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.

**[Ab]**H.Abels,*An example of a finitely presented solvable group*, in ``Homological group theory'', London Math. Society Lecture Notes Series, No. 36 (ed. C.T.C.Wall), Cambridge University Press (1979) 205-211. MR**82b:20047****[AnM]**J.W.Anderson, B.Maskit,*On the local connectivity of limit sets of kleinian groups*, Complex Variables**31**(1996) 177-183. MR**98a:30055****[BeF1]**M.Bestvina, M.Feighn,*Bounding the complexity of simplicial group actions on trees*, Invent. Math.**103**(1991) 449-469. MR**92c:20044****[BeF2]**M.Bestvina, M.Feighn,*A combination theorem for negatively curved groups*, J. Differential Geom.**35**(1992) 85-101. MR**93d:53053****[BeF3]**M.Bestvina, M.Feighn,*Stable actions of groups on real trees*, Invent. Math.**121**(1995) 287-321. MR**96h:20056****[BeM]**M.Bestvina, G.Mess,*The boundary of negatively curved groups*, J. Amer. Math. Soc.**4**(1991) 469-481. MR**93j:20076****[Bi]**R.Bieri,*Homological dimension of discrete groups*, Queen Mary College Mathematics Notes (1976). MR**57:6224****[Bo1]**B.H.Bowditch,*Discrete parabolic groups*, J. Differential Geom.**38**(1993) 559-583. MR**94h:53046****[Bo2]**B.H.Bowditch,*Geometrical finiteness with variable negative curvature*, Duke Math. J.**77**(1995) 229-274. MR**96b:53056****[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) 449-457. MR**87d:20037****[DS]**M.J.Dunwoody, M.E.Sageev,*JSJ splittings for finitely presented groups over slender subgroups*, Invent. Math.**135**(1999) 25-44.**[F]**E.M.Freden,*Negatively curved groups have the convergence property I*, Ann. Acad. Sci. Fenn. Ser. A Math.**20**(1995) 333-348. MR**96g:20054****[GeM]**F.W.Gehring, G.J.Martin,*Discrete quasiconformal groups I*, Proc. London Math. Soc.**55**(1987) 331-358. MR**88m:30057****[GhH]**E.Ghys, P.de la Harpe,*Sur les groupes hyperboliques d'après Mikhael Gromov*, Progress in Maths. 83, Birkhäuser (1990). MR**94m:53060****[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****[L]**G.Levitt,*Non-nesting actions on real trees*, Bull. London Math. Soc.**30**(1998) 46-54. MR**99a:20027****[M]**M.Mihalik,*Semistability at of finitely generated groups, and*, Topology and its Appl.*solvable groups***24**(1986) 259-269. MR**88c:57005****[Se]**Z.Sela,*Structure and rigidity in (Gromov) hyperbolic groups and discrete*, Geom. Funct. Anal.*groups in rank 1 Lie groups II***7**(1997) 561-593. MR**98j:20044****[Sw]**G.A.Swarup,*On the cut point conjecture*, Electron. Res. Announc. Amer. Math. Soc.**2**(1996) 98-100 (Electronic). MR**97f:20048****[T]**P.Tukia,*Convergence groups and Gromov's hyperbolic metric spaces*, New Zealand J. Math.**23**(1994) 157-187. MR**96c:30042**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (1991):
20F32

Retrieve articles in all journals with MSC (1991): 20F32

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:
https://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