Crystallographic Helly Groups

TL;DR

This paper explores the properties of Helly groups and spaces, proving their asymptotic cones are hyperconvex, characterizing virtually abelian Helly groups, and providing the first example of a systolic group that is not Helly.

Contribution

It establishes new geometric characterizations of Helly groups and spaces, and provides the first example of a systolic group that is not Helly.

Findings

Asymptotic cones of Helly graphs are countably hyperconvex.

Virtually nilpotent Helly groups are virtually abelian.

The 3-3-3-Coxeter group is not Helly or coarsely injective.

Abstract

We prove that asymptotic cones of Helly graphs are countably hyperconvex. We use this to show that virtually nilpotent Helly groups are virtually abelian and to characterize virtually abelian Helly groups via their point groups. In fact, we do this for the more general class of coarsely injective spaces and groups. We apply this to prove that the $3$-$3$-$3$-Coxeter group is not Helly (nor even coarsely injective), thus obtaining the first example of a systolic group that is not Helly, answering a question of Chalopin, Chepoi, Genevois, Hirai, and Osajda.