Homology Covers and Automorphisms: Examples

TL;DR

This paper explores the automorphism groups of homology covers of Riemann surfaces, analyzing when certain exact sequences split or do not, thereby deepening understanding of surface symmetries and automorphism extensions.

Contribution

It provides new insights into the conditions under which the automorphism group sequences of homology covers split or remain non-split, extending previous knowledge on surface automorphisms.

Findings

Identifies conditions for splitting of automorphism sequences

Provides examples where sequences do not split

Analyzes automorphism group structures of homology covers

Abstract

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.