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.