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.

Figures (6)

• Figure 1: The $3$-sun can be obtained from the amalgam of a triangle and a 3-fan over an edge.
• Figure 2: A house (left) and a $3$-deltoid (right).
• Figure 3: The proof of Lemma \ref{['lem:intersections']}. From left to right: (1), (2), (3).
• Figure 4: The proof of Lemma \ref{['lem:intersections']}(1).
• Figure 5: The proof of Lemma \ref{['l:c4t4strongH']}.

Theorems & Definitions (206)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Proposition 1.7
• Lemma 2.1
• Theorem 2.2
• Proposition 2.3