Hyperbolicity of curve graphs of Artin-Tits groups of type B and \tilde A

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 n-times punctured disc when A is the Artin braid group on n-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 question. In this talk we address this question for the groups of type B_n and \tilde A_n.

We show that the graph of irreducible parabolic subgroups associated to the Artin-Tits grup of type B_n is isomorphic to the curve graph of the (n+1)-times punctured disc, hence it is hyperbolic. For the type \tilde A_n we show that it is (not quasi-isometrically) embedded in the curve graph of the (n+2)-times punctured disc, nonetheless we prove that it is hyperbolic.

Joint work with Matthieu Calvez.