Connectedness properties of limit sets
Author:
B. H. Bowditch
Journal:
Trans. Amer. Math. Soc. 351 (1999), 3673-3686
MSC (1991):
Primary 20F32
DOI:
https://doi.org/10.1090/S0002-9947-99-02388-0
Published electronically:
April 20, 1999
MathSciNet review:
1624089
Full-text PDF Free Access
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.
- 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
- 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, DOI https://doi.org/10.1080/17476939608814957
- Mladen Bestvina and Mark Feighn, Bounding the complexity of simplicial group actions on trees, Invent. Math. 103 (1991), no. 3, 449–469. MR 1091614, DOI https://doi.org/10.1007/BF01239522
- M. Bestvina and M. Feighn, A combination theorem for negatively curved groups, J. Differential Geom. 35 (1992), no. 1, 85–101. MR 1152226
- Mladen Bestvina and Mark Feighn, Stable actions of groups on real trees, Invent. Math. 121 (1995), no. 2, 287–321. MR 1346208, DOI https://doi.org/10.1007/BF01884300
- Mladen Bestvina and Geoffrey Mess, The boundary of negatively curved groups, J. Amer. Math. Soc. 4 (1991), no. 3, 469–481. MR 1096169, DOI https://doi.org/10.1090/S0894-0347-1991-1096169-1
- Robert Bieri, Homological dimension of discrete groups, Mathematics Department, Queen Mary College, London, 1976. Queen Mary College Mathematics Notes. MR 0466344
- B. H. Bowditch, Discrete parabolic groups, J. Differential Geom. 38 (1993), no. 3, 559–583. MR 1243787
- B. H. Bowditch, Geometrical finiteness with variable negative curvature, Duke Math. J. 77 (1995), no. 1, 229–274. MR 1317633, DOI https://doi.org/10.1215/S0012-7094-95-07709-6
- B.H.Bowditch, Treelike structures arising from continua and convergence groups, Memoirs Amer. Math. Soc.
- B.H.Bowditch, Cut points and canonical splittings of hyperbolic groups, Acta Math. 180 (1998), no. 2, 145–186.
- B.H.Bowditch, Group actions on trees and dendrons, Topology 37 (1998), no. 6, 1275–1298.
- 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.
- 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.
- B.H.Bowditch, Relatively hyperbolic groups, preprint, Southampton (1997).
- B.H.Bowditch, Peripheral splittings of groups, preprint, Southampton (1997).
- B.H.Bowditch, Boundaries of geometrically finite groups, to appear in Math. Z.
- B.H.Bowditch, G.A.Swarup, Cut points in the boundaries of hyperbolic groups, in preparation.
- M. J. Dunwoody, The accessibility of finitely presented groups, Invent. Math. 81 (1985), no. 3, 449–457. MR 807066, DOI https://doi.org/10.1007/BF01388581
- M.J.Dunwoody, M.E.Sageev, JSJ splittings for finitely presented groups over slender subgroups, Invent. Math. 135 (1999) 25–44.
- 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
- F. W. Gehring and G. J. Martin, Discrete quasiconformal groups. I, Proc. London Math. Soc. (3) 55 (1987), no. 2, 331–358. MR 896224, DOI https://doi.org/10.1093/plms/s3-55_2.331
- 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
- M. Gromov, Hyperbolic groups, Essays in group theory, Math. Sci. Res. Inst. Publ., vol. 8, Springer, New York, 1987, pp. 75–263. MR 919829, DOI https://doi.org/10.1007/978-1-4613-9586-7_3
- Gilbert Levitt, Non-nesting actions on real trees, Bull. London Math. Soc. 30 (1998), no. 1, 46–54. MR 1479035, DOI https://doi.org/10.1112/S0024609397003561
- Michael Mihalik, Semistability at $\infty $ 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, DOI https://doi.org/10.1016/0166-8641%2886%2990069-6
- 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, DOI https://doi.org/10.1007/s000390050019
- G. A. Swarup, On the cut point conjecture, Electron. Res. Announc. Amer. Math. Soc. 2 (1996), no. 2, 98–100. MR 1412948, DOI https://doi.org/10.1090/S1079-6762-96-00013-3
- 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
Received by editor(s):
August 22, 1997
Published electronically:
April 20, 1999
Article copyright:
© Copyright 1999
American Mathematical Society