Asymptotic dimension and the disk graph I

TL;DR

This paper investigates the relationship between the asymptotic dimension of the disk graph associated with a 3-manifold and the curve graph of a boundary subsurface, using surgery techniques to establish a quasi-isometric embedding.

Contribution

It introduces a method to identify a disk graph embedded in the curve graph via surgery, linking the asymptotic properties of these complexes.

Findings

The disk graph is quasi-isometrically embedded in the curve graph.

The approach applies to 3-manifolds with boundary subsurfaces that are either empty or incompressible.

The work provides new insights into the geometric structure of disk and curve graphs.

Abstract

For a 3-manifold M and a subsurface $X$ of the boundary of M with empty or incompressible boundary we use surgery to identify a graph whose vertices are disks with boundary in X and which is quasi-isometrically embedded in the curve graph of X.