A short proof of Greenberg's Theorem

Gareth A. Jones

A short proof of Greenberg's Theorem

T0 review · 0 major / 6 minor · reviewed 2026-08-14 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Every finitely generated group is the full symmetry group of a Riemann surface.

desk verdict A clean, explicit proof of a known theorem; useful, honest about its limits, and sound enough to referee. read the letter →

arxiv 1908.06675 v2 pith:S6K37YNG submitted 2019-08-19 math.GR

Gareth A. Jones This is my paper · ORCID

classification math.GR MSC 20H1011F0611G3214H5720B2530F10

keywords RiemannsurfaceautomorphismgrouptriangleFuchsiancommensuratorfinitesurface-kernelepimorphismdessind'enfant

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper proves that every finitely generated group $A$ is isomorphic to the full automorphism group of a Riemann surface, and that the surface can be chosen compact whenever $A$ is finite. The result is a restricted form of a long-standing theorem, but the proof is new: instead of constructing maximal Fuchsian groups with prescribed signatures, it builds the surface from a hyperbolic triangle group and a finite quotient $\mathrm{PSL}_2(\mathbb{F}_q)$. A reader should care because the construction is explicit, algebraic, and arithmetically parameterized, so the theorem becomes something one can see and test rather than a distant existence result.

What carries the argument

The load-bearing object is a hyperbolic triangle group $\Delta(l,m,n)$, the group generated by $X,Y,Z$ with $X^l=Y^m=Z^n=XYZ=1$ and $l^{-1}+m^{-1}+n^{-1}<1$, acting by isometries on the hyperbolic plane. It is chosen maximal and non-arithmetic so that its commensurator in $\mathrm{PSL}_2(\mathbb{R})$ is itself: every element of the ambient group that maps a finite-index subgroup of $\Delta$ to another subgroup with finite-index intersection is already inside $\Delta$. That fact turns the normalizer of the kernel $M$ into a group computable inside $\Delta$, and the chain $\operatorname{Aut}(S)\cong N_{\mathrm{PSL}_2(\mathbb{R})}(M)/M=N_{\Delta}(M)/M\cong A$ carries the whole argument.

What would settle it

Compute the automorphism group of the surface $\mathbb{H}/M$ for a small group, say $A=C_2$, using the paper's explicit choices $(l,m,n)=(2,3,13)$ and $q=311$: if the conformal automorphism group is strictly larger than $C_2$, the normalizer equality that carries the proof is false for that instance. To disprove the theorem itself, one would need a finitely generated $A$ for which every admissible choice of triple, prime power, and epimorphism leaves the automorphism group larger than $A$.

Watch

Extended reading notes

Core claim

At the center of the paper is the claim that for any finitely generated group $A$ there is a Riemann surface $S$ whose automorphism group is exactly $A$, and if $A$ is finite the surface may be taken compact. The construction fixes a hyperbolic triangle group $\Delta(l,m,n)$ that is maximal and non-arithmetic, chooses a prime power $q$ for which $\Delta$ maps onto $\mathrm{PSL}_2(\mathbb{F}_q)$ with the three standard generators retaining orders $l,m,n$, and takes the inverse image $N$ of the stabilizer of $\infty$; this $N$ is a surface group of genus large enough to map onto $A$. The kernel $M$ of that epimorphism yields $S=\mathbb{H}/M$, and maximality plus non-arithmeticity imply that the normalizer of $M$ in $\mathrm{PSL}_2(\mathbb{R})$ equals its normalizer in $\Delta$, giving $\operatorname{Aut}(S)\cong N/M\cong A$.

Load-bearing premise

The proof needs the chosen triangle group to have no commensurability symmetries outside itself, a fact obtained by combining maximality with non-arithmeticity; if that guarantee failed, the final step equating the surface's automorphism group with $N/M$ would collapse.

Editorial extensions

If this is right

Every finite group occurs as the full automorphism group of a compact Riemann surface, which the paper notes is then defined over a number field.
Every infinite finitely generated group is realized as the exact symmetry group of a non-compact Riemann surface.
The proof is an arithmetic recipe: for a group of rank $d$, choose a maximal non-arithmetic triangle group and a prime power $q$ so that the resulting surface has genus at least $d$, then form the kernel of a map onto $A$.
Replacing the natural action of $\mathrm{PSL}_2(\mathbb{F}_q)$ by other primitive actions makes the genus grow cubically or super-exponentially, so the construction reaches groups of very large rank.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The paper leaves open whether arithmetic or non-maximal triangle groups can realize groups exactly; one plausible reading is that the maximal non-arithmetic condition is exactly the feature that prevents extra automorphisms, so any relaxation must replace the commensurability step with a new idea.
Because every step is algebraic and parameterized by $q$, a small computational search could turn the existence argument into an explicit generator of curves or dessins with a prescribed finite automorphism group.
For a fixed group $A$, varying the triple, the prime power, and the epimorphism should produce many non-isomorphic surfaces realizing $A$; counting such surfaces is a natural next question.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Colleague,

The main thing you should know: this is a new proof of a non-new theorem. Greenberg proved the statement for countable groups; Jones proves the finitely generated case, explicitly, using triangle groups, maximal non-arithmetic triangle groups, and Margulis. The proof is about three pages, and it is honest about being a restricted version. If you want a route to Greenberg for finite groups that does not follow Greenberg's original maximality machinery, this is the best I know.

