Spotted disk and sphere graphs I

TL;DR

This paper investigates the geometric properties of disk and sphere graphs associated with handlebodies, revealing the presence of quasi-isometrically embedded Euclidean planes under certain conditions.

Contribution

It demonstrates that for specific cases, the disk and sphere graphs contain quasi-isometrically embedded copies of R^2, highlighting their complex geometric structure.

Findings

Disk graph with one marked point contains quasi-isometric R^2

Sphere graph of double handlebody with even genus contains quasi-isometric R^2

Results reveal complex geometric embeddings in these graphs

Abstract

The disk graph of a handlebody H of genus g at least 2 with m marked points on the boundary is the graph whose vertices are isotopy classes of disks disjoint from the marked points and where two vertices are connected by an edge if they can be realized disjointly. We show that for m=1 the disk graph contains quasi-isometrically embedded copies of R^2. The same holds true for the sphere graph of the double handlebody with one marked point provided that g is even.