Algebraic characterisation of relatively hyperbolic special groups

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