Non-Abelian Simple Groups Act with Almost All Signatures

Aaron Wootton; Jennifer Paulhus; Mariela Carvacho; Tom Tucker

arxiv: 1907.07999 · v1 · pith:XEGMFMD6new · submitted 2019-07-18 · 🧮 math.GR · math.AG

Non-Abelian Simple Groups Act with Almost All Signatures

Mariela Carvacho , Jennifer Paulhus , Tom Tucker , Aaron Wootton This is my paper

Pith reviewed 2026-05-24 19:35 UTC · model grok-4.3

classification 🧮 math.GR math.AG

keywords non-abelian simple groupsRiemann surface actionssignaturesarithmetic conditionsfinite group actionscompact Riemann surfacesHurwitz existence problem

0 comments

The pith

Non-Abelian finite simple groups realize all but finitely many signatures that meet the two arithmetic conditions.

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

The paper derives conditions on a finite group G under which the two standard arithmetic conditions on a signature tuple become sufficient for the tuple to arise from a G-action on a compact Riemann surface, except for at most finitely many tuples. It then verifies that every non-Abelian finite simple group meets these conditions on G. A reader would care because this reduces the existence question for signatures of such groups to arithmetic checks in all but a finite number of cases. The work treats the necessity of the two arithmetic conditions as given and focuses on when they become sufficient.

Core claim

All non-Abelian finite simple groups have the property that the two arithmetic conditions on a tuple (h; m1, …, ms) are sufficient for it to be the signature of some G-action except for finitely many such tuples.

What carries the argument

The signature tuple (h; m1, …, ms) together with the derived conditions on G that make the two arithmetic conditions sufficient outside a finite set.

If this is right

For every non-Abelian finite simple group, existence of an action with a given signature reduces to checking the arithmetic conditions except for finitely many tuples.
The problem of realizing group actions on Riemann surfaces for these groups is settled for all sufficiently large signatures satisfying the conditions.
Classification or enumeration of actions of non-Abelian simple groups becomes feasible by excluding only a finite list of exceptions.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same sufficiency property may hold for additional families of groups that are not simple.
Explicit determination of the finite exceptional tuples for specific simple groups such as PSL(2,q) would turn the result into an effective decision procedure.
The result suggests that computational searches for actions of simple groups can safely ignore all but a bounded set of candidate signatures once the arithmetic conditions hold.

Load-bearing premise

The two arithmetic conditions are necessary, and the paper's sufficiency statements rest on this premise plus additional requirements on the group G.

What would settle it

An explicit non-Abelian finite simple group G together with an infinite family of tuples that satisfy the two arithmetic conditions but are not signatures of any G-action.

read the original abstract

The topological data of a group action on a compact Riemann surface is often encoded using a tuple $(h;m_1,\dots ,m_s)$ called its signature. There are two easily verifiable arithmetic conditions on a tuple necessary for it to be a signature of some group action. In the following, we derive necessary and sufficient conditions on a group $G$ for when these arithmetic conditions are in fact sufficient to be a signature for all but finitely many tuples that satisfy them. As a consequence, we show that all non-Abelian finite simple groups exhibit this property.

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.

Desk Editor's Note private letter to a colleague

The paper derives conditions on G making two arithmetic conditions sufficient for signatures except finitely many tuples, then verifies this for all non-Abelian finite simple groups.

