ALGEBRAIC CHARACTERISATION OF RELATIVELY HYPERBOLIC SPECIAL GROUPS

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 H, then G is hyperbolic relative to H if and only if (i) each subgroup of H is convex-cocompact, (ii) 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 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 F-2 x Z.