Subrepresentations in the homology of finite covers of graphs

Let p : Y -> X be a finite, regular cover of finite graphs with associated deck group G, and consider the first homology H-1(Y; C) of the cover as a G-representation. The main contribution of this article is to broaden the correspondence and dictionary between the representation theory of the deck group G on the one hand and topological properties of homology classes in H-1(Y; C) on the other hand. We do so by studying certain subrepresentations in the G-representation H-1(Y; C). The homology class of a lift of a primitive element in pi(1)(X) spans an induced subrepresentation in H-1(Y; C), and we show that this property is never sufficient to characterize such homology classes if G is Abelian. We study H-1(comm) (Y; C) <= H-1(Y; C)-the subrepresentation spanned by homology classes of lifts of commutators of primitive elements in pi(1)(X). Concretely, we prove that the span of such a homology class is isomorphic to the quotient of two induced representations. Furthermore, we construct examples of finite covers with H-1(comm) (Y; C) not equal ker(p(*)).