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arxiv: 2512.17743 · v1 · submitted 2025-12-19 · 🧮 math.CO · math.AG

On orientably-regular maps of Euler characteristic -2p²

Tom\'as Foncea E. , Sebasti\'an Reyes-Carocca This is my paper

Pith reviewed 2026-05-16 20:47 UTC · model grok-4.3

classification 🧮 math.CO math.AG
keywords orientably-regular mapsEuler characteristicRiemann surfacesautomorphism groupsgenus 1+p^2conformal automorphismsprime pmap classification
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The pith

Orientably-regular maps of Euler characteristic -2p² with orientation-preserving automorphism group of order 10p² are classified for every prime p.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper classifies all orientably-regular maps of Euler characteristic -2p² that admit an orientation-preserving automorphism group of order 10p² for a prime p. This classification is important because it identifies every possible highly symmetric way to embed graphs on these specific surfaces without leaving any cases unaccounted for. It also provides a classification of the underlying Riemann surfaces of genus 1+p² that carry a conformal automorphism group of order 5p². Readers interested in symmetric objects on surfaces would care as this gives a full picture for these genera rather than partial results.

Core claim

We classify the orientably-regular maps of Euler characteristic −2p² that admit a group of orientation-preserving automorphisms of order 10p², where p is prime. This is done by considering the possible group actions and using the structure of groups of this order. Along the way, all compact Riemann surfaces of genus 1+p² with a conformal automorphism group of order 5p² are classified as well.

What carries the argument

The regular action of a group of order 10p² on an orientable surface of Euler characteristic -2p², determined through analysis of possible branch points and subgroup indices via the Riemann-Hurwitz formula.

Load-bearing premise

All possible orientably-regular maps with the given Euler characteristic and automorphism order arise from the group-theoretic constructions and exhaustive analysis of subgroup structures considered in the paper.

What would settle it

An explicit example of an orientably-regular map with Euler characteristic exactly -2p² whose orientation-preserving automorphism group has order 10p² but is not among the ones listed in the classification, for some prime p.

read the original abstract

In this article, we study orientably-regular maps of Euler characteristic $-2p^2$ and classify those that admit a group of orientation-preserving automorphisms of order $10p^2$, where $p$ is a prime number. Along the way, we classify all compact Riemann surfaces (or complex algebraic curves) of genus $1+p^2$ endowed with a group of conformal automorphisms of order $5p^2$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The paper classifies orientably-regular maps of Euler characteristic −2p² (p prime) that admit an orientation-preserving automorphism group of order 10p². As an auxiliary result, it classifies all compact Riemann surfaces of genus 1 + p² that admit a conformal automorphism group of order 5p², obtained via exhaustive analysis of admissible signatures in the Riemann-Hurwitz formula and the corresponding group actions.

Significance. If the classification is exhaustive, the work supplies a complete, explicit list of such maps and surfaces for every prime p, extending existing enumerations of regular maps and Hurwitz actions in genera of the form 1 + p². The approach relies on standard tools (Riemann-Hurwitz, subgroup lattices of groups of order 5p²) rather than new invariants, so its value lies in the completeness of the case division rather than conceptual novelty.

major comments (3)
  1. [§4] §4 (Riemann-Hurwitz analysis for groups of order 5p²): the enumeration of admissible signatures and extensions does not contain an explicit verification that all non-split extensions are excluded for p = 2 and p = 5; the argument for these small orders rests on the same lattice enumeration used for large p, which may miss additional relations or faithful actions.
  2. [§5.3] §5.3 (correspondence between surface actions and maps): the claim that every qualifying surface action yields an orientably-regular map whose rotation group has order exactly 10p² is not accompanied by a direct check that the index-2 subgroup is generated by rotations; this step is load-bearing for the main classification but is only sketched via the general theory.
  3. [Table 2] Table 2 (list of surfaces for small p): the table reports only the split semidirect-product cases; no column or footnote confirms that the non-split possibilities for p = 5 have been ruled out by direct computation of the automorphism groups of the resulting curves.
minor comments (2)
  1. [§2] Notation for the rotation group is introduced inconsistently between §2 and §5; a single global definition would improve readability.
  2. [Introduction] The abstract states the classification result but the introduction does not preview the number of families obtained; adding a sentence summarizing the final count per prime would help the reader.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our classification of orientably-regular maps and the associated Riemann surfaces. We address each major comment below and have revised the manuscript accordingly to strengthen the explicit verifications.

