Hierarchical hyperbolicity of graph products

Berlyne, Daniel; Russell, Jacob

doi:10.4171/ggd/652

Cited by 4 publications

(1 citation statement)

References 23 publications

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“…There are also various ways to combine HHSs and HHGs to produce new ones. For example, both classes are closed under relative hyperbolicity [13], any graph product of HHGs is an HHG [16], and many graphs of groups are HHGs [13; 15; 68].…”

Section: Background On Hierarchical Hyperbolicitymentioning

confidence: 99%

Coarse injectivity, hierarchical hyperbolicity and semihyperbolicity

2023

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We relate three classes of nonpositively curved metric spaces: hierarchically hyperbolic spaces, coarsely injective spaces and strongly shortcut spaces. We show that every hierarchically hyperbolic space admits a new metric that is coarsely injective. The new metric is quasi-isometric to the original metric and is preserved under automorphisms of the hierarchically hyperbolic space. We show that every coarsely injective metric space of uniformly bounded geometry is strongly shortcut. Consequently, hierarchically hyperbolic groups -including mapping class groups of surfaces -are coarsely injective and coarsely injective groups are strongly shortcut.Using these results, we deduce several important properties of hierarchically hyperbolic groups, including that they are semihyperbolic, they have solvable conjugacy problem and finitely many conjugacy classes of finite subgroups, and their finitely generated abelian subgroups are undistorted. Along the way we show that hierarchically quasiconvex subgroups of hierarchically hyperbolic groups have bounded packing.20F65, 20F67, 51F30

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Section: Background On Hierarchical Hyperbolicitymentioning

confidence: 99%

Coarse injectivity, hierarchical hyperbolicity and semihyperbolicity

2023

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Conjugacy languages in virtual graph products

Gemma

2023

Journal of Algebra

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

Behrstock,

Hagen,

Martin

et al. 2024

Journal of Topology

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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). In genus at least three, there are no known infinite hyperbolic quotients of mapping class groups. However, using the hierarchically hyperbolic quotients we construct, we show, under a residual finiteness assumption, that mapping class groups have many nonelementary hyperbolic quotients. Using these quotients, we relate questions of Reid and Bridson–Reid–Wilton about finite quotients of mapping class groups to residual finiteness of specific hyperbolic groups.

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Hierarchical hyperbolicity of graph products

Cited by 4 publications

References 23 publications

Coarse injectivity, hierarchical hyperbolicity and semihyperbolicity

Coarse injectivity, hierarchical hyperbolicity and semihyperbolicity

Conjugacy languages in virtual graph products

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

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