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 \({\mathbb{F}_2} \times \mathbb{Z} \).

本文致力于描述Haglund和Wise的特殊群体的相对双曲性。更确切地说，我们引入了一种新的组合形式主义来研究（虚拟地）特殊组，并且我们证明，给定一个紧紧的特殊组G和一个有限子集\（{\ cal H} \），则G相对于\（{\ cal H} \）当且仅当（i）\（{\ cal H} \）的每个子组是凸-紧紧的，（ii）\（{\ cal H} \}是几乎不正常的集合，以及（iii）G的每个非虚拟循环阿贝尔亚群都包含在\（{\ cal H} \）的一组共轭中。作为一个应用程序，我们证明，当且仅当一个不包含cocompact的特殊组不包含\（{\ mathbb {F} _2} \ times \ mathbb {Z} \）时，它才相对于abelian子组是双曲的。