Helly groups

Chalopin, Jérémie¹ ; Chepoi, Victor² ; Genevois, Anthony³ ; Hirai, Hiroshi⁴ ; Osajda, Damian⁵

¹ Laboratoire d’Informatique et Systèmes, CNRS, Aix-Marseille Université, Marseille, France
² Laboratoire d’Informatique et Systèmes, Aix-Marseille Université, CNRS, Marseille, France
³ Département de Mathématiques, Faculté des Sciences d’Orsay Université Paris-Sud, Orsay, France, Institut Montpelliérain Alexander Grothendieck, Université de Montpellier, Montpellier, France
⁴ Graduate School of Mathematics, Nagoya University, Nagoya, Japan
⁵ Institute of Mathematics, Polish Academy of Sciences, Warszawa, Poland, Instytut Matematyczny, Uniwersytet Wrocławski, Wrocław, Poland

Geometry & topology, Tome 29 (2025) no. 1, pp. 1-70.

Résumé

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, $\mathrm{CAT}(0)$ cubical groups, finitely presented graphical $C(4)$–$T(4)$ small cancellation groups and type-preserving uniform lattices in Euclidean buildings of type $C_n$ 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.

Détail

DOI : 10.2140/gt.2025.29.1

Keywords: Helly group, injective space, hyperbolic group, $\mathrm{CAT}(0)$ cubical group, biautomaticity, EZ-boundary, Baum–Connes conjecture

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