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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 | |
| LIS, CNRS Aix-Marseille Universite Marseille France |
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| Victor Chepoi | |
| LIS, Aix-Marseille Universite,
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). |
This article is currently available only to readers at paying institutions. If enough institutions subscribe to this Subscribe to Open journal for 2025, the article will become Open Access in early 2025. Otherwise, this article (and all 2025 articles) will be available only to paid subscribers.
