Title: Generating tuples of Fuchsian groups

Abstract: Generating n-tuples of a group $G$, or in other words epimorphisms $F_n\to G$ are usually studied up to the natural right action of $Aut(F_n)$ on $Epi(F_n,G)$; here $F_n$ is the free group of $n$ generators. The orbits are then called Nielsen classes. It is a classic result of Nielsen that for any $n\ge k$ there is exactly one Nielsen class of generating $n$-tuples of $F_k$. This result was generalized to surface groups by Louder.

In this talk the case of Fuchsian groups is discussed. It turns out that the situation is much more involved and interesting. While uniqueness does not hold in general one can show that each class is represented by some unique geometric object called an "almost orbifold covers". This can be thought of as a classification of Nielsen classes. This is joint work with Ederson Dutra.