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A combinatorial take on hierarchical hyperbolicity and applications to quotients of mapping class groups

Jason Behrstock, Mark Hagen, Alexandre Martin, Alessandro Sisto

Abstract

We give a simple combinatorial criterion, in terms of an action on a hyperbolic simplicial complex, for a group to be hierarchically hyperbolic. We apply this to show that quotients of mapping class groups by large powers of Dehn twists are hierarchically hyperbolic (and even relatively hyperbolic in the genus 2 case). Under residual finiteness assumptions, we construct many non-elementary hyperbolic quotients of mapping class groups. Using these quotients, we reduce questions of Reid and Bridson-Reid-Wilton about finite quotients of mapping class groups to residual finiteness of specific hyperbolic groups.

A combinatorial take on hierarchical hyperbolicity and applications to quotients of mapping class groups

Abstract

We give a simple combinatorial criterion, in terms of an action on a hyperbolic simplicial complex, for a group to be hierarchically hyperbolic. We apply this to show that quotients of mapping class groups by large powers of Dehn twists are hierarchically hyperbolic (and even relatively hyperbolic in the genus 2 case). Under residual finiteness assumptions, we construct many non-elementary hyperbolic quotients of mapping class groups. Using these quotients, we reduce questions of Reid and Bridson-Reid-Wilton about finite quotients of mapping class groups to residual finiteness of specific hyperbolic groups.

Paper Structure

This paper contains 42 sections, 58 theorems, 38 equations, 1 figure.

Table of Contents

  1. Combinatorial HHS
  2. Basic definitions
  3. On the various parts of the definition
  4. Projections to links
  5. Statement of Theorem \ref{['thm:hhs_links']}
  6. User's guide and simple examples
  7. Amalgamated free product example
  8. Relative hyperbolicity
  9. Mapping class groups and blow-ups
  10. Background on hierarchical hyperbolicity
  11. Axioms
  12. Useful facts
  13. Hyperbolicity of $Y_\Delta$
  14. Preliminary lemma about hyperbolic graphs
  15. $Y_\Delta$ is hyperbolic for $\mathcal{C}(\Delta)$ large
  16. ...and 27 more sections

Key Result

Theorem 1

Let the group $G$ act cocompactly on the flag simplicial complex $X$, and suppose that maximal simplices have finite stabilizers. Suppose that: Then $G$ is a hierarchically hyperbolic group.

Figures (1)

  • Figure 1: A portion of the complex $X$. The vertices in red correspond to elements of $C$, and all have the same link (red edge $e$ in the picture). The link of that edge in $X$ is a discrete set in bijection with $C$. However, due to the choice of $W$, the augmented link $\mathcal{C}(e)$ is the Cayley graph of $C$ with respect to $S_C$ (red dotted lines in the picture).

Theorems & Definitions (206)

  • Theorem 1
  • Theorem 2
  • Theorem 4
  • Corollary 5
  • Theorem 6
  • Remark 7
  • Remark 8
  • Theorem 9
  • Theorem 10
  • Remark 13
  • ...and 196 more