The moduli spaces of curves, via the arithmetic of the Knudsen-Mumford

stratification and their étale fundamental groups representation, have a

long tradition in the study of the absolute Galois group of rationals.

This led in particular to the creation of Grothendieck-Teichmüller theory,

that in return provides a group theoretic approach to arithmetic questions.

The goal of this talk is to present how the GT and arithmetic approaches

lead to new arithmetic results in the study of the stack stratification

given by the automorphism group of curves.

I will begin by showing how Grothendieck-Teichmüller theory, in relation

with explicit presentations and fundamental properties of the mapping class

group of surfaces of low genus, gives a result for the first stack strata.

I will then explain how the switch to a geometric context extends this

result to the generic cyclic strata in every genus.

My presentation will also illustrate how this stack arithmetic is organized

around two essential arithmetic questions i.e., for the moduli spaces to be

an étale K(\pi,1) and the rationality of irreducible components in Hurwitz

spaces.

### Galois Seminar

Friday, February 9, 2018 - 3:15pm

#### Benjamin Collas

Univ. Bayreuth