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Helly groups
Jérémie Chalopin, Victor Chepoi, Anthony Genevois, Hiroshi Hirai and Damian Osajda |
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Geometry & Topology 29 (2025) 1–70
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DOI: 10.2140/gt.2025.29.1
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Abstract |
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Helly graphs are graphs in which every family of pairwise-intersecting balls has a nonempty intersection. This is a classical and widely studied class of graphs. We focus on groups acting geometrically on Helly graphs — Helly groups. We provide numerous examples of such groups: all (Gromov) hyperbolic groups, cubical groups, finitely presented graphical – small cancellation groups and type-preserving uniform lattices in Euclidean buildings of type are Helly; free products of Helly groups with amalgamation over finite subgroups, graph products of Helly groups, some diagram products of Helly groups, some right-angled graphs of Helly groups and quotients of Helly groups by finite normal subgroups are Helly. We show many properties of Helly groups: biautomaticity, existence of finite-dimensional models for classifying spaces for proper actions, contractibility of asymptotic cones, existence of EZ-boundaries, satisfiability of the Farrell–Jones conjecture and satisfiability of the coarse Baum–Connes conjecture. This leads to new results for some classical families of groups (eg for FC-type Artin groups) and to a unified approach to results obtained earlier. |
Keywords
Helly group, injective space, hyperbolic group,
$\mathrm{CAT}(0)$ cubical group, biautomaticity,
EZ-boundary, Baum–Connes conjecture
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Mathematical Subject Classification
Primary: 20F06, 20F65, 20F67
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References |
Publication
Received: 17 June 2020
Revised: 4 April 2023
Accepted: 6 May 2023
Published: 1 January 2025
Proposed: David Fisher
Seconded: Urs Lang, Mladen Bestvina
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Authors |
| Jérémie Chalopin | |
| Laboratoire d’Informatique et
Systèmes CNRS Aix-Marseille Université Marseille France |
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| Victor Chepoi | |
| Laboratoire d’Informatique et
Systèmes Aix-Marseille Université CNRS Marseille France |
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| Anthony Genevois | |
| Département de Mathématiques Faculté des Sciences d’Orsay Université Paris-Sud Orsay France |
Institut Montpelliérain Alexander
Grothendieck Université de Montpellier Montpellier France |
| Hiroshi Hirai | |
| Graduate School of Mathematics Nagoya University Nagoya Japan |
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| Damian Osajda | |
| Institute of Mathematics Polish Academy of Sciences Warszawa Poland |
Instytut Matematyczny Uniwersytet Wrocławski Wrocław Poland |
| © 2025 MSP (Mathematical Sciences Publishers). Distributed under the Creative Commons Attribution License 4.0 (CC BY). |
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