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Splittings of finitely generated groups over two-ended subgroups
Author(s):
Brian
H.
Bowditch
Journal:
Trans. Amer. Math. Soc.
354
(2002),
1049-1078.
MSC (2000):
Primary 20F65, 20E08
Posted:
October 26, 2001
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Abstract:
We describe a means of constructing splittings of a one-ended finitely generated group over two-ended subgroups, starting with a finite collection of codimension-one two-ended subgroups. In the case where all the two-ended subgroups have two-ended commensurators, we obtain an annulus theorem, and a form of the JSJ splitting of Rips and Sela. The construction uses ideas from the work of Dunwoody, Sageev and Swenson. We use a particular kind of order structure which combines cyclic orders and treelike structures. In the special case of hyperbolic groups, this provides a link between combinarorial constructions, and constructions arising from the topological structure of the boundary. In this context, we recover the annulus theorem of Scott and Swarup. We also show that a one-ended finitely generated group which contains an infinite-order element, and such that every infinite cyclic subgroup is (virtually) codimension-one is a virtual surface group.
References:
- [AN]
- S.A.Adeleke, P.M.Neumann, Relations related to betweenness: their structure and automorphisms Memoirs Amer. Math. Soc. Volume 131, no. 623 (1998). MR 98h:20008
- [BeF]
- M.Bestvina, M.Feighn, Bounding the complexity of simplicial actions on trees, Invent. Math. 103 (1991) 449-469. MR 92c:20044
- [Bo1]
- B.H.Bowditch, Cut points and canonical splitting of hyperbolic groups, Acta Math. 180 (1998) 145-186. MR 99g:20069
- [Bo2]
- B.H.Bowditch, Convergence groups and configuration spaces, in ``Group Theory Down Under'', (ed. J.Cossey, C.F.Miller III, W.D.Neumann, M.Shapiro), de Gruyter (1999) 23-54. MR 2001d:20035
- [Bo3]
- B.H.Bowditch, Treelike structures arising from continua and convergence groups, Memoirs Amer. Math. Soc. Volume 139, no. 662 (1999). MR 2000c:20061
- [Bo4]
- B.H.Bowditch, Planar groups and the Seifert conjecture, preprint, Southampton (1999).
- [CJ]
- A.Casson, D.Jungreis, Convergence groups and Seifert fibered 3-manifolds, Invent. Math. 118 (1994) 441-456. MR 96f:57011
- [DSa]
- M.J.Dunwoody, M.E.Sageev, JSJ-splittings for finitely presented groups over slender subgroups, Invent. Math. 135 (1999) 25-44. MR 2000b:20050
- [DSw]
- M.J.Dunwoody, E.L.Swenson, The algebraic torus theorem, Invent. Math. 140 (2000) 605-637. MR 2001d:20039
- [FP]
- K.Fujiwara, P.Papasoglu, JSJ decompositions of finitely presented groups and complexes of groups, preprint (1997).
- [Ga]
- D.Gabai, Convergence groups are fuchsian groups, Ann. of Math. 136 (1992) 447-510. MR 93m:20065
- [Geo]
- R.Geoghegan, Topological methods in group theory, manuscript, Binghamton (2000).
- [GerM]
- F.W.Gehring, G.J.Martin, Discrete quasiconformal groups I, Proc. London Math. Soc. 55 (1987) 331-358. MR 88m:30057
- [Gr]
- M.Gromov, Hyperbolic groups, in ``Essays in Group Theory'' (ed. S.M.Gersten) M.S.R.I. Publications No. 8, Springer-Verlag (1987) 75-263. MR 89e:20070
- [K]
- P.H.Kropholler, A group theoretic proof of the torus theorem, in ``Geometric Group Theory, Volume 1'', London Math. Soc. Lecture Note Series No. 181, (ed. G.A.Niblo, M.A.Roller), Cambridge University Press (1993) 138-158. MR 94i:57029
- [KR]
- P.H.Kropholler, M.A.Roller, Relative ends and duality groups, J. Pure Appl. Algebra 61 (1989) 197-210. MR 91b:20069
- [M]
- G.Mess, The Seifert conjecture and groups that are coarse quasiisometric to planes, preprint, UCLA (1988).
- [P]
- P.Papasoglu, Quasi-isometry invariance of group splittings, preprint, Orsay (2000).
- [RS]
- E.Rips, Z.Sela, Cyclic splittings of finitely presented groups and the canonical JSJ decomposition, Ann. of Math. 146 (1997) 53-109. MR 98m:20044
- [ScS1]
- P.Scott, G.A.Swarup, An algebraic annulus theorem, Pacific J. Math. 196 (2000) 461-506. CMP 2001:05
- [ScS2]
- P.Scott, G.A.Swarup, The number of ends of a pair of groups, preprint, Ann Arbor/Melbourne (2000).
- [Se]
- Z.Sela, Structure and rigidity in (Gromov) hyperbolic groups and discrete, groups in rank 1 Lie groups II Geom. Funct. Anal. 7 (1997) 561-593. MR 98j:20044
- [Sw]
- E.L.Swenson, Axial pairs and convergence groups on
 , Topology 39 (2000) 229-237. MR 2001c:20097 - [T1]
- P.Tukia, Homeomorphic conjugates of fuchsian groups, J. Reine Angew. Math. 391 (1988) 1-54. MR 89m:30047
- [T2]
- P.Tukia, Convergence groups and Gromov's metric hyperbolic spaces, New Zealand J. Math. 23 (1994) 157-187. MR 96c:30042
- [W]
- L.E.Ward, Axioms for cutpoints, in ``General topology and modern analysis'', Proceedings, University of California, Riverside (ed. L.F.McAuley, M.M.Rao), Academic Press (1980) 327-336. MR 82g:54053
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Additional Information:
Brian
H.
Bowditch
Affiliation:
Faculty of Mathematical Studies, University of Southampton, Highfield, Southampton SO17 1BJ, Great Britain
DOI:
10.1090/S0002-9947-01-02907-5
PII:
S 0002-9947(01)02907-5
Received by editor(s):
January 31, 2001
Received by editor(s) in revised form:
July 01, 2001
Posted:
October 26, 2001
Copyright of article:
Copyright
2001,
American Mathematical Society
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