Homology Covers and Automorphisms: Examples

Let $S$ be a Riemann surface with a non-abelian fundamental group and for each integer $k\ge 2$ or $k=\mathrm{\infty }$, let ${\tilde{S}}_k$ be its $k$-homology cover. The surface ${\tilde{S}}_k$ admits a group of conformal automorphisms $M_k\cong {\mathrm{H}}_1(S;{\mathbb{Z}}_k)$, where ${\mathbb{Z}}_{\mathrm{\infty }}:=\mathbb{Z}$, such that $S={\tilde{S}}_k\mathrm{/}{M}_k$. If $L\le \mathrm{A}\mathrm{u}\mathrm{t}(S)$, then there is a short exact sequence $1\to M_k\to {\tilde{L}}_k\to L\to 1$, where ${\tilde{L}}_k$ is a subgroup of conformal automorphisms of ${\tilde{S}}_k$. In general, the above exact sequence does not need to be split. This paper investigates situations when the splitting is or is not obtained.