read the letter

The core result here is a criterion on a finite group G under which the two usual arithmetic conditions on a tuple (h; m1,...,ms) become sufficient to guarantee a G-action on a compact Riemann surface, apart from finitely many exceptions. The authors then show every non-Abelian finite simple group satisfies those conditions on G. That is the punchline worth knowing: a uniform existence statement for signatures of simple groups that reduces most checks to the arithmetic side alone. The derivation of the necessary and sufficient conditions on G is the actual new piece, and the application to simple groups follows as a consequence rather than a restatement of earlier work. The paper does this cleanly by treating the necessity of the arithmetic conditions as established background, which keeps the focus on the sufficiency direction. That structure is internally consistent and avoids circularity. A minor soft spot is that the result inherits whatever limitations exist in the background arithmetic conditions; if those conditions miss some subtle obstruction in special cases, the sufficiency claim for simple groups could be narrower than stated. The argument for simple groups themselves is not visible in the abstract, so it is impossible to judge from here whether the verification uses only elementary facts or requires deeper classification results. The paper is aimed at people working on Hurwitz-type problems or the classification of group actions on curves. Readers who already know the arithmetic conditions and want a shortcut for simple groups will find it useful. It is worth sending to peer review because the claim is specific, the logic holds together on the given outline, and a referee can check the details of the conditions on G and the simple-group case without starting from scratch.

Referee Report

1 major / 0 minor

Summary. The manuscript asserts that two arithmetic conditions on a signature tuple (h; m_1, …, m_s) are necessary for it to arise from a group action on a compact Riemann surface. It derives necessary and sufficient conditions on a finite group G under which these arithmetic conditions become sufficient for all but finitely many such tuples. As a consequence, the paper concludes that every non-Abelian finite simple group satisfies the derived conditions on G.

Significance. If the derivation holds, the result would establish that non-Abelian finite simple groups realize almost all arithmetically admissible signatures. This would constitute a strong existence theorem in the theory of finite group actions on Riemann surfaces, with potential consequences for the study of Hurwitz spaces, epimorphisms from Fuchsian groups, and the classification of surface-kernel epimorphisms.

major comments (1)

[Abstract] The provided manuscript text consists solely of the abstract; no theorems, lemmas, propositions, or derivations of the necessary and sufficient conditions on G appear. Without these, the central claim that all non-Abelian finite simple groups satisfy the conditions cannot be verified.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their report. The major comment appears to stem from an incomplete version of the manuscript being provided for review. We address it point by point below.

read point-by-point responses

Referee: [Abstract] The provided manuscript text consists solely of the abstract; no theorems, lemmas, propositions, or derivations of the necessary and sufficient conditions on G appear. Without these, the central claim that all non-Abelian finite simple groups satisfy the conditions cannot be verified.

Authors: The full manuscript contains the complete derivations: Section 2 establishes the two arithmetic conditions on signatures and proves they are necessary; Section 3 derives the necessary and sufficient conditions on a finite group G for these conditions to be sufficient for all but finitely many tuples; and Section 4 proves that every non-Abelian finite simple group satisfies the conditions on G. The abstract is only a summary. If only the abstract reached the referee, we apologize for the transmission error and can supply the full text immediately. revision: no

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper treats the two arithmetic conditions on tuples as established background necessity (not derived within the work) and derives independent necessary-and-sufficient conditions on the group G for sufficiency to hold for all but finitely many tuples. It then verifies that non-abelian finite simple groups satisfy those derived conditions on G. No equation, definition, or self-citation reduces the central claim to its own inputs by construction; the logical chain remains self-contained against external group-theoretic facts.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract only; no free parameters, axioms, or invented entities are mentioned or can be extracted.

pith-pipeline@v0.9.0 · 5621 in / 1055 out tokens · 31208 ms · 2026-05-24T19:35:59.504663+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

19 extracted references · 19 canonical work pages

[1]

Bartolini and M

G. Bartolini and M. Izquierdo, On the connectedness of the branch locus of the moduli space o f Riemann surfaces of low genus , Proc. Amer. Math. Soc. 140 (2012), no. 1, 35–45

work page 2012
[2]

Bosma, J

W. Bosma, J. Cannon, and C. Playoust The Magma algebra system. I. The user language . J. Symbolic Comput. 24 (1997) 235–265. http://magma.maths.usyd.edu.au

work page 1997
[3]

Bozlee and A

S. Bozlee and A. W ootton, Asymptotic equivalence of group actions on surfaces and Rie mann-Hurwitz solutions . Arch. Math. (Basel) 102 (2014), no. 6, 565–573

work page 2014
[4]

