Unbounded domains in hierarchically hyperbolic groups

Petyt, Harry; Davide, Spriano,

doi:10.4171/ggd/706

Cited by 4 publications

(2 citation statements)

References 7 publications

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“…For example, the property of being an HHS is invariant under quasi-isometries, but there are groups that are virtually HHGs but not HHGs themselves. Indeed, the .3; 3; 3/ triangle group is virtually abelian, but, as mentioned in the introduction, it is not coarsely injective [50], and it therefore cannot be an HHG by Corollary H. A more direct proof, not relying on the results of this paper, is given in [66]. On the other hand, any group that is an HHS can be equipped with a coarse median [13], but this may fail to be equivariant if the structure is only an HHS structure.…”

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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“…In more recent work, Petyt and Spriano show that for the class of crystallographic groups, being hierarchically hyperbolic is also equivalent to being cocompactly cubulated [24].…”

Section: Introductionmentioning

confidence: 99%

Crystallographic Helly groups

Hoda

2023

Bulletin of London Math Soc

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We prove that asymptotic cones of Helly graphs are countably hyperconvex. We use this to show that virtually nilpotent Helly groups are virtually abelian and to characterize virtually abelian Helly groups via their point groups. In fact, we do this for the more general class of coarsely injective spaces and groups. We apply this to prove that the 3‐3‐3‐Coxeter group is not Helly (nor even coarsely injective), thus obtaining the first example of a systolic group that is not Helly, answering a question of Chalopin, Chepoi, Genevois, Hirai, and Osajda.

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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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Unbounded domains in hierarchically hyperbolic groups

Cited by 4 publications

References 7 publications

Coarse injectivity, hierarchical hyperbolicity and semihyperbolicity

Coarse injectivity, hierarchical hyperbolicity and semihyperbolicity

Crystallographic Helly groups

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

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