Homology Covers and Automorphisms: Examples
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866913595056455680 |
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| author | Hidalgo, Rubén A. |
| author_facet | Hidalgo, Rubén A. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_05442 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Homology Covers and Automorphisms: Examples Hidalgo, Rubén A. Geometric Topology Complex Variables 30F10, 30F40, 14H37 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. |
| title | Homology Covers and Automorphisms: Examples |
| topic | Geometric Topology Complex Variables 30F10, 30F40, 14H37 |
| url | https://arxiv.org/abs/2407.05442 |