Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-28T14:50:08.141Z Has data issue: false hasContentIssue false
  • English
  • Français

A combination theorem for combinatorially non-positively curved complexes of hyperbolic groups

Part of: Special aspects of infinite or finite groups

Published online by Cambridge University Press:  09 March 2020

ALEXANDRE MARTIN  and
DAMIAN OSAJDA
ALEXANDRE MARTIN
Affiliation:
Dept. of Mathematics, Heriot-Watt University, EH14 4ASEdinburgh, e-mail: [email protected]
DAMIAN OSAJDA
Affiliation:
Instytut Matematyczny, Universytet PL Grunwaldzki 2/4, Wroclawski, 50–384Wroclaw, Poland Institute of Mathematics, Polish Academy of Sciences, Śniadeckich 8, 00–656Warszawa, Poland, e-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We prove a combination theorem for hyperbolic groups, in the case of groups acting on complexes displaying combinatorial features reminiscent of non-positive curvature. Such complexes include for instance weakly systolic complexes and C'(1/6) small cancellation polygonal complexes. Our proof involves constructing a potential Gromov boundary for the resulting groups and analyzing the dynamics of the action on the boundary in order to use Bowditch’s characterisation of hyperbolicity. A key ingredient is the introduction of a combinatorial property that implies a weak form of non-positive curvature, and which holds for large classes of complexes.

As an application, we study the hyperbolicity of groups obtained by small cancellation over a graph of hyperbolic groups.

MSC classification

Primary: 20F67: Hyperbolic groups and nonpositively curved groups
Secondary: 20F65: Geometric group theory
Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 170 , Issue 3 , May 2021 , pp. 445 - 477
Copyright
© Cambridge Philosophical Society 2020

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Bandelt, H. J. and Chepoi, V.. Metric graph theory and geometry: a survey. In Surveys on discrete and computational geometry, volume 453 of Contemp. Math. (Amer. Math. Soc., Providence, RI, 2008), (pages 49–86).CrossRefGoogle Scholar
Bestvina, M. and Feighn, M.. A combination theorem for negatively curved groups. J. Differential Geom. 35(1) (1992), 85–101.CrossRefGoogle Scholar
Bestvina, M. and Feighn, M.. Hyperbolicity of the complex of free factors. Adv. Math. 256 (2014), 104–155.CrossRefGoogle Scholar
Bowditch, B. H.. A topological characterisation of hyperbolic groups. J. Amer. Math. Soc. 11(3) (1998), 643–667.CrossRefGoogle Scholar
Bowditch, B. H.. Tight geodesics in the curve complex. Invent. Math. 171(2) (2008), 281–300.CrossRefGoogle Scholar
Bridson, M. R. and Haefliger, A.. Metric Spaces of Non-Positive Curvature (Springer-Verlag, Berlin, 1999).CrossRefGoogle Scholar
Chalopin, J., Chepoi, V., Hirai, H. and Osajda, D.. Weakly modular graphs and nonpositive curvature. Mem. Amer. Math. Soc., to appear (2019).Google Scholar
Chatterji, I. and Martin, A.. A note on the acylindrical hyperbolicity of groups acting on CAT(0) cube complexes. London Math. Soc. Lecture Note Series (Cambridge University Press, 2019), pages 160–178.Google Scholar
Chepoi, V.. Classification of graphs by means of metric triangles. Metody Diskret. Analiz. (49) (1989), 75–93, 96.Google Scholar
Chepoi, V. and Osajda, D.. Dismantlability of weakly systolic complexes and applications. Trans. Amer. Math. Soc. 367(2) (2015), 1247–1272.CrossRefGoogle Scholar
Dahmani, F.. Combination of convergence groups. Geom. Topol. 7 (2003), (electronic), 933–963.CrossRefGoogle Scholar
Elsner, T.. Flats and the flat torus theorem in systolic spaces. Geom. Topol. 13(2) (2009), 661–698.CrossRefGoogle Scholar
Freden, E. M.. Negatively curved groups have the convergence property. I. Ann. Acad. Sci. Fenn. Ser. A I Math. 20(2) (1995), 333–348.Google Scholar
Gromov, M.. Hyperbolic groups. In Essays in group theory, volume 8 of Math. Sci. Res. Inst. Publ. (Springer, New York, 1987), (pages 75–263).CrossRefGoogle Scholar
Gromov, M.. CAT (κ)-spaces: construction and concentration. Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), 280(Geom. i Topol. 7) (2001), 100–140, 299–300.Google Scholar
Gruber, D. and Sisto, A.. Infinitely presented graphical small cancellation groups are acylindrically hyperbolic. Ann. Inst. Fourier (Grenoble), 68(6) (2018), 2501–2552.CrossRefGoogle Scholar
Januszkiewicz, T. and Światkowski, J.. Simplicial nonpositive curvature. Publ. Math. Inst. Hautes Études Sci. (104) (2006), 1–85.CrossRefGoogle Scholar
Lyndon, R. C. and Schupp, P. E.. Combinatorial Group Theory (Springer-Verlag, Berlin, 1977).Google Scholar
Martin, A.. Non-positively curved complexes of groups and boundaries. Geom. Topol. 18(1) (2014), 31–102.CrossRefGoogle Scholar
Martin, A.. Combination of universal spaces for proper actions. J. Homotopy Relat. Struct. 10(4) (2015), 803–820.CrossRefGoogle Scholar
Martin, A.. Complexes of groups and geometric small cancelation over graphs of groups. Bull. Soc. Math. France 145(2) (2017), 193–223.CrossRefGoogle Scholar
Martin, A.. On the acylindrical hyperbolicity of the tame automorphism group of SL2(ℂ). Bull. Lond. Math. Soc. 49(5) (2017), 881–894.CrossRefGoogle Scholar
Martin, A. and Steenbock, M.. A combination theorem for cubulation in small cancellation theory over free products. Ann. Inst. Fourier (Grenoble) 67(4) (2017), 1613–1670.CrossRefGoogle Scholar
McCammond, J. P. and Wise, D. T.. Fans and ladders in small cancellation theory. Proc. London Math. Soc. (3) 84(3) (2002), 599–644.CrossRefGoogle Scholar
Osajda, D.. A combinatorial non-positive curvature I: weak systolicity. arXiv:1305.4661 (2013).Google Scholar
Pal, A. and Paul, S.. Complex of relatively hyperbolic groups. Glasg. Math. J. 61(3) (2019), 657–672.CrossRefGoogle Scholar
Strebel, R.. Appendix. Small cancellation groups. In Sur les groupes hyperboliques d’après Mikhael Gromov (Bern, 1988), volume 83 of Progr. Math. (Birkhäuser Boston, Boston, MA, 1990), (pages 227–273).Google Scholar
Zeeman, E. C.. Relative simplicial approximation. Proc. Cambridge Philos. Soc. 60 (1964), 39–43.CrossRefGoogle Scholar