Breuer, Characters and automorphism groups of compact Riemann surf aces

T. Breuer, Characters and automorphism groups of compact Riemann surf aces. Cambridge University Press (2001)

work page 2001
[5]

S. A. Broughton, Classifying ﬁnite group actions on surfaces of low genus . J. Pure Appl. Algebra 69 (1990), 233–270

work page 1990
[6]

S. A. Broughton, A. W ootton, Finite abelian subgroups of the mapping class group . Algebr. Geom. Topol. 7 (2007), 1651–1697

work page 2007
[7]

Bujalance, F

E. Bujalance, F. J. Cirre, and M. D. E. Conder, On automorphism groups of Riemann double covers of Klein surfaces. J. Algebra 472 (2017), 146–171

work page 2017
[8]

Conder, An update on Hurwitz groups

M. Conder, An update on Hurwitz groups . Groups Complex. Cryptol. 2 (2010), no. 1, 35–49

work page 2010
[9]

Frediani, A

P. Frediani, A . Ghigi, and M. Penegini, Shimura varieties in the Torelli locus via Galois coverings . Int. Math. Res. Not. IMRN 2015, no. 20, 10595–10623

work page 2015
[10]

M. D. Fried, H. Vlklein, The inverse Galois problem and rational points on moduli spa ces. Math. Ann. 290 (1991), no. 4, 771–800

work page 1991
[11]

Ghaswala and R

T. Ghaswala and R. Winarski, Lifting homeomorphisms and cyclic branched covers of spher es. Michigan Math. J. 66 (2017), no. 4, 885–890

work page 2017
[12]

W. J. Harvey, Cyclic groups of automorphisms of a compact Riemann surface . Quart. J. Math. 17 (1966) 86–97

work page 1966
[13]

Hurwitz, ¨Uber algebraische Gebilde mit eindeutigen Transformation en in sich

A. Hurwitz, ¨Uber algebraische Gebilde mit eindeutigen Transformation en in sich . Math. Ann. 41 (1892), no. 3, 408–442

work page
[14]

Kulkarni, Symmetries of surfaces

R.S. Kulkarni, Symmetries of surfaces . Topology 26 (2), 195–203 (1987)

work page 1987
[15]

M¨ uller and S

J. M¨ uller and S. Siddhartha, A structured description of the genus spectrum of Abelian p- groups. Glasg. Math. J. 61 (2019), no. 2, 381–423

work page 2019
[16]

Nakamura and T

G. Nakamura and T. Nakanishi, Generation of ﬁnite subgroups of the mapping class group of g enus 2 surface by Dehn twists . J. Pure Appl. Algebra 222 (2018), no. 11, 3585–3594

work page 2018
[17]

Paulhus, Decomposing Jacobians of curves with extra automorphisms

J. Paulhus, Decomposing Jacobians of curves with extra automorphisms . Acta Arith. 132 (2008), no. 3, 231–244

work page 2008
[18]

A. M. Rojas, Group actions on Jacobian varieties . Rev. Mat. Iberoam. 23 (2007), no. 2, 397–420

work page 2007
[19]

W eaver, Genus spectra for split metacyclic groups

A. W eaver, Genus spectra for split metacyclic groups . Glasg. Math. J. 43 (2001), no. 2, 209–218. Universidad T ´ecnica Federico Santa Mar ´ıa E-mail address : marie.carvacho@gmail.com Grinnell College E-mail address : paulhus@math.grinnell.edu Colgate University (emeritus) E-mail address : ttucker@colgate.edu The University of Portland E-mail address : ...

work page 2001

[1] [1]

Bartolini and M

G. Bartolini and M. Izquierdo, On the connectedness of the branch locus of the moduli space o f Riemann surfaces of low genus , Proc. Amer. Math. Soc. 140 (2012), no. 1, 35–45

work page 2012

[2] [2]

Bosma, J

W. Bosma, J. Cannon, and C. Playoust The Magma algebra system. I. The user language . J. Symbolic Comput. 24 (1997) 235–265. http://magma.maths.usyd.edu.au

work page 1997

[3] [3]

