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

Abstract

Let $S$ be an infinite-type surface and let $G\le \mathrm{Map}(S)$ 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 $G\ge {\mathrm{PMap}}_c(S)$ is boundedly generated or $G\in \{\overline{{\mathrm{PMap}}_c(S)},\mathrm{PMap}(S),\mathrm{Map}(S)\}$ and is locally bounded, then $\mathrm{asdim}M=\mathrm{asdim}G=\infty$. This result completes the classification of the asymptotic dimension of stable boundedly generated infinite-type surface mapping class groups begun by Grant--Rafi--Verberne.

18 pages, 3 figures