Helly groups

By Jérémie Chalopin, Victor Chepoi, Anthony Genevois, Hiroshi Hirai, Damian Osajda. Geometry & Topology, 2020

47 citations recorded by the source corpus.

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.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/6e86261140348d53d20b6ca43c75ce7c51ed56bc

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In Discrete Mathematics and Combinatorics

  1. Damian Osajda
  2. Jingyin Huang

In Geometry and Topology

  1. Jingyin Huang
  2. Damian Osajda

In Mathematical Physics

  1. Jingyin Huang
  2. Damian Osajda

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