Homology Covers and Automorphisms: Examples

Abstract

Let $S$ be a Riemann surface with a non-abelian fundamental group and for each integer $k\ge 2$ or $k=\infty$, 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 ${\mathbb{Z}}_{\infty }:=\mathbb{Z}$, such that $S={\tilde{S}}_k/M_k$. If $L\le \mathrm{Aut}(S)$, 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.

Changes in the Abstract and the Introduction. Part of the original introduction is now in Section 2. Some edition in Section 3. Some mistakes in Example 5 were corrected