What's good: the construction is concrete. Pick a hyperbolic triangle group Δ(l,m,n) with Δ maximal non-arithmetic. Use Dirichlet plus Macbeath/Garion to get a Fuchsian kernel N of genus g ≥ d, then map N onto A, take kernel M. The delicate step is showing Aut(H/M) = N(M)/M equals A, i.e., that the normalizer of M in PSL2(R) does not exceed Δ. The argument via maximality of Δ and Margulis' commensurator theorem is sound. I checked the normalizer equality; the logic holds. The proof does not smuggle in Greenberg's theorem—all tools are independent classifications and commensurability facts.

Soft spots: the title is slightly overbroad, because the paper proves only finitely generated groups, not all countable groups. The abstract and Remark 1 make that clear, but the title invites quibbles. The proof leans on heavy external results—Singerman, Takeuchi, Margulis, Macbeath—so "short" means short in execution, not elementary. For a finite A the surface is compact; for infinite finitely generated A you get a non-compact surface, as expected. There are a few typos (e.g., 'nornaliser', 'B1 /nelementK') but no mathematical errors I could find. The remarks about variants are a nice bonus, especially the super-exponential genus growth via symmetric groups.

Who this is for: people working on automorphism groups of Riemann surfaces, dessins, and Fuchsian groups. It is a useful reference and could be assigned to students as a modern route to a classical theorem. The theorem's age and the heavy machinery cap the significance, but the proof is rigorous and reusable. I would not desk reject it; it deserves a serious referee, and I expect a good referee would accept with minor revisions.

Recommendation: send it to review. It is not a breakthrough, but it is a correct, well-cited, clearly written proof that improves accessibility of a known result.

Referee Report

0 major / 6 minor

Summary. The paper gives a short algebraic proof of Greenberg's theorem for finitely generated groups: every such group A occurs as the automorphism group of a Riemann surface, and the surface can be taken compact when A is finite. The construction chooses a hyperbolic triangle group Δ(l,m,n), uses Dirichlet's theorem and Macbeath's theorem to map Δ onto PSL2(Fq) for a suitable prime power q, and takes N to be the preimage in Δ of the stabilizer of a point. The subgroup N is a torsion-free surface group of genus g; for a given A, a normal subgroup M⊲N is chosen with N/M≅A. The main technical step is proving that the normalizer of M in PSL2(R) coincides with its normalizer in Δ. This is achieved by choosing Δ to be maximal and non-arithmetic, so that Margulis' theorem identifies the commensurator of Δ with Δ itself, and then showing that any element normalizing M lies in that commensurator.

Significance. If the proof is correct, the paper provides a notably short and explicit route to a restricted version of a classical theorem, avoiding the N-equivalence and N-maximality machinery of Greenberg's original proof. The construction is transparent, yields explicit genus bounds, and gives concrete examples of usable triangle groups. The proof is not elementary, but the paper is honest about this and precisely cites the external inputs: Dirichlet, Macbeath, Singerman, Takeuchi, and Margulis. The value of the paper is mainly methodological and expository: it shows how standard tools about triangle groups, finite quotients, and commensurability combine to realize automorphism groups of Riemann surfaces.

minor comments (6)

[Section 2, final paragraph] The symbol g is used both for the genus in Eq. (1) and for an arbitrary element of N(M) in the final paragraph; please use a different letter (for instance γ) for the group element to avoid confusion.
[Section 2, transition before the final step] The triple (l,m,n) is not fixed to be maximal and non-arithmetic until after N and M have already been constructed; please move this choice earlier or state explicitly that from that point onward the triple is chosen with these properties and q is then selected so that g ≥ d.
[Section 2, final paragraph] The notation for the commensurator appears to be lost in the typeset text (it reads 'its commensurator Δ in PSL2(R)'); use a distinct symbol such as \tilde{\Delta}, and add one sentence explaining that a finite-index subgroup N of Δ has the same commensurator as Δ.
[Section 2, typos and notation] There are several small presentation issues: 'nornaliser' should be 'normaliser'; 'generators Ai of Δ' should be 'generators Ai of N'; and the subgroup M and the hypermap M are denoted by the same symbol, which is confusing.
[Section 2, paragraph on elliptic elements] The claim that N has no elliptic elements would be clearer if it explicitly invoked the standard fact that every elliptic element of a triangle group is conjugate to a power of X, Y, or Z, so the semiregularity of the images rules out elliptic elements in N.
[Section 2, final paragraph] The statement that N and N^g have finite index in N(M) because they are cocompact is correct but terse; one sentence indicating that a cocompact Fuchsian subgroup of a Fuchsian group necessarily has finite index would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: all load-bearing steps reduce to external theorems independent of the target result.

