Helly groups

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

Abstract A note on r-dominating cliques 1

Let M be a finite subset of vertices of a connected graph G and assume that every vertex v E M has a dominating radius r(v)E N U {0}. A complete subgraph C is an r-dominating clique of M if every vertex vEM is at distance at most r(v) from C. Even for r(v) ~ 1 the problem whether or not a given graph has an r-dominating clique is NP-complete. Evidently, if M admits an r-dominating clique then d(u, v) < ~ r(u) + r(v) + 1 for any u, v E M. We characterize the graphs G for which this condition guarantees the existence of r-dominating cliques not only in G but also in all isometric subgraphs of G comprising M. These are the graphs which do not contain the house, the 3-deltoid, or any n-cycle with n ~> 5 as an isometric subgraph. I. Introduction and the result In recent years several kinds of domination problems in graphs have been investi-gated. In all of them it is necessary to find a subgraph Q with a prescribed structure which dominates all or certain vertices of a graph G. Recall that Q dominates a subset of vertices M if every vertex of M outside Q is adjacent to some vertex of Q. In