On the asymptotic dimension of the curve complex

TL;DR

This paper establishes a linear bound related to the complexity of the surface on the asymptotic dimension of the curve complex and the capacity dimension of the ending lamination space.

Contribution

It provides the first linear bounds on the asymptotic and capacity dimensions in these geometric structures.

Findings

Linear bound on the asymptotic dimension of the curve complex

Linear bound on the capacity dimension of the ending lamination space

Improves understanding of the geometric complexity of surface-related spaces

Abstract

We give a bound, linear in the complexity of the surface, on the asymptotic dimension of the curve complex as well as the capacity dimension of the ending lamination space.