Helly groups

Abstract

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.

References in corpus (9)

• Weakly modular graphs and nonpositive curvature
• Residually finite non-exact groups
• Combinatorics and geometry of finite and infinite squaregraphs
• Large-type Artin groups are systolic
• A coarse Cartan-Hadamard theorem with application to the coarse Baum-Connes conjecture
• Realisation and dismantlability
• Obstructions to a small hyperbolicity in Helly graphs
• The Farrell-Jones conjecture for hyperbolic and CAT(0)-groups revisited
• Gluing hyperconvex metric spaces

Cited by in corpus (1)

• Asymptotic cones of snowflake groups and the strong shortcut property