Asymptotic Dimension of Big Mapping Class Groups

TL;DR

This paper investigates the large-scale geometric properties of big mapping class groups, showing that certain infinite-type surface groups have infinite asymptotic dimension, contrasting with finite-type cases.

Contribution

It establishes that big mapping class groups with essential shifts have infinite asymptotic dimension, providing a topological characterization of these shifts.

Findings

Big mapping class groups with essential shifts have infinite asymptotic dimension.

Finite-type surface mapping class groups have finite asymptotic dimension.

Provides a topological characterization of essential shifts.

Abstract

Even though big mapping class groups are not countably generated, certain big mapping class groups can be generated by a coarsely bounded set and have a well defined quasi-isometry type. We show that the big mapping class group of a stable surface of infinite type with a coarsely bounded generating set that contains an essential shift has infinite asymptotic dimension. This is in contrast with the mapping class groups of surfaces of finite type where the asymptotic dimension is always finite. We also give a topological characterization of essential shifts.