# Nikolaus Conference 2018

Speaker: Xenia Flamm (Bonn)

Title: Homology of Finite Covers of Graphs

Abstract:

Let $$N$$ be a normal subgroup of finite index in the free group $$F_n$$. Then the finite group $$G := F_n/N$$ acts on the free abelian group $$N/N'$$ and thus also on $$N/N' \otimes \mathbb{C}$$. A result by Gaschütz shows that this representation is isomorphic to $$n-1$$ copies of the regular representation and one copy of the trivial representation. We will define subrepresentations coming from interesting subsets of $$F_n$$, namely the subset of primitive elements and the subset of commutators of primitive elements. A natural question to ask is whether these subrepresentations generate $$N/N' \otimes \mathbb{C}$$. I will present some results by Farb-Hensel and Malestein-Putman, and talk about recent contributions of my master's thesis. The results are motivated by the investigation of finite covers of graphs.