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 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 MCG$(\Sigma)$ consisting of those elements in the Torelli group which have compact support. Next, we prove the abstract commensurator group of ${\mathcal I}(\Sigma)$ coincides with MCG$(\Sigma)$. This extends the results for finite-type surfaces \[9, 6, 7, 16] to the setting of infinite-type surfaces.