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

Abstract

Abstract The 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 of the graph of irreducible parabolic subgroups for more general Artin–Tits groups is an important open question. In this paper, we give a partial affirmative answer. For n⩾3, we show that the graph of irreducible parabolic subgroups associated to the Artin–Tits group of spherical type Bn is also isomorphic to the curve graph of the (n+1)‐times punctured disk; hence, it is hyperbolic. For n⩾2, we show that the graphs of irreducible parabolic subgroups associated to the Artin–Tits groups of euclidean type A∼n and 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.

Keywords

• 20F36
• 20F65 (primary)
• 20F67 (secondary)