Automorphisms and subdivisions of Helly graphs

TL;DR

This paper investigates the structure of Helly graphs with finite combinatorial dimension, providing explicit simplicial models of their injective hulls and analyzing automorphisms, including their classification and properties.

Contribution

It introduces explicit simplicial models of the injective hulls of Helly graphs and characterizes automorphisms as elliptic or hyperbolic with rational translation lengths.

Findings

Automorphisms are classified as elliptic or hyperbolic.

Hyperbolic automorphisms have axes in Helly subdivisions.

Translation lengths of hyperbolic automorphisms are rational with bounded denominators.

Abstract

We study Helly graphs of finite combinatorial dimension, i.e. whose injective hull is finite-dimensional. We describe very simple fine simplicial subdivisions of the injective hull of a Helly graph, following work of Lang. We also give a very explicit simplicial model of the injective hull of a Helly graphs, in terms of cliques which are intersections of balls. We use these subdivisions to prove that any automorphism of a Helly graph with finite combinatorial dimension is either elliptic or hyperbolic. Moreover, every such hyperbolic automorphism has an axis in an appropriate Helly subdivision, and its translation length is rational with uniformly bounded denominator.