I will discuss structures for relative completions of mapping class groups. In higher genus (genus at least 3) cases, presentations are known, and will be reviewed first. The only unknown case is in genus two, where I am jointly working with Tatsunari Watanabe. In this case, the generators are related to Collino cycles. This is similar in higher genus cases where generators are related to Ceresa cycles. Moreover in our case, we provide evidence to generate relations via arithmetic considerations, while in higher genus cases, relations are quite topological.

罗马助理教授（华东师范大学）