The non-peripheral curve graph and divergence in big mapping class groups

TL;DR

This paper introduces a new invariant and a non-peripheral curve graph to analyze the large-scale geometry of big mapping class groups, revealing hyperbolic properties and divergence behavior.

Contribution

It defines the invariant $z( extSigma)$ and the non-peripheral curve graph to study the coarse geometry of big mapping class groups, establishing hyperbolicity and divergence results.

Findings

The non-peripheral curve graph is connected and Gromov hyperbolic.

Mapping class groups have infinite coarse rank when $z( extSigma)  4$.

Groups have at most quadratic divergence when $z( extSigma)  5$.

Abstract

We introduce a numerical invariant $\zeta(\Sigma)$ measuring the end-complexity of $\Sigma$ and use it to organize coarse-geometric features of Map($\Sigma$). Our main tool is the \emph{non-peripheral curve graph} $C_{\rm np}(\Sigma)$, whose vertices are those essential simple closed curves that cannot be pushed out of every compact subsurface, with edges given by disjointness. Assuming Map($\Sigma$) is CB-generated and $\zeta(\Sigma)\ge 5$, we prove that $C_{\rm np}(\Sigma)$ is connected, has infinite diameter, is Gromov hyperbolic, and that the Map($\Sigma$)-action has unbounded orbits. As applications, we show that if $\zeta(\Sigma)\ge 4$ then Map($\Sigma$) has infinite coarse rank, and if $\zeta(\Sigma)\ge 5$ then Map($\Sigma$) has at most quadratic divergence, hence is one-ended.