Crystallographic Helly Groups

Hoda, Nima

doi:10.48550/arxiv.2010.07407

Cited by 5 publications

(10 citation statements)

References 15 publications

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“…Remark 3.4 (Concrete description of octahedral flat 3-manifolds). Combining [Hag14] and [Hod20], a crystallographic group is octahedral (in any dimension) if and only if it is cocompactly cubulated if and only if it is Helly. In [PS20], it is shown that for crystallographic groups, this is equivalent to being an HHG.…”

Section: Statements Of the Main Resultsmentioning

confidence: 99%

Equivariant hierarchically hyperbolic structures for 3-manifold groups via quasimorphisms

Hagen¹,

Russell²,

Sisto³

et al. 2022

Preprint

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Behrstock, Hagen, and Sisto classified 3-manifold groups admitting a hierarchically hyperbolic space structure. However, these structures were not always equivariant with respect to the group. In this paper, we classify 3-manifold groups admitting equivariant hierarchically hyperbolic structures. The key component of our proof is that the admissible groups introduced by Croke and Kleiner always admit equivariant hierarchically hyperbolic structures. For non-geometric graph manifolds, this is contrary to a conjecture of Behrstock, Hagen, and Sisto and also contrasts with results about CAT(0) cubical structures on these groups. Perhaps surprisingly, our arguments involve the construction of suitable quasimorphisms on the Seifert pieces, in order to construct actions on quasi-lines.

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Section: Statements Of the Main Resultsmentioning

confidence: 99%

Equivariant hierarchically hyperbolic structures for 3-manifold groups via quasimorphisms

Hagen¹,

Russell²,

Sisto³

et al. 2022

Preprint

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“…It is natural to ask what happens for SL(n, K). Inspired by the work of Hoda on crystallographic Helly groups (see [Hod20]), we prove the following.…”

Section: Symmetric Spacesmentioning

confidence: 86%

“…We now turn to the case of the special linear group. We will prove that it is not coarsely Helly, inspired by the result of Hoda that the (3, 3, 3) triangle Coxeter group W , which is virtually Z 2 , is not Helly (see [Hod20]). However, the group W is a subgroup of Z 3 ⋊ S 3 , which is Helly.…”

Section: The Special Linear Group Is Not Coarsely Hellymentioning

confidence: 95%

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Injective metrics on buildings and symmetric spaces

Haettel¹

2021

Preprint

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In this article, we show that the Goldman-Iwahori metric on the space of all norms on a fixed vector space satisfies the Helly property for balls. On the non-Archimedean side, we deduce that most classical Bruhat-Tits buildings may be endowed with a natural piecewise ℓ ∞ metric which is injective. We also prove that most classical semisimple groups over non-Archimedean local fields act properly and cocompactly on Helly graphs. This gives another proof of biautomaticity for their uniform lattices. On the Archimedean side, we deduce that most classical symmetric spaces of non-compact type may be endowed with a natural piecewise ℓ ∞ metric which is coarsely Helly. We also prove that most classical semisimple groups over Archimedean local fields act properly and cocompactly on injective metric spaces. The only exception is the special linear group: if n 3 and K is a local field, we show that SL(n, K) does not act properly and coboundedly on an injective metric space.

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“…Note that one may wonder whether it is necessary to consider the direct product with R. In fact, Hoda proved (see [Hod20]) that the Euclidean Coxeter group of type A n is not Helly for n 2, even though its direct product with Z is. We made a similar distinction for automorphism groups of Euclidean buildings of type A n in [Hae21].…”

Section: Introductionmentioning

confidence: 99%

Lattices, injective metrics and the $K(π,1)$ conjecture

Haettel¹

2021

Preprint

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Starting with a lattice with an action of Z or R, we build a Helly graph or an injective metric space. We deduce that the ℓ ∞ orthoscheme complex of any bounded graded lattice is injective. We also prove a Cartan-Hadamard result for locally injective metric spaces. We apply this to show that any Garside group acts on an injective metric space and on a Helly graph. We also deduce that the natural piecewise ℓ ∞ metric on any Euclidean building of type A n extended, B n , C n or D n is injective, and its thickening is a Helly graph. Concerning Artin groups of Euclidean types A n and C n , we show that the natural piecewise ℓ ∞ metric on the Deligne complex is injective, the thickening is a Helly graph, and it admits a convex bicombing. This gives a metric proof of the K(π, 1) conjecture, as well as several other consequences usually known when the Deligne complex has a CAT(0) metric.

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Crystallographic Helly Groups

Cited by 5 publications

References 15 publications

Equivariant hierarchically hyperbolic structures for 3-manifold groups via quasimorphisms

Equivariant hierarchically hyperbolic structures for 3-manifold groups via quasimorphisms

Injective metrics on buildings and symmetric spaces

Lattices, injective metrics and the $K(π,1)$ conjecture

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