Nikolaus Conference 2018

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.