Asymptotic dimension and the disk graph II

TL;DR

This paper establishes that the asymptotic dimension of hyperbolic relatively hyperbolic graphs is finite under certain conditions and applies this to show the disk graph of a handlebody has at most quadratic asymptotic dimension in the genus.

Contribution

It proves the finiteness of the asymptotic dimension for a class of hyperbolic graphs and determines an upper bound for the disk graph of handlebodies.

Findings

Asymptotic dimension of hyperbolic relatively hyperbolic graphs is finite.

The asymptotic dimension of the disk graph of a handlebody is at most quadratic in the genus.

Provides a new connection between hyperbolic geometry and topological properties of handlebodies.

Abstract

We show that the asymptotic dimension of a hyperbolic relatively hyperbolic graph is finite provided that this holds true uniformly for the peripheral subgraphs and for the electrifiation. We use this to show that the asymptotic dimension of the disk graph of a handlebody of genus at least two is at most quadratic in the genus.