Big Torelli groups: generation and commensuration

Abstract

For any surface $\Sigma$ of infinite topological type, we study the Torelli subgroup $\mathcal{I}(\Sigma )$ of the mapping class group $\mathrm{MCG}(\Sigma )$, whose elements are those mapping classes that act trivially on the homology of $\Sigma$. Our first result asserts that $\mathcal{I}(\Sigma )$ is topologically generated by the subgroup of $\mathrm{MCG}(\Sigma )$ consisting of those elements in the Torelli group which have compact support. In particular, using results of Birman, Powell, and Putman we deduce that $\mathcal{I}(\Sigma )$ is topologically generated by separating twists and bounding pair maps. Next, we prove the abstract commensurator group of $\mathcal{I}(\Sigma )$ coincides with $\mathrm{MCG}(\Sigma )$. This extends the results for finite-type surfaces of Farb-Ivanov, Brendle-Margalit and KIda to the setting of infinite-type surfaces.

Made changes suggested by the referee. To appear in Groups, Geometry, and Dynamics