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 ${\rm MCG}(\Sigma)$, 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 ${\rm MCG}(\Sigma)$ 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 ${\rm MCG}(\Sigma)$. This extends the results for finite-type surfaces of Farb-Ivanov, Brendle-Margalit and KIda to the setting of infinite-type surfaces.