Subrepresentations in the homology of finite covers of graphs

Let $p\text{\hspace{0.17em}⁣}:Y\to X$ be a finite, regular cover of finite graphs with associated deck group $G$, and consider the first homology $H_1(Y;\mathbb{C})$ 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 $H_1(Y;\mathbb{C})$ on the other hand. We do so by studying certain subrepresentations in the $G$-representation $H_1(Y;\mathbb{C})$. The homology class of a lift of a primitive element in ${\pi }_1(X)$ spans an induced subrepresentation in $H_1(Y;\mathbb{C})$, and we show that this property is never sufficient to characterize such homology classes if $G$ is Abelian. We study $H_1^{\text{comm}}(Y;\mathbb{C})\le H_1(Y;\mathbb{C})$ – 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 $H_1^{\text{comm}}(Y;\mathbb{C})\mathrm{\ne }\ker (p_{\ast })$.