Monday 21 & Monday 28 November 2022
Martin Palmer-Anghel (IMAR) — On the homology of big mapping class groups, I, II
A surface has "infinite type" if its fundamental group is not finitely generated. This may be because it has infinite genus or because its space of ends is infinite, such as for the Cantor tree surface. Mapping class groups of surfaces of infinite type ("big mapping class groups") arise naturally in dynamics and are also related to the study of Thompson's groups and other "fractal groups".
A natural challenge is to understand the homology of big mapping class groups: this has been studied in degree 1 in many cases and computed (by Calegari-Chen) up to degree 2 in the case of the Cantor tree surface. I will describe recent joint work with Xiaolei Wu (arxiv:2211.07470) in which we completely compute the homology of many big mapping class groups (in all degrees), in particular for the plane minus a Cantor set. In the cases where we compute the homology, it is either zero or periodic but finitely generated in each degree. To contrast this, I will also describe further joint work in progress with Xiaolei Wu in which we prove that the homology of many (other) big mapping class groups is uncountable in all degrees.
* Eastern European Time, i.e. UTC+2 in winter and UTC+3 in summer.