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Blown-up corona of relatively hyperbolic groups

Tomohiro Fukaya

Abstract

We show that under appropriate assumptions, a blown-up corona of a relatively hyperbolic group is equivariant and the compactification of the universal space for proper action by the blown-up corona is contractible. As a corollary, we establish the formula to determine the covering dimension of the blown-up corona by the cohomological dimension of the group. We also show that the blown-up corona of a hyperbolic group with respect to an almost malnormal family of quasiconvex subgroups is homeomorphic to the Gromov boundary of the group.

Blown-up corona of relatively hyperbolic groups

Abstract

We show that under appropriate assumptions, a blown-up corona of a relatively hyperbolic group is equivariant and the compactification of the universal space for proper action by the blown-up corona is contractible. As a corollary, we establish the formula to determine the covering dimension of the blown-up corona by the cohomological dimension of the group. We also show that the blown-up corona of a hyperbolic group with respect to an almost malnormal family of quasiconvex subgroups is homeomorphic to the Gromov boundary of the group.
Paper Structure (25 sections, 67 equations, 2 tables)

This paper contains 25 sections, 67 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Cohomological dimension
  3. Blown-up corona of a hyperbolic group with an almost malnormal family of quasiconvex subgroups
  4. Organisation of the paper
  5. Coarse space and $\mathcal{Z}$-structure
  6. Coarse compactification
  7. Coarse metric simplicial complex
  8. $\mathcal{Z}$-structure and the condition $\mathbf{BM}$
  9. Relatively hyperbolic groups
  10. Combinatorial horoball
  11. Augmented space
  12. $\mathcal{Z}$-structure of relatively hyperbolic group
  13. Setting
  14. Rips complex of the augmented space and universal space
  15. One-point blow up
  16. ...and 10 more sections

Theorems & Definitions (21)

  • proof
  • proof
  • proof
  • proof
  • proof
  • proof
  • proof : Proof of \ref{['prop:BM-single']}
  • proof
  • proof
  • proof
  • ...and 11 more