Hostname: page-component-6bf8c574d5-2jptb Total loading time: 0 Render date: 2025-02-17T02:21:03.113Z Has data issue: false hasContentIssue false
  • English
  • Français

Continuity of critical exponent of quasiconvex-cocompact groups under Gromov–Hausdorff convergence

Part of: Global differential geometry Special aspects of infinite or finite groups

Published online by Cambridge University Press:  10 February 2022

NICOLA CAVALLUCCI Open the ORCID record for NICOLA CAVALLUCCI [Opens in a new window]
NICOLA CAVALLUCCI*
Affiliation:
Mathematisches Institut, Universitat Köln, Weyertal 86–90, 50931 Köln, Germany
*
e-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We show continuity under equivariant Gromov–Hausdorff convergence of the critical exponent of discrete, non-elementary, torsion-free, quasiconvex-cocompact groups with uniformly bounded codiameter acting on uniformly Gromov-hyperbolic metric spaces.

Keywords

Gromov hyperbolicityGromov–Hausdorff convergenceentropycontinuityconvex-cocompact

MSC classification

Primary: 53C23: Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces
Secondary: 20F67: Hyperbolic groups and nonpositively curved groups
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 43 , Issue 4 , April 2023 , pp. 1189 - 1221
Check for updates
Copyright
© The Author(s), 2022. Published by Cambridge University Press

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

Besson, G., Courtois, G., Gallot, S. and Sambusetti, A.. Curvature-free Margulis lemma for Gromov-hyperbolic spaces. Preprint, 2020, arXiv:1712.08386.Google Scholar
Besson, G., Courtois, G., Gallot, S. and Sambusetti, A.. Finiteness theorems for Gromov-hyperbolic spaces and groups. Preprint, 2021, arXiv:2109.13025.10.1007/978-3-030-86695-2_6CrossRefGoogle Scholar
Bridson, M. and Haefliger, A.. Metric Spaces of Non-positive Curvature (Grundlehren der mathematischen Wissenschaften, 319). Springer, Berlin, 2013.Google Scholar
Bishop, C. and Jones, P.. Hausdorff dimension and Kleinian groups. Acta Math. 179(1) (1997), 1–39.10.1007/BF02392718CrossRefGoogle Scholar
Bartels, A. and Lück, W.. Geodesic flow for $\mathrm{CAT}(0)$ -groups. Geom. Topol. 16(3) (2012), 1345–1391.10.2140/gt.2012.16.1345CrossRefGoogle Scholar
Cavallucci, N.. Entropies of non positively curved metric spaces. Preprint, 2021, arXiv:2102.07502.10.1007/s10711-022-00719-1CrossRefGoogle Scholar
Cavallucci, N.. Topological entropy of the geodesic flow of non-positively curved metric spaces. Preprint, 2021, arXiv:2105.11774.10.1007/s10711-022-00719-1CrossRefGoogle Scholar
Coornaert, M., Delzant, T. and Papadopoulos, A.. Géométrie et théorie des groupes: Les groupes hyperboliques de Gromov (Lecture Notes in Mathematics, 1441). Springer, Berlin, 1990.CrossRefGoogle Scholar
Coornaert, M.. Mesures de Patterson–Sullivan sur le bord d’un espace hyperbolique au sens de Gromov. Pacific J. Math. 159(2) (1993), 241–270.CrossRefGoogle Scholar
Cavallucci, N. and Sambusetti, A.. Discrete groups of packed, non-positively curved, Gromov hyperbolic metric spaces. Preprint, 2020, arXiv:2102.09829.Google Scholar
Cavallucci, N. and Sambusetti, A.. Packing and doubling in metric spaces with curvature bounded above. Math. Z. (2021). https://doi.org/10.1007/s00209-021-02905-5 Google Scholar
Druţu, C. and Kapovich, M.. Geometric Group Theory (Colloquium Publications, 63). American Mathematical Society, Providence, RI, 2018.CrossRefGoogle Scholar
Das, T., Simmons, D. and Urbański, M.. Geometry and Dynamics in Gromov Hyperbolic Metric Spaces (Mathematical Surveys and Monographs, 218). American Mathematical Society, Providence, RI, 2017.10.1090/surv/218CrossRefGoogle Scholar
Fukaya, K.. Theory of convergence for Riemannian orbifolds. Jpn. J. Math. New Ser. 12(1) (1986), 121–160.10.4099/math1924.12.121CrossRefGoogle Scholar
Gromov, M.. Groups of polynomial growth and expanding maps (with an appendix by Jacques Tits). Publ. Math. Inst. Hautes Études Sci. 53 (1981), 53–78.10.1007/BF02698687CrossRefGoogle Scholar
Herron, D. A.. Gromov–Hausdorff distance for pointed metric spaces. J. Anal. 24(1) (2016), 1–38.10.1007/s41478-016-0001-xCrossRefGoogle Scholar
Jansen, D.. Notes on pointed Gromov–Hausdorff convergence. Preprint, 2017, arXiv:1703.09595.Google Scholar
Kelley, J. L.. General Topology. Dover Publications, Mineola, NY, 2017.Google Scholar
Paulin, F.. Un groupe hyperbolique est déterminé par son bord. J. Lond. Math. Soc. (2) 54(1) (1996), 50–74.10.1112/jlms/54.1.50CrossRefGoogle Scholar
Paulin, F.. On the critical exponent of a discrete group of hyperbolic isometries. Differential Geom. Appl. 7(3) (1997), 231–236.10.1016/S0926-2245(96)00051-4CrossRefGoogle Scholar