Algebraic characterisation of relatively hyperbolic special groups

TL;DR

This paper characterizes when special groups are relatively hyperbolic using a new combinatorial approach, linking subgroup properties to hyperbolicity relative to a collection of subgroups.

Contribution

It introduces a novel combinatorial formalism to determine relative hyperbolicity of special groups based on subgroup convex-cocompactness, malnormality, and containment conditions.

Findings

A cocompact special group is relatively hyperbolic iff subgroups are convex-cocompact and almost malnormal.

Virtually cocompact special groups are relatively hyperbolic relative to abelian subgroups iff they lack _2 imes Z.

New criteria connect subgroup properties to the hyperbolic structure of special groups.

Abstract

This article is dedicated to the characterisation of the relative hyperbolicity of Haglund and Wise's special groups. More precise, we introduce a new combinatorial formalism to study (virtually) special groups, and we prove that, given a cocompact special group $G$ and a finite collection of subgroups $\mathcal{H}$, then $G$ is hyperbolic relative to $\mathcal{H}$ if and only if (i) each subgroup of $\mathcal{H}$ is convex-cocompact, (ii) $\mathcal{H}$ is an almost malnormal collection, and (iii) every non-virtually cyclic abelian subgroup of $G$ is contained in a conjugate of some group of $\mathcal{H}$. As an application, we show that a virtually cocompact special group is hyperbolic relative to abelian subgroups if and only if it does not contain $\mathbb{F}_2 \times \mathbb{Z}$.