Homology Covers and Automorphisms: Examples

Let $S$ be a Riemann surface with a non-abelian fundamental group and for each integer $k \geq 2$ or $k=\infty$, let $\widetilde{S}_{k}$ be its $k$-homology cover. The surface $\widetilde{S}_{k}$ admits a group of conformal automorphisms $M_{k} \cong {\rm H}_{1}(S;{\mathbb Z}_{k})$, where ${\mathbb Z}_{\infty}:={\mathbb Z}$, such that $S=\widetilde{S}_{k}/M_{k}$. If $L \leq {\rm Aut}(S)$, then there is a short exact sequence $1 \to M_{k} \to \widetilde{L}_{k} \to L \to 1$, where $\widetilde{L}_{k}$ is a subgroup of conformal automorphisms of $\widetilde{S}_{k}$. In general, the above exact sequence does not need to be split. This paper investigates situations when the splitting is or is not obtained.