Algebraic & Geometric Topology

Local cut points and splittings of relatively hyperbolic groups

Matthew Haulmark

Full-text: Access denied (no subscription detected)

However, an active subscription may be available with MSP at

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


We show that the existence of a nonparabolic local cut point in the Bowditch boundary (G,) of a relatively hyperbolic group (G,) implies that G splits over a 2–ended subgroup. This theorem generalizes a theorem of Bowditch from the setting of hyperbolic groups to relatively hyperbolic groups. As a consequence we are able to generalize a theorem of Kapovich and Kleiner by classifying the homeomorphism type of 1–dimensional Bowditch boundaries of relatively hyperbolic groups which satisfy certain properties, such as no splittings over 2–ended subgroups and no peripheral splittings.

In order to prove the boundary classification result we require a notion of ends of a group which is more general than the standard notion. We show that if a finitely generated discrete group acts properly and cocompactly on two generalized Peano continua X and Y, then Ends(X) is homeomorphic to Ends(Y). Thus we propose an alternative definition of Ends(G) which increases the class of spaces on which G can act.

Article information

Algebr. Geom. Topol., Volume 19, Number 6 (2019), 2795-2836.

Received: 18 August 2017
Revised: 3 October 2018
Accepted: 23 January 2019
First available in Project Euclid: 29 October 2019

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 20F65: Geometric group theory [See also 05C25, 20E08, 57Mxx] 20F67: Hyperbolic groups and nonpositively curved groups

relatively hyperbolic groups splittings local cut points JSJ splittings relatively hyperbolic groups ends of Spaces group Boundaries


Haulmark, Matthew. Local cut points and splittings of relatively hyperbolic groups. Algebr. Geom. Topol. 19 (2019), no. 6, 2795--2836. doi:10.2140/agt.2019.19.2795.

