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From cut sets to cube complexes

Matthew Haulmark, Jason Fox Manning

Abstract

In this paper, we obtain an action on a cube complex from an action on a path-connected topological space with a system of divisions. In the settings of hyperbolic groups or relatively hyperbolic groups with no peripheral splittings, our result provides an alternate route to Sageev's construction of a cube complex action from a collection of (relatively) quasiconvex subgroups of a (relatively) hyperbolic group.

From cut sets to cube complexes

Abstract

In this paper, we obtain an action on a cube complex from an action on a path-connected topological space with a system of divisions. In the settings of hyperbolic groups or relatively hyperbolic groups with no peripheral splittings, our result provides an alternate route to Sageev's construction of a cube complex action from a collection of (relatively) quasiconvex subgroups of a (relatively) hyperbolic group.
Paper Structure (17 sections, 41 theorems, 23 equations)

This paper contains 17 sections, 41 theorems, 23 equations.

Table of Contents

  1. Introduction
  2. Acknowledgments
  3. Preliminaries
  4. The Cube Complex Dual to a Space with Walls
  5. Cubes from divisions
  6. Convergence groups and relative hyperbolicity
  7. Convergence groups in general
  8. Relative Hyperbolicity and relative quasiconvexity
  9. Splittings of Relatively Hyperbolic Groups
  10. Limit Sets of Vertex Stabilizers
  11. The hyperbolic case
  12. The relatively hyperbolic case
  13. Canonical Divisions
  14. Canonical divisions and $X(\mathcal{D}(\mathcal{C}))$
  15. A subdivided tree $\mathcal{T}(\mathcal{C})$ and its vertex stabilizers when $M = \partial(G,\mathcal{P})$
  16. ...and 2 more sections

Key Result

Theorem A

Under Assumption as:main, there is a $G$--action on a $\mathop{\mathrm{CAT}}\nolimits(0)$ cube complex $X=X(\mathcal{D})$ so that the set of hyperplane stabilizers is equal to the set of division stabilizers.

Theorems & Definitions (113)

  • Definition 1.1
  • Definition 1.2
  • Remark 1.3
  • Definition 1.4
  • Definition 1.5
  • Remark 1.6
  • Theorem A
  • Theorem B
  • Theorem C
  • Definition 2.1
  • ...and 103 more