Connectedness properties of limit sets

Author:
B. H. Bowditch

Journal:
Trans. Amer. Math. Soc. **351** (1999), 3673-3686

MSC (1991):
Primary 20F32

Published electronically:
April 20, 1999

MathSciNet review:
1624089

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Abstract | References | Similar Articles | Additional Information

Abstract: We study convergence group actions on continua, and give a criterion which ensures that every global cut point is a parabolic fixed point. We apply this result to the case of boundaries of relatively hyperbolic groups, and consider implications for connectedness properties of such spaces.

**[Ab]**Herbert Abels,*An example of a finitely presented solvable group*, Homological group theory (Proc. Sympos., Durham, 1977) London Math. Soc. Lecture Note Ser., vol. 36, Cambridge Univ. Press, Cambridge-New York, 1979, pp. 205–211. MR**564423****[AnM]**James W. Anderson and Bernard Maskit,*On the local connectivity of limit set of Kleinian groups*, Complex Variables Theory Appl.**31**(1996), no. 2, 177–183. MR**1423249****[BeF1]**Mladen Bestvina and Mark Feighn,*Bounding the complexity of simplicial group actions on trees*, Invent. Math.**103**(1991), no. 3, 449–469. MR**1091614**, 10.1007/BF01239522**[BeF2]**M. Bestvina and M. Feighn,*A combination theorem for negatively curved groups*, J. Differential Geom.**35**(1992), no. 1, 85–101. MR**1152226****[BeF3]**Mladen Bestvina and Mark Feighn,*Stable actions of groups on real trees*, Invent. Math.**121**(1995), no. 2, 287–321. MR**1346208**, 10.1007/BF01884300**[BeM]**Mladen Bestvina and Geoffrey Mess,*The boundary of negatively curved groups*, J. Amer. Math. Soc.**4**(1991), no. 3, 469–481. MR**1096169**, 10.1090/S0894-0347-1991-1096169-1**[Bi]**Robert Bieri,*Homological dimension of discrete groups*, Mathematics Department, Queen Mary College, London, 1976. Queen Mary College Mathematics Notes. MR**0466344****[Bo1]**B. H. Bowditch,*Discrete parabolic groups*, J. Differential Geom.**38**(1993), no. 3, 559–583. MR**1243787****[Bo2]**B. H. Bowditch,*Geometrical finiteness with variable negative curvature*, Duke Math. J.**77**(1995), no. 1, 229–274. MR**1317633**, 10.1215/S0012-7094-95-07709-6**[Bo3]**B.H.Bowditch,*Treelike structures arising from continua and convergence groups*, Memoirs Amer. Math. Soc. CMP**98:05****[Bo4]**B.H.Bowditch,*Cut points and canonical splittings of hyperbolic groups*, Acta Math.**180**(1998), no. 2, 145-186. CMP**98:17****[Bo5]**B.H.Bowditch,*Group actions on trees and dendrons*, Topology**37**(1998), no. 6, 1275-1298. CMP**98:15****[Bo6]**B.H.Bowditch,*Boundaries of strongly accessible hyperbolic groups*, in ``The Epstein Birthday Schrift'', Geometry and Topology Monographs, Vol. 1 (ed. I.Rivin, C.Rourke, C.Series) International Press (1998) 59-97.**[Bo7]**B.H.Bowditch,*Convergence groups and configuration spaces*, to appear in ``Group Theory Down Under'' (ed. J.Cossey, C.F.Miller, W.D.Neumann, M.Shapiro), de Gruyter.**[Bo8]**B.H.Bowditch,*Relatively hyperbolic groups*, preprint, Southampton (1997).**[Bo9]**B.H.Bowditch,*Peripheral splittings of groups*, preprint, Southampton (1997).**[Bo10]**B.H.Bowditch,*Boundaries of geometrically finite groups*, to appear in Math. Z.**[BoS]**B.H.Bowditch, G.A.Swarup,*Cut points in the boundaries of hyperbolic groups*, in preparation.**[D]**M. J. Dunwoody,*The accessibility of finitely presented groups*, Invent. Math.**81**(1985), no. 3, 449–457. MR**807066**, 10.1007/BF01388581**[DS]**M.J.Dunwoody, M.E.Sageev,*JSJ splittings for finitely presented groups over slender subgroups*, Invent. Math.**135**(1999) 25-44.**[F]**Eric M. Freden,*Negatively curved groups have the convergence property. I*, Ann. Acad. Sci. Fenn. Ser. A I Math.**20**(1995), no. 2, 333–348. MR**1346817****[GeM]**F. W. Gehring and G. J. Martin,*Discrete quasiconformal groups. I*, Proc. London Math. Soc. (3)**55**(1987), no. 2, 331–358. MR**896224**, 10.1093/plms/s3-55_2.331**[GhH]**È. Gis and P. de lya Arp (eds.),*Giperbolicheskie gruppy po Mikhailu Gromovu*, “Mir”, Moscow, 1992 (Russian). Papers from the Swiss Seminar on Hyperbolic Groups held in Bern, 1988; Translation of Sur les groupes hyperboliques d’après Mikhael Gromov (Bern, 1988), Birkhäuser Boston, Boston, 1990 [ MR1086648 (92f:53050)]. MR**1266631****[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**, 10.1007/978-1-4613-9586-7_3**[L]**Gilbert Levitt,*Non-nesting actions on real trees*, Bull. London Math. Soc.**30**(1998), no. 1, 46–54. MR**1479035**, 10.1112/S0024609397003561**[M]**Michael Mihalik,*Semistability at ∞ of finitely generated groups, and solvable groups*, Topology Appl.**24**(1986), no. 1-3, 259–269. Special volume in honor of R. H. Bing (1914–1986). MR**872498**, 10.1016/0166-8641(86)90069-6**[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**, 10.1007/s000390050019**[Sw]**G. A. Swarup,*On the cut point conjecture*, Electron. Res. Announc. Amer. Math. Soc.**2**(1996), no. 2, 98–100 (electronic). MR**1412948**, 10.1090/S1079-6762-96-00013-3**[T]**Pekka Tukia,*Convergence groups and Gromov’s metric hyperbolic spaces*, New Zealand J. Math.**23**(1994), no. 2, 157–187. MR**1313451**

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Additional Information

**B. H. Bowditch**

Affiliation:
Faculty of Mathematical Studies, University of Southampton, Highfield, Southampton SO17 1BJ, Great Britain

Email:
bhb@maths.soton.ac.uk

DOI:
http://dx.doi.org/10.1090/S0002-9947-99-02388-0

Received by editor(s):
August 22, 1997

Published electronically:
April 20, 1999

Article copyright:
© Copyright 1999
American Mathematical Society