Geometry & Topology

Non-positively curved complexes of groups and boundaries

Alexandre Martin

Full-text: Open access


Given a complex of groups over a finite simplicial complex in the sense of Haefliger, we give conditions under which it is possible to build an EZ–structure in the sense of Farrell and Lafont for its fundamental group out of such structures for its local groups. As an application, we prove a combination theorem that yields a procedure for getting hyperbolic groups as fundamental groups of simple complexes of hyperbolic groups. The construction provides a description of the Gromov boundary of such groups.

Article information

Geom. Topol., Volume 18, Number 1 (2014), 31-102.

Received: 22 February 2012
Revised: 20 March 2013
Accepted: 16 July 2013
First available in Project Euclid: 20 December 2017

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 20F69: Asymptotic properties of groups

complexes of groups boundaries of groups hyperbolic groups


Martin, Alexandre. Non-positively curved complexes of groups and boundaries. Geom. Topol. 18 (2014), no. 1, 31--102. doi:10.2140/gt.2014.18.31.

Export citation


  • J Behrstock, C Druţu, L Mosher, Thick metric spaces, relative hyperbolicity, and quasi-isometric rigidity, Math. Ann. 344 (2009) 543–595
  • M Bestvina, Local homology properties of boundaries of groups, Michigan Math. J. 43 (1996) 123–139
  • M Bestvina, M Feighn, A combination theorem for negatively curved groups, J. Differential Geom. 35 (1992) 85–101
  • M Bestvina, G Mess, The boundary of negatively curved groups, J. Amer. Math. Soc. 4 (1991) 469–481
  • B H Bowditch, A topological characterisation of hyperbolic groups, J. Amer. Math. Soc. 11 (1998) 643–667
  • B H Bowditch, Convergence groups and configuration spaces, from: “Geometric group theory down under”, (J Cossey, C F Miller III, W D Neumann, M Shapiro, editors), de Gruyter, Berlin (1999) 23–54
  • B H Bowditch, Tight geodesics in the curve complex, Invent. Math. 171 (2008) 281–300
  • M R Bridson, Geodesics and curvature in metric simplicial complexes, from: “Group theory from a geometrical viewpoint”, (E Ghys, A Haefliger, A Verjovsky, editors), World Sci. Publ. (1991) 373–463
  • M R Bridson, A Haefliger, Metric spaces of non-positive curvature, Grundl. Math. Wissen. 319, Springer, Berlin (1999)
  • P-E Caprace, A Lytchak, At infinity of finite-dimensional CAT(0) spaces, Math. Ann. 346 (2010) 1–21
  • G Carlsson, E K Pedersen, Controlled algebra and the Novikov conjectures for $K$– and $L$–theory, Topology 34 (1995) 731–758
  • M Coornaert, T Delzant, A Papadopoulos, Géométrie et théorie des groupes: Les groupes hyperboliques de Gromov, Lecture Notes in Mathematics 1441, Springer, Berlin (1990)
  • J M Corson, Complexes of groups, Proc. London Math. Soc. 65 (1992) 199–224
  • F Dahmani, Classifying spaces and boundaries for relatively hyperbolic groups, Proc. London Math. Soc. 86 (2003) 666–684
  • F Dahmani, Combination of convergence groups, Geom. Topol. 7 (2003) 933–963
  • T Delzant, Sur l'accessibilité acylindrique des groupes de présentation finie, Ann. Inst. Fourier $($Grenoble$)$ 49 (1999) 1215–1224
  • C H Dowker, Topology of metric complexes, Amer. J. Math. 74 (1952) 555–577
  • A N Dranishnikov, On Bestvina–Mess formula, from: “Topological and asymptotic aspects of group theory”, (R Grigorchuk, M Mihalik, M Sapir, Z Šuni\'k, editors), Contemp. Math. 394, Amer. Math. Soc. (2006) 77–85
  • F T Farrell, J-F Lafont, $E\!\mathcal{Z}$–structures and topological applications, Comment. Math. Helv. 80 (2005) 103–121
  • E M Freden, Negatively curved groups have the convergence property, I, Ann. Acad. Sci. Fenn. Ser. A I Math. 20 (1995) 333–348
  • R Gitik, M Mitra, E Rips, M Sageev, Widths of subgroups, Trans. Amer. Math. Soc. 350 (1998) 321–329
  • M Gromov, Hyperbolic groups, from: “Essays in group theory”, (S M Gersten, editor), Math. Sci. Res. Inst. Publ. 8, Springer, New York (1987) 75–263
  • M Gromov, Asymptotic invariants of infinite groups, from: “Geometric group theory, Vol. 2”, (G A Niblo, M A Roller, editors), London Math. Soc. Lecture Note Ser. 182, Cambridge Univ. Press (1993) 1–295
  • A Haefliger, Extension of complexes of groups, Ann. Inst. Fourier $($Grenoble$)$ 42 (1992) 275–311
  • I Kapovich, The combination theorem and quasiconvexity, Internat. J. Algebra Comput. 11 (2001) 185–216
  • H A Masur, Y N Minsky, Geometry of the complex of curves, I: Hyperbolicity, Invent. Math. 138 (1999) 103–149
  • D Meintrup, T Schick, A model for the universal space for proper actions of a hyperbolic group, New York J. Math. 8 (2002) 1–7
  • M Mj, P Sardar, A combination theorem for metric bundles, Geom. Funct. Anal. 22 (2012) 1636–1707
  • D Osajda, P Przytycki, Boundaries of systolic groups, Geom. Topol. 13 (2009) 2807–2880
  • 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
  • Z Sela, Acylindrical accessibility for groups, Invent. Math. 129 (1997) 527–565
  • C J Tirel, $\mathcal Z$–structures on product groups, Algebr. Geom. Topol. 11 (2011) 2587–2625
  • P Tukia, Conical limit points and uniform convergence groups, J. Reine Angew. Math. 501 (1998) 71–98