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

2020/05/01 by Jason Behrstock, Mark Hagen, Behrstock, Jason +5 · 1 citation
Mathematics · #Advanced Combinatorial Mathematics #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Group Theory (math.GR) #Homotopy and Cohomology in Algebraic Topology

paper · doi:10.48550/arxiv.2005.00567

openalex publication_date 2020/05/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01

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.

Citations

• Dehn filling in relatively hyperbolic groups

Cited by

• Embedability of right-angled Artin groups into hierarchically hyperbolic groups

Related

• Short hierarchically hyperbolic groups II: quotients and the Hopf property for Artin groups
• A Combinatorial Structure for Many Hierarchically Hyperbolic Spaces
• A refined combination theorem for hierarchically hyperbolic groups
• Extra-large type Artin groups are hierarchically hyperbolic
• Property (QT) of relatively hierarchically hyperbolic groups