On the biggest purely non-free conformal actions on compact Riemann surfaces and their asymptotic properties
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| Format: | Preprint |
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2026
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| _version_ | 1866915982158594048 |
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| author | Bagiński, C. Gromadzki, G. Hidalgo, R. A. |
| author_facet | Bagiński, C. Gromadzki, G. Hidalgo, R. A. |
| contents | A continuous action of a finite group $G$ on a closed orientable surface $X$ is said to be gpnf (Gilman purely non-free) if every element of $G$ has a fixed point on $X$. We prove that the biggest order {$μ(g)$}, of a gpnf-action on a surface of even genus $g \geq 2$, is bounded below by $8g$ and that this bound is sharp for infinitely many even $g$ as well. This provides, for even genera, a gpnf-action analog of the celebrated Accola-Maclachlan bound $8g+8$ for arbitrary finite continuous actions. We also describe the asymptotic behavior of $μ$. We define $\mathcal{M}$ as the set of values of the form $$\widetildeμ(g)=\frac{μ(g)}{g+1},$$ and its subsets
$\mathcal{M}_+$ and $\mathcal{M}_-$ corresponding to even and odd genera $g$. We show that the set $\mathcal{M}_+^d$, of accumulation points of $\mathcal{M}_+$, consists of a single number $8$. If $g$ is odd, then we prove that $4g \leq μ(g)<8g$. We conjecture that this lower bound is sharp for infinitely many odd $g$. Finally, we prove that this conjecture implies that $4$ is the only element of $\mathcal{M}_-^d$, leading to $\mathcal{M}^d=\{4,8\}.$ |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_04214 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On the biggest purely non-free conformal actions on compact Riemann surfaces and their asymptotic properties Bagiński, C. Gromadzki, G. Hidalgo, R. A. Geometric Topology Algebraic Geometry 57M60, 20H10, 30F10 A continuous action of a finite group $G$ on a closed orientable surface $X$ is said to be gpnf (Gilman purely non-free) if every element of $G$ has a fixed point on $X$. We prove that the biggest order {$μ(g)$}, of a gpnf-action on a surface of even genus $g \geq 2$, is bounded below by $8g$ and that this bound is sharp for infinitely many even $g$ as well. This provides, for even genera, a gpnf-action analog of the celebrated Accola-Maclachlan bound $8g+8$ for arbitrary finite continuous actions. We also describe the asymptotic behavior of $μ$. We define $\mathcal{M}$ as the set of values of the form $$\widetildeμ(g)=\frac{μ(g)}{g+1},$$ and its subsets $\mathcal{M}_+$ and $\mathcal{M}_-$ corresponding to even and odd genera $g$. We show that the set $\mathcal{M}_+^d$, of accumulation points of $\mathcal{M}_+$, consists of a single number $8$. If $g$ is odd, then we prove that $4g \leq μ(g)<8g$. We conjecture that this lower bound is sharp for infinitely many odd $g$. Finally, we prove that this conjecture implies that $4$ is the only element of $\mathcal{M}_-^d$, leading to $\mathcal{M}^d=\{4,8\}.$ |
| title | On the biggest purely non-free conformal actions on compact Riemann surfaces and their asymptotic properties |
| topic | Geometric Topology Algebraic Geometry 57M60, 20H10, 30F10 |
| url | https://arxiv.org/abs/2605.04214 |