Splittings of finitely generated groups over twoended subgroups
Author:
Brian H. Bowditch
Journal:
Trans. Amer. Math. Soc. 354 (2002), 10491078
MSC (2000):
Primary 20F65, 20E08
Published electronically:
October 26, 2001
MathSciNet review:
1867372
Fulltext PDF Free Access
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Abstract: We describe a means of constructing splittings of a oneended finitely generated group over twoended subgroups, starting with a finite collection of codimensionone twoended subgroups. In the case where all the twoended subgroups have twoended 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 oneended finitely generated group which contains an infiniteorder element, and such that every infinite cyclic subgroup is (virtually) codimensionone is a virtual surface group.
 [AN]
S.
A. Adeleke and Peter
M. Neumann, Relations related to betweenness: their structure and
automorphisms, Mem. Amer. Math. Soc. 131 (1998),
no. 623, viii+125. MR 1388893
(98h:20008), http://dx.doi.org/10.1090/memo/0623
 [BeF]
Mladen
Bestvina and Mark
Feighn, Bounding the complexity of simplicial group actions on
trees, Invent. Math. 103 (1991), no. 3,
449–469. MR 1091614
(92c:20044), http://dx.doi.org/10.1007/BF01239522
 [Bo1]
Brian
H. Bowditch, Cut points and canonical splittings of hyperbolic
groups, Acta Math. 180 (1998), no. 2,
145–186. MR 1638764
(99g:20069), http://dx.doi.org/10.1007/BF02392898
 [Bo2]
B.
H. Bowditch, Convergence groups and configuration spaces,
Geometric group theory down under (Canberra, 1996) de Gruyter, Berlin,
1999, pp. 23–54. MR 1714838
(2001d:20035)
 [Bo3]
B.
H. Bowditch, Treelike structures arising from continua and
convergence groups, Mem. Amer. Math. Soc. 139 (1999),
no. 662, viii+86. MR 1483830
(2000c:20061), http://dx.doi.org/10.1090/memo/0662
 [Bo4]
B.H.Bowditch, Planar groups and the Seifert conjecture, preprint, Southampton (1999).
 [CJ]
Andrew
Casson and Douglas
Jungreis, Convergence groups and Seifert fibered 3manifolds,
Invent. Math. 118 (1994), no. 3, 441–456. MR 1296353
(96f:57011), http://dx.doi.org/10.1007/BF01231540
 [DSa]
M.
J. Dunwoody and M.
E. Sageev, JSJsplittings for finitely presented groups over
slender groups, Invent. Math. 135 (1999), no. 1,
25–44. MR
1664694 (2000b:20050), http://dx.doi.org/10.1007/s002220050278
 [DSw]
M.
J. Dunwoody and E.
L. Swenson, The algebraic torus theorem, Invent. Math.
140 (2000), no. 3, 605–637. MR 1760752
(2001d:20039), http://dx.doi.org/10.1007/s002220000063
 [FP]
K.Fujiwara, P.Papasoglu, JSJ decompositions of finitely presented groups and complexes of groups, preprint (1997).
 [Ga]
David
Gabai, Convergence groups are Fuchsian groups, Ann. of Math.
(2) 136 (1992), no. 3, 447–510. MR 1189862
(93m:20065), http://dx.doi.org/10.2307/2946597
 [Geo]
R.Geoghegan, Topological methods in group theory, manuscript, Binghamton (2000).
 [GerM]
F.
W. Gehring and G.
J. Martin, Discrete quasiconformal groups. I, Proc. London
Math. Soc. (3) 55 (1987), no. 2, 331–358. MR 896224
(88m:30057), http://dx.doi.org/10.1093/plms/s355_2.331
 [Gr]
M.
Gromov, Hyperbolic groups, Essays in group theory, Math. Sci.
Res. Inst. Publ., vol. 8, Springer, New York, 1987,
pp. 75–263. MR 919829
(89e:20070), http://dx.doi.org/10.1007/9781461395867_3
 [K]
Peter
H. Kropholler, A grouptheoretic proof of the torus theorem,
Geometric group theory, Vol.\ 1 (Sussex, 1991) London Math. Soc. Lecture
Note Ser., vol. 181, Cambridge Univ. Press, Cambridge, 1993,
pp. 138–158. MR 1238522
(94i:57029), http://dx.doi.org/10.1017/CBO9780511661860.013
 [KR]
P.
