Dyer groups: Centres, hyperbolicity, and acylindrical hyperbolicity

TL;DR

This paper characterizes the centers of Dyer groups and provides a complete classification of their hyperbolic and acylindrically hyperbolic properties based on the Dyer graph structure.

Contribution

It offers a comprehensive description of centers and a clear classification criteria for hyperbolicity in Dyer groups, advancing understanding of their geometric properties.

Findings

Centers of all Dyer groups are described.

Conditions for hyperbolicity are fully classified.

Conditions for acylindrical hyperbolicity are fully classified.

Abstract

In this article we describe the centres of all Dyer groups. We also give a complete classification of when a Dyer group $D(\Gamma)$ is hyperbolic or acylindricality hyperbolic, with conditions that can easily be read on the Dyer graph $\Gamma$.