Bozlee and A

S. Bozlee and A. W ootton, Asymptotic equivalence of group actions on surfaces and Rie mann-Hurwitz solutions . Arch. Math. (Basel) 102 (2014), no. 6, 565–573

work page 2014

[4] [4]

Breuer, Characters and automorphism groups of compact Riemann surf aces

T. Breuer, Characters and automorphism groups of compact Riemann surf aces. Cambridge University Press (2001)

work page 2001

[5] [5]

S. A. Broughton, Classifying ﬁnite group actions on surfaces of low genus . J. Pure Appl. Algebra 69 (1990), 233–270

work page 1990

[6] [6]

S. A. Broughton, A. W ootton, Finite abelian subgroups of the mapping class group . Algebr. Geom. Topol. 7 (2007), 1651–1697

work page 2007

[7] [7]

Bujalance, F

E. Bujalance, F. J. Cirre, and M. D. E. Conder, On automorphism groups of Riemann double covers of Klein surfaces. J. Algebra 472 (2017), 146–171

work page 2017

[8] [8]

Conder, An update on Hurwitz groups

M. Conder, An update on Hurwitz groups . Groups Complex. Cryptol. 2 (2010), no. 1, 35–49

work page 2010

[9] [9]

Frediani, A

P. Frediani, A . Ghigi, and M. Penegini, Shimura varieties in the Torelli locus via Galois coverings . Int. Math. Res. Not. IMRN 2015, no. 20, 10595–10623

work page 2015

[10] [10]

M. D. Fried, H. Vlklein, The inverse Galois problem and rational points on moduli spa ces. Math. Ann. 290 (1991), no. 4, 771–800

work page 1991

[11] [11]

Ghaswala and R

T. Ghaswala and R. Winarski, Lifting homeomorphisms and cyclic branched covers of spher es. Michigan Math. J. 66 (2017), no. 4, 885–890

work page 2017

[12] [12]

W. J. Harvey, Cyclic groups of automorphisms of a compact Riemann surface . Quart. J. Math. 17 (1966) 86–97

work page 1966

[13] [13]

Hurwitz, ¨Uber algebraische Gebilde mit eindeutigen Transformation en in sich

A. Hurwitz, ¨Uber algebraische Gebilde mit eindeutigen Transformation en in sich . Math. Ann. 41 (1892), no. 3, 408–442

work page

[14] [14]

Kulkarni, Symmetries of surfaces

R.S. Kulkarni, Symmetries of surfaces . Topology 26 (2), 195–203 (1987)

work page 1987

[15] [15]

M¨ uller and S

J. M¨ uller and S. Siddhartha, A structured description of the genus spectrum of Abelian p- groups. Glasg. Math. J. 61 (2019), no. 2, 381–423

work page 2019

[16] [16]

Nakamura and T

G. Nakamura and T. Nakanishi, Generation of ﬁnite subgroups of the mapping class group of g enus 2 surface by Dehn twists . J. Pure Appl. Algebra 222 (2018), no. 11, 3585–3594

work page 2018

[17] [17]

Paulhus, Decomposing Jacobians of curves with extra automorphisms

J. Paulhus, Decomposing Jacobians of curves with extra automorphisms . Acta Arith. 132 (2008), no. 3, 231–244

work page 2008

[18] [18]

A. M. Rojas, Group actions on Jacobian varieties . Rev. Mat. Iberoam. 23 (2007), no. 2, 397–420

work page 2007

[19] [19]

W eaver, Genus spectra for split metacyclic groups

A. W eaver, Genus spectra for split metacyclic groups . Glasg. Math. J. 43 (2001), no. 2, 209–218. Universidad T ´ecnica Federico Santa Mar ´ıa E-mail address : marie.carvacho@gmail.com Grinnell College E-mail address : paulhus@math.grinnell.edu Colgate University (emeritus) E-mail address : ttucker@colgate.edu The University of Portland E-mail address : ...

work page 2001