Dyer groups have the falsification by fellow-traveller property

TL;DR

This paper proves that Dyer groups possess the falsification by fellow-traveller property by analyzing their Cayley graphs, highlighting their structural similarities to Coxeter groups and graph products of cyclic groups.

Contribution

It introduces a finite generating set for Dyer groups with a Cayley graph as a locally finite mediangle graph, establishing the FFTP for these groups.

Findings

Dyer groups have the FFTP.

Dyer groups have finitely many cone types.

Dyer groups unify properties of Coxeter groups and graph products.

Abstract

This paper is devoted to the study of the falsification by fellow-traveller property (FFTP) in Dyer groups. We exhibit a finite generating set for which the associated Cayley graph is a locally finite mediangle graph, and leverage its properties to prove that Dyer groups have the FFTP. It follows that Dyer groups have finitely many cone types, emphasising their role in providing a unified approach to Coxeter groups and graph products of cyclic groups.