Spotted disk and sphere graphs. II (Q6561671)

This article studies the coarse geometry of disk graphs associated to spotted handlebodies. A \textit{spotted handlebody} is the three-manifold obtained as the regular neighbourhood of a finite graph in \(3\)-space, together with a finite number of marked disks (``spots'') on the boundary. The \textit{disk graph} of such a handlebody is the graph whose vertices correspond to isotopy classes of properly embedded disks in the handlebody, whose boundary curves are essential in the boundary surface, and with edges corresponding to disjoint disks.\N\NThe main result (Theorem 2) of the paper shows that these graphs are not Gromov hyperbolic for handlebodies of genus \(g \geq 2\) with two spots (extending previous results for handlebodies with one spot [\textit{U. Hamenstädt}, New York J. Math. 27, 881--902 (2021; Zbl 1471.57022)]). A similar result (Theorem 3) holds for sphere graphs of doubled handlebodies with two marked points. In both cases, the reason for non-hyperbolicity is the existence of quasi-isometrically embedded Euclidean spaces.\N\NIn the sphere graph case, the dimension of such spaces in unbounded, showing that the asymptotic dimension is infinite. These results are in sharp contrast to the case of handlebodies (or doubled handlebodies) without spots, in which case the corresponding graphs are known to be hyperbolic [\textit{H. Masur} and \textit{S. Schleimer}, J. Am. Math. Soc. 26, No. 1, 1--62 (2013; Zbl 1272.57015); \textit{U. Hamenstädt}, Groups Geom. Dyn. 10, No. 1, 365--405 (2016; Zbl 1332.05045); J. Topol. 12, No. 2, 658--673 (2019; Zbl 1423.57038); \textit{M. Handel} and \textit{L. Mosher}, Geom. Topol. 17, No. 3, 1581--1672 (2013; Zbl 1278.20053)].