Hostname: page-component-78c5997874-j824f Total loading time: 0 Render date: 2024-11-10T15:09:55.287Z Has data issue: false hasContentIssue false
  • English
  • Français

Geodesics in Trees of Hyperbolic and Relatively Hyperbolic Spaces

Part of: Special aspects of infinite or finite groups Low-dimensional topology

Published online by Cambridge University Press:  20 November 2015

François Gautero
François Gautero*
Affiliation:
Université de Nice Sophia Antipolis, Parc Valrose, Laboratoire de Mathématiques J. A. Dieudonné, UMR CNRS 7351, 06108 Nice Cedex 02, France (francois.gautero@unice.fr)
Get access Rights & Permissions [Opens in a new window]

Abstract

We present a careful approximation of the quasi-geodesics of trees of hyperbolic and relatively hyperbolic spaces. As an application we prove a dynamical and geometric combination theorem for trees of relatively hyperbolic spaces, with both Farb's and Gromov's definitions.

Keywords

Gromov hyperbolicityFarb and Gromov relative hyperbolicitytrees of spacesmapping toricombination theorem

MSC classification

Secondary: 20F65: Geometric group theory 57M50: Geometric structures on low-dimensional manifolds
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 59 , Issue 3 , August 2016 , pp. 701 - 740
Copyright
Copyright © Edinburgh Mathematical Society 2015 

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

1. Alibegovic, E., A combination theorem for relatively hyperbolic groups, Bull. Lond. Math. Soc. 37(3) (2005), 459–466.Google Scholar
2. Bestvina, M. and Feighn, M., A combination theorem for negatively curved groups, J. Diff. Geom. 35(1) (1992), 85–101 (addendum and correction: J. Diff. Geom. 43(4) (1996), 783–788).Google Scholar
3. Bigdely, H. and Wise, D. T., Quasiconvexity and relatively hyperbolic groups that split, Michigan Math. J. 62 (2013), 387–406.CrossRefGoogle Scholar
4. Bowditch, B. H., Relatively hyperbolic groups, Preprint, University of Southampton (http://www.warwick.ac.uk/~masgak/preprints.html; 1999).Google Scholar
5. Bowditch, B. H., Stacks of hyperbolic spaces and ends of 3-manifolds, Preprint, University of Southampton (http://www.warwick.ac.uk/~masgak/preprints.html; 2002).Google Scholar
6. Bridson, M. and Haefliger, A., Metric spaces of non-positive curvature, Grundlehren der Mathematischen Wissenschaften, Volume 319 (Springer, 1999).CrossRefGoogle Scholar
7. Bumagin, I., On definitions of relatively hyperbolic groups, in Geometric methods in group theory, Contemporary Mathematics, Volume 372 (American Mathematical Society, Providence, RI, 2005).Google Scholar
8. Coornaert, M., Delzant, T. and Papadopoulos, A., Géométrie et théorie des groupes, Lecture Notes in Mathematics, Volume 1441 (Springer, 1990).CrossRefGoogle Scholar
9. Dahmani, F., Combination of convergence groups, Geom. Topology 7 (2003), 933–963.CrossRefGoogle Scholar
10. Dowdall, S. and Taylor, S. J., Hyperbolic extensions of free groups, Preprint (arxiv.org/abs/1406.2567; 2014).Google Scholar
11. Farb, B., Relatively hyperbolic groups, Geom. Funct. Analysis 8(5) (1998), 810–840.Google Scholar
12. Gautero, F., Hyperbolicity of mapping torus groups and spaces, L’Enseignement Math. 49 (2003), 263–305.Google Scholar
13. Gautero, F., Quatre problèmes dynamiques, géométriques ou algébriques autour de la suspension, Habilitation thesis, Université Blaise Pascal (2006).Google Scholar
14. Gautero, F. and Lustig, M., Relative hyperbolization of (one-ended hyperbolic)-bycyclic groups, Math. Proc. Camb. Phil. Soc. 137 (2004), 595–611.CrossRefGoogle Scholar
15. Gitik, R., On the combination theorem for negatively curved groups, Int. J. Alg. Comput. 7(2) (1997), 267–276.Google Scholar
16. Gromov, M., Hyperbolic groups, in Essays in group theory, Mathematical Sciences Research Institute Publications, Volume 8, pp. 75–263 (Springer, 1987).Google Scholar
17. Kapovich, I., A non-quasiconvexity embedding theorem for hyperbolic groups, Math. Proc. Camb. Phil. Soc. 127 (1999), 461–486.Google Scholar
18. Kapovich, I., Mapping tori of endomorphisms of free groups, Commun. Alg. 28(6) (2000), 2895–2917.Google Scholar
19. Kharlampovich, O. and Myasnikov, A., Hyperbolic groups and free constructions, Trans. Am. Math. Soc. 350(2) (1998), 571–613.CrossRefGoogle Scholar
20. Mj, M. and Reeves, L., A combination theorem for strong relative hyperbolicity, Geom. Topology 12(3) (2008), 1777–1798.Google Scholar
21. Osin, D., Weak hyperbolicity and free constructions, in Group theory, statistics, and cryptography, Contemporary Mathematics, Volume 360, pp. 103–111 (American Mathematical Society, Providence, RI, 2004).Google Scholar
22. Osin, D., Relatively hyperbolic groups: intrinsic geometry, algebraic properties and algorithmic problems, Memoirs of the American Mathematical Society, Volume 179 (American Mathematical Society, Providence, RI, 2006).Google Scholar
23. Osin, D., Relative Dehn functions of amalgated products and HNN extensions, in Topological and asymptotic aspects of group theory, Contemporary Mathematics, Volume 394, pp. 209–220 (American Mathematical Society, Providence, RI, 2006).CrossRefGoogle Scholar
24. Szczepański, A., Relatively hyperbolic groups, Michigan Math. J. 45 (1998), 611–618.Google Scholar