447.57005 · doi:10.1016/0040-9383(80)90009-9
[11]	[K]M. Keane, Interval exchange transformations, Math. Zeit. 141 (1975), 25–31. · doi:10.1007/BF01236981
[12]	[L]G. Levitt, Pantalons et feuilletages des surfaces, Topology 21 (1982), 9–33. · Zbl 0473.57014 · doi:10.1016/0040-9383(82)90039-8
[13]	[Li]W.B.R. Lickorish, A finite set of generators for the homeotopy group of a 2-manifold, Proc. Cambridge Phil. Soc. 60 (1964), 769–778. · Zbl 0131.20801 · doi:10.1017/S030500410003824X
[14]	[M]J. McCool, Some finitely presented subgroups of the automorphism group of a free group, J. of Algebra 35 (1975), 205–213. · Zbl 0325.20025 · doi:10.1016/0021-8693(75)90045-9
[15]	[Mo1]J. Morgan, Actions of groups on real trees, notes on lectures by E. Rips.
[16]	[Mo2]J. Morgan, Ergodic theory and free actions of groups onR-trees, Inventiones Math. 94 (1988), 605–622. · Zbl 0676.57001 · doi:10.1007/BF01394277
[17]	[MoSh1]J.W. Morgan, P.B. Shalen, Degeneration of hyperbolic structures III, Annals of Math. 127 (1988), 457–519. · Zbl 0661.57004 · doi:10.2307/2007003
[18]	[MoSh2]J.W. Morgan, P.B. Shalen, Free actions of surface groups onR-trees, Topology 30 (1991), 143–154. · Zbl 0726.57001 · doi:10.1016/0040-9383(91)90002-L
[19]	[P]F. Paulin, Outer automorphisms of hyperbolic groups and small actions onR-trees, Arboreal Group Theory (R.C. Alperin, ed.), MSRI publications no. 19 (1991), 331–341. · Zbl 0804.57002
[20]	[R]E. Rips, Group actions onR-trees, in preparation.
[21]	[Se1]Z. Sela, On the isomorphism problem for hyperbolic groups and the classification of 3-manifolds, IHES preprint.
[22]	[Se2]Z. Sela, On the isomorphism problem for hyperbolic groups II, in preparation.
[23]	[Se3]Z. Sela, Structure and rigidity in hyperbolic groups and discrete groups in rank 1 Lie groups II, preprint.
[24]	[Se4]Z. Sela, Acylindrical accessibility for groups, MPI preprint.
[25]	[Ser]J-P. Serre, Trees, Springer-Verlag, 1980.
[26]	[W]B. Wajnryb, A simple presentation for the mapping class group of an orientable surface, Israel Journal of Math. 45 (1983), 157–174. · Zbl 0533.57002 · doi:10.1007/BF02774014
[27]	[ZVC]H. Zieschang, E. Vogt, H. Coldeway, Surfaces and Planar Discontinuous Groups, Lecture Notes in Math. 835, Springer-Verlag, 1980.