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  Deforming cubulations of hyperbolic groups

Fioravanti, E., & Hagen, M. (2021). Deforming cubulations of hyperbolic groups. Journal of Topology, 14(3), 877-912. doi:10.1112/topo.12201.

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Fioravanti-Hagen_Deforming cubulations of hyperbolic groups_2021.pdf (Publisher version), 531KB
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© 2021 The Authors. Journal of Topology is copyright © London Mathematical Society. This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.
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https://doi.org/10.1112/topo.12201 (Publisher version)
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 Creators:
Fioravanti, Elia1, Author           
Hagen, Mark, Author
Affiliations:
1Max Planck Institute for Mathematics, Max Planck Society, ou_3029201              

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Free keywords: Mathematics, Geometric Topology, Group Theory
 Abstract: We describe a procedure to deform cubulations of hyperbolic groups by
"bending hyperplanes". Our construction is inspired by related constructions
like Thurston's Mickey Mouse example, walls in fibred hyperbolic $3$-manifolds
and free-by-$\mathbb Z$ groups, and Hsu-Wise turns.
As an application, we show that every cocompactly cubulated Gromov-hyperbolic
group admits a proper, cocompact, essential action on a ${\rm CAT}(0)$ cube
complex with a single orbit of hyperplanes. This answers (in the negative) a
question of Wise, who proved the result in the case of free groups.
We also study those cubulations of a general group $G$ that are not
susceptible to trivial deformations. We name these "bald cubulations" and
observe that every cocompactly cubulated group admits at least one bald
cubulation. We then apply the hyperplane-bending construction to prove that
every cocompactly cubulated hyperbolic group $G$ admits infinitely many bald
cubulations, provided $G$ is not a virtually free group with ${\rm Out}(G)$
finite. By contrast, we show that the Burger-Mozes examples each admit a unique
bald cubulation.

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Language(s): eng - English
 Dates: 2021
 Publication Status: Issued
 Pages: 36
 Publishing info: -
 Table of Contents: -
 Rev. Type: Peer
 Identifiers: arXiv: 1912.10999
DOI: 10.1112/topo.12201
 Degree: -

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Title: Journal of Topology
Source Genre: Journal
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Publ. Info: Wiley
Pages: - Volume / Issue: 14 (3) Sequence Number: - Start / End Page: 877 - 912 Identifier: -