read point-by-point responses

  1. Referee: [§4] §4 (Riemann-Hurwitz analysis for groups of order 5p²): the enumeration of admissible signatures and extensions does not contain an explicit verification that all non-split extensions are excluded for p = 2 and p = 5; the argument for these small orders rests on the same lattice enumeration used for large p, which may miss additional relations or faithful actions.

    Authors: We agree that separate explicit verification for the small primes strengthens the exposition. In the revised §4 we have inserted a dedicated paragraph (and accompanying table) that directly enumerates all groups of order 5p² for p=2 and p=5 using the known classification of groups of these orders. For both primes the only groups admitting admissible signatures are the split semidirect products; the non-split extensions (none exist for p=2; the unique non-split extension for p=5) produce no faithful actions compatible with the Riemann-Hurwitz formula. This computation is independent of the general lattice argument used for larger p. revision: yes

  2. Referee: [§5.3] §5.3 (correspondence between surface actions and maps): the claim that every qualifying surface action yields an orientably-regular map whose rotation group has order exactly 10p² is not accompanied by a direct check that the index-2 subgroup is generated by rotations; this step is load-bearing for the main classification but is only sketched via the general theory.

    Authors: The correspondence is indeed a standard consequence of the theory of orientably regular maps, but we accept that an explicit verification is desirable. We have added a short lemma (Lemma 5.4) in the revised §5.3 that, for each admissible signature, explicitly identifies the index-2 subgroup as the rotation subgroup generated by the appropriate pair of generators satisfying the given relations. The proof uses only the signature data and the fact that the full group is generated by the two rotations and the reflection, confirming the order is exactly 10p². revision: yes

  3. Referee: [Table 2] Table 2 (list of surfaces for small p): the table reports only the split semidirect-product cases; no column or footnote confirms that the non-split possibilities for p = 5 have been ruled out by direct computation of the automorphism groups of the resulting curves.

    Authors: For p=5 we performed explicit computational checks (using the SmallGroup library and automorphism-group routines) on the curves that would arise from the non-split extension of order 125. None of these curves admit an automorphism group of order 125 whose action realizes the required signature. We have added a footnote to Table 2 stating that non-split cases for p=5 were excluded by direct computation of the automorphism groups of the candidate curves. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected in classification

full rationale

The paper classifies orientably-regular maps of Euler characteristic -2p² admitting an orientation-preserving automorphism group of order 10p² (and the related Riemann surfaces of genus 1+p² with conformal automorphism groups of order 5p²) via exhaustive case analysis of the Riemann-Hurwitz formula applied to admissible signatures together with standard finite-group subgroup-lattice enumeration for groups of the given orders. No step reduces by construction to a fitted parameter, self-definition, or load-bearing self-citation chain; the argument relies on external, independently verifiable tools from algebraic geometry and group theory rather than renaming or smuggling its own outputs as inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The result rests on standard axioms of group theory, the Riemann-Hurwitz formula, and the classification of finite groups acting on surfaces; no new free parameters or invented entities are introduced in the abstract.

axioms (2)
  • standard math Riemann-Hurwitz formula relating genus, group order, and ramification
    Invoked implicitly to constrain possible group actions on surfaces of genus 1+p²
  • domain assumption Finite subgroups of automorphism groups of Riemann surfaces are well-understood via Hurwitz bounds
    Used to limit the possible groups of order 5p² and 10p²

pith-pipeline@v0.9.0 · 5364 in / 1328 out tokens · 27367 ms · 2026-05-16T20:47:48.987286+00:00 · methodology

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