Purely non-free actions on Riemann surfaces
Abstract
A group of conformal automorphisms of a Riemann surface is said to act purely non-free, or to produce a gpnf-action, if each of its finite-order elements has fixed points. It is known that every finite group admits a gpnf-action on a suitable closed Riemann surface of genus at least two. If G is an infinite countable group, and X is an infinite-type orientable surface, without planar ends and whose space of ends is self-similar, then we observe that there is a Riemann surface structure on X admitting a gpnf-action by G. We obtain upper bounds for those finite groups producing a gpnf-action on a closed Riemann surface of genus g ≥ 2. If G is a finite group, then its purely non-free genus σpnfhyp(G) is the minimum genus of at least two of a closed Riemann surface admitting a gpnf-action by G. This parameter is already known for cyclic, abelian non-cyclic groups, 𝒜4, 𝒜5 and S4. For general G, we provide the value of σpnfhyp(G), correcting a previous statement on this value. In particular, we observe that every genus is the purely non-free genus of some group. As an application, we describe the minimal purely non-free actions for the family of metacyclic groups G0s(n)=⟨ x,y: xn=y2=1, yxy=xs⟩, where n ≥ 3 and 2 ≤ s ≤ n−1 is such that s2 ≡ 1 mod(n).
Related
- Dimers and M-curves: Limit shapes from Riemann surfaces
- Prescribed mean curvature surfaces of Riemann minimal type
- Finiteness of Fuchsian equations with unitarizable monodromy on compact Riemann surfaces
- Uniform Green packing on compact Riemann surfaces
- Eynard-Orantin Topological Recursions for Higher Weil-Petersson Volumes on Moduli Spaces of Super Riemann Surfaces
BibTeX
@misc{hidalgo2026purely,
title = {{Purely non-free actions on Riemann surfaces}},
author = {Rubén A. Hidalgo and Jhony S. Leyca},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.17405},
eprint = {2610.17405},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.17405},
}
RIS
TY - UNPB
TI - Purely non-free actions on Riemann surfaces
AU - Rubén A. Hidalgo
AU - Jhony S. Leyca
PY - 2026
DA - 2026/10/12
PB - arXiv
DO - 10.48550/arxiv.2610.17405
UR - https://arxiv.org/abs/2610.17405
AB - A group of conformal automorphisms of a Riemann surface is said to act purely non-free, or to produce a gpnf-action, if each of its finite-order elements has fixed points. It is known that every finite group admits a gpnf-action on a suitable closed Riemann surface of genus at least two. If G is an infinite countable group, and X is an infinite-type orientable surface, without planar ends and whose space of ends is self-similar, then we observe that there is a Riemann surface structure on X admitting a gpnf-action by G. We obtain upper bounds for those finite groups producing a gpnf-action on a closed Riemann surface of genus g ≥ 2. If G is a finite group, then its purely non-free genus σ^pnf_hyp(G) is the minimum genus of at least two of a closed Riemann surface admitting a gpnf-action by G. This parameter is already known for cyclic, abelian non-cyclic groups, 𝒜_4, 𝒜_5 and S_4. For general G, we provide the value of σ^pnf_hyp(G), correcting a previous statement on this value. In particular, we observe that every genus is the purely non-free genus of some group. As an application, we describe the minimal purely non-free actions for the family of metacyclic groups G^0_s(n)=⟨ x,y: x^n=y^2=1, yxy=x^s⟩, where n ≥ 3 and 2 ≤ s ≤ n−1 is such that s^2 ≡ 1 mod(n).
ER -