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

Abstract

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

37 pages, 3 figures