Helly groups

Jérémie Chalopin (1) , Victor Chepoi (2) , Anthony Genevois (3) , Hirai Hiroshi (4) , Damian Osajda (5)

1 DALGO - Algorithmique Distribuée
2 ACRO - Algorithmique, Combinatoire et Recherche Opérationnelle
3 IMAG - Institut Montpelliérain Alexander Grothendieck
4 UTokyo - The University of Tokyo
5 Uniwersytet Wroclawski = University of Wroclaw

Helly graphs are graphs in which every family of pairwise intersecting balls has a non-empty intersection. This is a classical and widely studied class of graphs. In this article we focus on groups acting geometrically on Helly graphs -- Helly groups. We provide numerous examples of such groups: all (Gromov) hyperbolic, CAT(0) cubical, 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 of the coarse Baum-Connes conjecture. This leads to new results for some classical families of groups (e.g. for FC-type Artin groups) and to a unified approach to results obtained earlier.

Domaines: Mathématique discrète [cs.DM]; Géométrie algorithmique [cs.CG]; Algorithme et structure de données [cs.DS]; Combinatoire [math.CO]; Géométrie métrique [math.MG]

Jérémie Chalopin, Victor Chepoi, Anthony Genevois, Hirai Hiroshi, Damian Osajda. Helly groups. 2022. ⟨hal-03550002⟩