Monodromy kernels for strata of abelian differentials

by Prof. Riccardo Giannini (University of Glasgow)

Wednesday 20 Sept 2023, 15:00 → 16:00 Europe/Rome

Sala Conferenze - Palazzo Puteano (CRM - SNS)

In this talk, we are going to talk about the topology of the moduli space of translation surfaces. These are closed Riemann surfaces with a flat metric on the complement of some finite number of points, around which the metric is given by cyclically gluing a finite number of half-planes. Their moduli space can be stratified in orbifolds, and there are connected components consisting of hyperelliptic Riemann surfaces that are topologically well-understood. However, studying the topology of the non-hyperelliptic components proves to be more intricate. It has been conjectured that the (orbifold) fundamental groups are commensurable with some mapping class groups, but it is not known whether or not the natural monodromy map $\rho_\mathcal{C}:\pi_1^{orb}(\mathcal{C})\rightarrow\operatorname{Mod}(\Sigma_g,\mathcal{Z})$ of a non-hyperelliptic component $\mathcal{C}$ is an isomorphism onto its image. In this talk, we are going to show why the kernel of $\rho_\mathcal{C}$ in genus $3$ is huge for some connected components $\mathcal{C}$, as it contains a free group of rank $2$. In particular, the result is a consequence of a more general phenomenon related to Artin groups.