Geometry of the augmented disk graph

Abstract

AbstractFor a handlebodyH, we define two graphs, the augmented disk graph$\mathcal{ADG}(H)$and the truncated augmented disk graph$\mathcal{TADG}(H)$, and we show they are hyperbolic in the sense of Gromov. In the process, we show they are quasi-isometric to two other disk graphs defined by U. Hamenstädt, the super conducting disk graph$\mathcal{SDG}(H)$and the electrified disk graph$\mathcal{EDG}(H)$respectively. So we reprove two theorems of Hamenstädt [12].Our approach uses techniques from Masur–Schleimer's study on the hyperbolicity of the disk graph$\mathcal{DG}(H)$[21].