H. Kropholler and M.
A. Roller, Relative ends and duality groups, J. Pure Appl.
Algebra 61 (1989), no. 2, 197–210. MR 1025923
(91b:20069), http://dx.doi.org/10.1016/00224049(89)900145
 [M]
G.Mess, The Seifert conjecture and groups that are coarse quasiisometric to planes, preprint, UCLA (1988).
 [P]
P.Papasoglu, Quasiisometry invariance of group splittings, preprint, Orsay (2000).
 [RS]
E.
Rips and Z.
Sela, Cyclic splittings of finitely presented groups and the
canonical JSJ decomposition, Ann. of Math. (2) 146
(1997), no. 1, 53–109. MR 1469317
(98m:20044), http://dx.doi.org/10.2307/2951832
 [ScS1]
P.Scott, G.A.Swarup, An algebraic annulus theorem, Pacific J. Math. 196 (2000) 461506. 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), no. 3, 561–593. MR 1466338
(98j:20044), http://dx.doi.org/10.1007/s000390050019
 [Sw]
E.
L. Swenson, Axial pairs and convergence groups on
𝑆¹, Topology 39 (2000), no. 2,
229–237. MR 1722040
(2001c:20097), http://dx.doi.org/10.1016/S00409383(98)000688
 [T1]
Pekka
Tukia, Homeomorphic conjugates of Fuchsian groups, J. Reine
Angew. Math. 391 (1988), 1–54. MR 961162
(89m:30047), http://dx.doi.org/10.1515/crll.1988.391.1
 [T2]
Pekka
Tukia, Convergence groups and Gromov’s metric hyperbolic
spaces, New Zealand J. Math. 23 (1994), no. 2,
157–187. MR 1313451
(96c:30042)
 [W]
L.
E. Ward Jr., Axioms for cutpoints, General topology and modern
analysis (Proc. Conf., Univ. California, Riverside, Calif., 1980)
Academic Press, New YorkLondon, 1981, pp. 327–336. MR 619058
(82g:54053)
 [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) 449469. MR 92c:20044
 [Bo1]
 B.H.Bowditch, Cut points and canonical splitting of hyperbolic groups, Acta Math. 180 (1998) 145186. 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) 2354. 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 3manifolds, Invent. Math. 118 (1994) 441456. MR 96f:57011
 [DSa]
 M.J.Dunwoody, M.E.Sageev, JSJsplittings for finitely presented groups over slender subgroups, Invent. Math. 135 (1999) 2544. MR 2000b:20050
 [DSw]
 M.J.Dunwoody, E.L.Swenson, The algebraic torus theorem, Invent. Math. 140 (2000) 605637. 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) 447510. 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) 331358. MR 88m:30057
 [Gr]
 M.Gromov, Hyperbolic groups, in ``Essays in Group Theory'' (ed. S.M.Gersten) M.S.R.I. Publications No. 8, SpringerVerlag (1987) 75263. 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) 138158. MR 94i:57029
 [KR]
 P.H.Kropholler, M.A.Roller, Relative ends and duality groups, J. Pure Appl. Algebra 61 (1989) 197210. MR 91b:20069
 [M]
 G.Mess, The Seifert conjecture and groups that are coarse quasiisometric to planes, preprint, UCLA (1988).
 [P]
 P.Papasoglu, Quasiisometry 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) 53109. MR 98m:20044
 [ScS1]
 P.Scott, G.A.Swarup, An algebraic annulus theorem, Pacific J. Math. 196 (2000) 461506. 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) 561593. MR 98j:20044
 [Sw]
 E.L.Swenson, Axial pairs and convergence groups on , Topology 39 (2000) 229237. MR 2001c:20097
 [T1]
 P.Tukia, Homeomorphic conjugates of fuchsian groups, J. Reine Angew. Math. 391 (1988) 154. MR 89m:30047
 [T2]
 P.Tukia, Convergence groups and Gromov's metric hyperbolic spaces, New Zealand J. Math. 23 (1994) 157187. 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) 327336. 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:
http://dx.doi.org/10.1090/S0002994701029075
PII:
S 00029947(01)029075
Received by editor(s):
January 31, 2001
Received by editor(s) in revised form:
July 1, 2001
Published electronically:
October 26, 2001
Article copyright:
© Copyright 2001
American Mathematical Society
