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 precisely, 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 \({\cal H}\), then G is hyperbolic relative to \({\cal H}\) if and only if (i) each subgroup of \({\cal H}\) is convex-cocompact, (ii) \({\cal H}\) is an almost malnormal collection, and (iii) every non-virtually cyclic abelian subgroup of G is contained in a conjugate of some group of \({\cal H}\). 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 \({\mathbb{F}_2} \times \mathbb{Z} \).