Export citation


  • R D Anderson, A characterization of the universal curve and a proof of its homogeneity, Ann. of Math. 67 (1958) 313–324
  • R D Anderson, One-dimensional continuous curves and a homogeneity theorem, Ann. of Math. 68 (1958) 1–16
  • H-J Baues, A Quintero, Infinite homotopy theory, $K$–Monographs in Mathematics 6, Kluwer, Dordrecht (2001)
  • B H Bowditch, Cut points and canonical splittings of hyperbolic groups, Acta Math. 180 (1998) 145–186
  • B H Bowditch, A topological characterisation of hyperbolic groups, J. Amer. Math. Soc. 11 (1998) 643–667
  • B H Bowditch, Boundaries of geometrically finite groups, Math. Z. 230 (1999) 509–527
  • B H Bowditch, Connectedness properties of limit sets, Trans. Amer. Math. Soc. 351 (1999) 3673–3686
  • B H Bowditch, Convergence groups and configuration spaces, from “Geometric group theory down under” (J Cossey, W D Neumann, M Shapiro, editors), de Gruyter, Berlin (1999) 23–54
  • B H Bowditch, Treelike structures arising from continua and convergence groups, Mem. Amer. Math. Soc. 662, Amer. Math. Soc., Providence, RI (1999)
  • B H Bowditch, Peripheral splittings of groups, Trans. Amer. Math. Soc. 353 (2001) 4057–4082
  • B H Bowditch, Relatively hyperbolic groups, Internat. J. Algebra Comput. 22 (2012) art. id. 1250016
  • M R Bridson, A Haefliger, Metric spaces of non-positive curvature, Grundl. Math. Wissen. 319, Springer (1999)
  • S Claytor, Topological immersion of Peanian continua in a spherical surface, Ann. of Math. 35 (1934) 809–835
  • F Dahmani, Combination of convergence groups, Geom. Topol. 7 (2003) 933–963
  • F Dahmani, V Guirardel, P Przytycki, Random groups do not split, Math. Ann. 349 (2011) 657–673
  • R J Daverman, Decompositions of manifolds, Pure and Applied Mathematics 124, Academic, Orlando, FL (1986)
  • H Freudenthal, Über die Enden topologischer Räume und Gruppen, Math. Z. 33 (1931) 692–713
  • R Geoghegan, Topological methods in group theory, Graduate Texts in Mathematics 243, Springer (2008)
  • V Gerasimov, Expansive convergence groups are relatively hyperbolic, Geom. Funct. Anal. 19 (2009) 137–169
  • E Ghys, P de la Harpe (editors), Sur les groupes hyperboliques d'après Mikhael Gromov, Progress in Mathematics 83, Birkhäuser, Boston, MA (1990)
  • B W Groff, Quasi-isometries, boundaries and JSJ-decompositions of relatively hyperbolic groups, J. Topol. Anal. 5 (2013) 451–475
  • M Gromov, Hyperbolic groups, from “Essays in group theory” (S M Gersten, editor), Math. Sci. Res. Inst. Publ. 8, Springer (1987) 75–263
  • D Groves, J F Manning, Dehn filling in relatively hyperbolic groups, Israel J. Math. 168 (2008) 317–429
  • D Groves, J F Manning, Dehn fillings and elementary splittings, Trans. Amer. Math. Soc. 370 (2018) 3017–3051
  • C R Guilbault, Ends, shapes, and boundaries in manifold topology and geometric group theory, from “Topology and geometric group theory” (M W Davis, J Fowler, J-F Lafont, I J Leary, editors), Springer Proc. Math. Stat. 184, Springer (2016) 45–125
  • C R Guilbault, M A Moran, Proper homotopy types and $\mathcal Z$–boundaries of spaces admitting geometric group actions, Expo. Math. 37 (2019) 292–313
  • D P Guralnik, Ends of cusp-uniform groups of locally connected continua, I, Internat. J. Algebra Comput. 15 (2005) 765–798
  • M Haulmark, Boundary classification and two-ended splittings of groups with isolated flats, J. Topol. 11 (2018) 645–665
  • G C Hruska, Relative hyperbolicity and relative quasiconvexity for countable groups, Algebr. Geom. Topol. 10 (2010) 1807–1856
  • G C Hruska, B Kleiner, Hadamard spaces with isolated flats, Geom. Topol. 9 (2005) 1501–1538
  • G C Hruska, K Ruane, Connectedness properties and splittings of groups with isolated flats, preprint (2017)
  • M Kapovich, B Kleiner, Hyperbolic groups with low-dimensional boundary, Ann. Sci. École Norm. Sup. 33 (2000) 647–669
  • M Mihalik, S Tschantz, Visual decompositions of Coxeter groups, Groups Geom. Dyn. 3 (2009) 173–198
  • R Myers, Excellent $1$–manifolds in compact $3$–manifolds, Topology Appl. 49 (1993) 115–127
  • D V Osin, Elementary subgroups of relatively hyperbolic groups and bounded generation, Internat. J. Algebra Comput. 16 (2006) 99–118
  • P Papasoglu, E Swenson, From continua to $\mathbb R$–trees, Algebr. Geom. Topol. 6 (2006) 1759–1784
  • P Papasoglu, E Swenson, Boundaries and JSJ decompositions of $\mathrm{CAT}(0)$–groups, Geom. Funct. Anal. 19 (2009) 559–590
  • P Scott, T Wall, Topological methods in group theory, from “Homological group theory” (C T C Wall, editor), London Math. Soc. Lecture Note Ser. 36, Cambridge Univ. Press (1979) 137–203
  • J-P Serre, Trees, Springer (2003)
  • E L Swenson, A cutpoint tree for a continuum, from “Computational and geometric aspects of modern algebra” (M Atkinson, N Gilbert, J Howie, S Linton, E Robertson, editors), London Math. Soc. Lecture Note Ser. 275, Cambridge Univ. Press (2000) 254–265
  • H C Tran, Relations between various boundaries of relatively hyperbolic groups, Internat. J. Algebra Comput. 23 (2013) 1551–1572
  • P Tukia, Homeomorphic conjugates of Fuchsian groups, J. Reine Angew. Math. 391 (1988) 1–54
  • P Tukia, Convergence groups and Gromov's metric hyperbolic spaces, New Zealand J. Math. 23 (1994) 157–187
  • P Tukia, Conical limit points and uniform convergence groups, J. Reine Angew. Math. 501 (1998) 71–98
  • G T Whyburn, Topological characterization of the Sierpiński curve, Fund. Math. 45 (1958) 320–324
  • S Willard, General topology, Addison-Wesley, Reading, MA (1970)
  • A Yaman, A topological characterisation of relatively hyperbolic groups, J. Reine Angew. Math. 566 (2004) 41–89
  • W-y Yang, Peripheral structures of relatively hyperbolic groups, J. Reine Angew. Math. 689 (2014) 101–135