Geometric models and asymptotic dimension for infinite-type surface mapping class groups

TL;DR

This paper constructs a geometric model for infinite-type surface mapping class groups, showing that under certain conditions, these groups have infinite asymptotic dimension, extending previous classifications.

Contribution

It introduces a new metric graph model for these groups and establishes their infinite asymptotic dimension in broad cases, completing prior classification efforts.

Findings

Constructed a metric graph preserved by the group action.

Proved the asymptotic dimension of the group and the graph is infinite.

Extended classification of asymptotic dimensions for infinite-type surface groups.

Abstract

Let $S$ be an infinite-type surface and let $G \leq \operatorname{Map}(S)$ be a locally bounded Polish subgroup. We construct a metric graph $M$ of simple arcs and curves on $S$ preserved by the action of $G$ and for which the vertex orbit map $G \to V(M)$ is a coarse equivalence; if $G$ is boundedly generated, then $M$ is a Cayley--Abels--Rosendal graph for $G$ and the orbit map is a quasi-isometry. In particular, if $S$ contains a non-displaceable subsurface and $G \geq \operatorname{PMap}_c(S)$ is boundedly generated or $G \in \{\overline{\operatorname{PMap}_c(S)}, \operatorname{PMap}(S), \operatorname{Map}(S) \}$ and is locally bounded, then $\operatorname{asdim} M = \operatorname{asdim} G = \infty$. This result completes the classification of the asymptotic dimension of stable boundedly generated infinite-type surface mapping class groups begun by Grant--Rafi--Verberne.