Algebraic characterisation of relatively hyperbolic 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 ${\mathbb{F}}_2\times \mathbb{Z}$.

arXiv comment28 pages, 3 figures. The section dedicated to graph braid groups in the first version is now contained in arXiv:1912.10674. To appear in Israel J. Math