Spotted disk and sphere graphs

TL;DR

This paper investigates the geometric properties of disk and sphere graphs associated with handlebodies, revealing the existence of high-dimensional Euclidean quasi-isometric embeddings, which inform the understanding of their large-scale geometry.

Contribution

It demonstrates that the disk graph for certain handlebodies contains quasi-isometric embeddings of  and that the sphere graph of doubled handlebodies contains embeddings of ^n, highlighting their complex geometric structure.

Findings

Disk graph contains quasi-isometric  embeddings for m=2.

Sphere graph contains quasi-isometric ^n embeddings for genus .

Reveals high-dimensional Euclidean structures within these graphs.

Abstract

The disk graph of a handlebody H of gneus $g\geq 2$ with $m\geq 0$ 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 of length one if they can be realized disjointly. We show that for m=2 the disk graph contains quasi-isometrically embedded copies of $\mathbb{R}^2$. Furthermore, the sphere graph of the doubled handlebody of genus $g\geq 4$ with two marked points contains for every $n\geq 1$ a quasi-isometrically embedded copy of $\mathbb{R}^n$.