Curve graphs for Artin–Tits groups of type B, A∼ and C∼ are hyperbolic

Abstract

The \emph{graph of irreducible parabolic subgroups} is a combinatorial object associated to an Artin-Tits group $A$ defined so as to coincide with the curve graph of the $(n+1)$-times punctured disk when $A$ is Artin's braid group on $(n+1)$ strands. In this case, it is a hyperbolic graph, by the celebrated Masur-Minsky's theorem. Hyperbolicity for more general Artin-Tits groups is an important open question. In this paper, we give a partial affirmative answer. For $n\geqslant 3$, we prove that the graph of irreducible parabolic subgroups associated to the Artin-Tits group of spherical type $B_n$ is also isomorphic to the curve graph of the $(n+1)$-times punctured disk, hence it is hyperbolic. For $n\geqslant 2$, we show that the graphs of irreducible parabolic subgroups associated to the Artin-Tits groups of euclidean type $\widetilde A_n$ and $\widetilde C_n$ are isomorphic to some subgraphs of the curve graph of the $(n+2)$-times punctured disk which are not quasi-isometrically embedded. We prove nonetheless that these graphs are hyperbolic.