Big Torelli groups: generation and commensuration

TL;DR

This paper investigates the Torelli subgroup of the mapping class group for infinite-type surfaces, showing it is generated by specific elements and that its commensurator group equals the entire mapping class group.

Contribution

It extends finite-type surface results to infinite-type surfaces, demonstrating generation by separating twists and bounding pair maps, and characterizing the commensurator group.

Findings

Torelli subgroup is topologically generated by compact support elements.

Generated by separating twists and bounding pair maps.

Commensurator group equals the entire mapping class group.

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.