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Maps between Boundaries of Relatively Hyperbolic Groups

Abhijit Pal, Rana Sardar

Abstract

F. Paulin proved that if the Gromov boundaries of two hyperbolic groups are quasi-Mobius equivalent, then the groups themselves are quasi-isometric. The goal of this article is to extend Paulin's result to the setting of relatively hyperbolic groups by introducing the notion of relative quasi-Mobius maps between the Bowditch boundaries of relatively hyperbolic groups. We show that any coarsely cusp-preserving quasi-isometry between two relatively hyperbolic groups induces a homeomorphism between their Bowditch boundaries, and that this induced homeomorphism is relative quasi-Mobius and linearly distorts the exit points of bi-infinite geodesics into combinatorial horoballs. Conversely, we prove that if a homeomorphism between the Bowditch boundaries of two relatively hyperbolic groups preserves parabolic fixed points and is either relative quasi-Mobius or linearly distorts the exit points of bi-infinite geodesics into combinatorial horoballs, then it arises from a coarsely cusp-preserving quasi-isometry between the groups.

Maps between Boundaries of Relatively Hyperbolic Groups

Abstract

F. Paulin proved that if the Gromov boundaries of two hyperbolic groups are quasi-Mobius equivalent, then the groups themselves are quasi-isometric. The goal of this article is to extend Paulin's result to the setting of relatively hyperbolic groups by introducing the notion of relative quasi-Mobius maps between the Bowditch boundaries of relatively hyperbolic groups. We show that any coarsely cusp-preserving quasi-isometry between two relatively hyperbolic groups induces a homeomorphism between their Bowditch boundaries, and that this induced homeomorphism is relative quasi-Mobius and linearly distorts the exit points of bi-infinite geodesics into combinatorial horoballs. Conversely, we prove that if a homeomorphism between the Bowditch boundaries of two relatively hyperbolic groups preserves parabolic fixed points and is either relative quasi-Mobius or linearly distorts the exit points of bi-infinite geodesics into combinatorial horoballs, then it arises from a coarsely cusp-preserving quasi-isometry between the groups.
Paper Structure (20 sections, 39 theorems, 108 equations, 15 figures)

This paper contains 20 sections, 39 theorems, 108 equations, 15 figures.

Table of Contents

  1. Introduction
  2. Outline of the Paper
  3. Notations.
  4. Acknowledgements.
  5. Preliminaries
  6. Hyperbolic Metric Spaces
  7. Gromov Boundary of a Hyperbolic Metric Space
  8. Relatively Hyperbolic Groups and Boundaries
  9. Quasi-centres, Cross Ratios and Quasi-Möbius maps
  10. Quasi-centres
  11. Cross-Ratios and Quasi-Möbius Maps
  12. Quasi-Isometries Between Relatively Hyperbolic Groups and Relative Quasi-Möbius Homeomorphisms Between Their Bowditch Boundaries
  13. Relative Quasi-Möbius Maps
  14. Relative Quasi-Möbius Maps Induced by Quasi-isometries
  15. Coarse Denseness of Quasi-centres
  16. ...and 5 more sections

Key Result

Theorem 1

Suppose the numbers $\delta, K,\epsilon \geq 0$ and $\lambda\geq 1$ are given. Let $(G_1,\mathcal{H}_{G_1})$ and $(G_2,\mathcal{H}_{G_2})$ be two $\delta$-relatively hyperbolic groups. If $\varphi: G_1 \to G_2$ is a $K$-coarsely cusp-preserving $(\lambda,\epsilon)$-quasi-isometry, then its boundary

Figures (15)

  • Figure 2.1:
  • Figure 2.2:
  • Figure 3.1:
  • Figure 4.1:
  • Figure 4.2:
  • ...and 10 more figures

Theorems & Definitions (77)

  • Theorem 1
  • Theorem 2
  • Corollary 1: Pau96
  • Theorem 3
  • Theorem 4
  • Corollary 2
  • Definition 2.1
  • Definition 2.2: Quasi-isometry
  • Definition 2.3: Quasigeodesic
  • Example 2.4
  • ...and 67 more