full rationale

The proof is self-contained after citing standard, external results that are not equivalent to the theorem being proved. The construction selects a maximal non-arithmetic hyperbolic triangle group Δ using Singerman's and Takeuchi's classifications, invokes Macbeath's theorem to obtain surface-kernel epimorphisms Δ → PSL2(Fq), uses Dirichlet's theorem for the prime powers q, and then applies Margulis' commensurability theorem to conclude that the PSL2(R)-normalizer of the constructed surface subgroup M coincides with its normalizer inside Δ. Each of these inputs is an independent mathematical result about classification, generation, or commensurability of Fuchsian groups; none of them assumes Greenberg's theorem or asserts arbitrary groups are automorphism groups of surfaces. The author's earlier paper [12] is cited only in Remark 5 as the source of the adapted method and is contrasted with the present argument: [12] gives dessins whose underlying surfaces have automorphism group containing A, whereas the new proof supplies the extra normalizer argument needed to force equality. No parameter is fitted to the target, no 'prediction' is obtained by construction, and no uniqueness claim is imported from the author's own prior work. Hence there is no significant circularity.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

The proof is a short argument built on standard external theorems in number theory, finite group theory, and Fuchsian groups. No free parameters are fitted to data and no new entities are postulated. The heaviest external input is Margulis' commensurability theorem, which carries the final equality of automorphism groups.

assumptions (7)

standard math Dirichlet's theorem on primes in arithmetic progressions gives infinitely many prime powers q ≡ -1 mod k.
Invoked in Section 2 to generate the finite quotients PSL2(F_q). Classical result, not proved in the paper.
domain assumption For q ≡ -1 mod lcm(2l,2m,2n), there is a surface-kernel epimorphism Δ(l,m,n) → PSL2(F_q) with generators of orders l,m,n.
Section 2 cites Macbeath and Garion for this fact. It is the bridge from triangle groups to finite quotients, and the proof does not establish it.
domain assumption There exist hyperbolic triples (l,m,n) for which Δ(l,m,n) is maximal and non-arithmetic.
Section 2 invokes Singerman and Takeuchi; Remark 2 gives examples. Needed so that Margulis' commensurability theorem can be applied.
domain assumption For a non-arithmetic Fuchsian group of finite covolume, its commensurator is a Fuchsian group; for a maximal such group, the commensurator is the group itself.
Used in the last paragraph of Section 2 to move from an element of the normalizer of M to an element of Δ. This is the key external theorem for the equality Aut S = A.
domain assumption For a torsion-free Fuchsian group M, Aut(H/M) is isomorphic to N(M)/M, where N(M) is the normalizer in PSL2(R).
Section 2 cites this standard result to identify the automorphism group of the surface with the normalizer quotient.
standard math Riemann-Hurwitz formula: the surface subgroup N of index q+1 in Δ(l,m,n) has genus g = (q+1)/2 (1 - 1/l - 1/m - 1/n) + 1.
Equation (1) in Section 2; used to choose q large enough so that g is at least the rank of A.
domain assumption PSL2(F_q) acts primitively on P^1(F_q); elements of order dividing (q+1)/2 are semiregular; the point stabilizer is maximal of index q+1.
Section 2 uses these facts to ensure the inverse image N of the point stabilizer is a maximal torsion-free surface subgroup.

how reviews work

0 comments

Cite this review

Pith. "Pith review of A short proof of Greenberg's Theorem." pith.science (2026). https://pith.science/paper/S6K37YNG

@misc{pith2026190806675,
  author       = {Pith},
  title        = {Pith review of: A short proof of Greenberg's Theorem},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/S6K37YNG}},
  note         = {Machine review of arXiv:1908.06675}
}

abstract

Greenberg proved that every countable group $A$ is isomorphic to the automorphism group of a Riemann surface, which can be taken to be compact if $A$ is finite. We give a short and explicit algebraic proof of this for finitely generated groups $A$.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

22 extracted references · 22 canonical work pages

[1]

Allcock, Hyperbolic surfaces with prescribed inﬁnit e symmetry groups, Proc

D. Allcock, Hyperbolic surfaces with prescribed inﬁnit e symmetry groups, Proc. Amer. Math. Soc. 134 (2006), 3057–3057

work page 2006
[2]

G. V . Bely˘ ı, On Galois extensions of a maximal cyclotomi c ﬁeld, Izv. Akad. Nauk SSSR Ser. Mat. 43 (1979), 267–276, 479

work page 1979
[3]

L. E. Dickson, Linear Groups, Dover, New Y ork, 1958

work page 1958
[4]

Garion, On Beauville structures for PSL2(q), J

S. Garion, On Beauville structures for PSL2(q), J. Group Theory 18 (2015), 981–1019

work page 2015
[5]

Girondo, D

E. Girondo, D. Torres-Teigell and J. Wolfart, Shimura cu rves with many uniform dessins, Math. Z. 271 (2012), 757–779

work page 2012
[6]

Greenberg, Conformal transformations of Riemann sur faces, Amer

L. Greenberg, Conformal transformations of Riemann sur faces, Amer. J. Math. 82 (1960), 749–760

work page 1960
[7]

Greenberg, Discrete groups of motions, Canad

L. Greenberg, Discrete groups of motions, Canad. J. Math. 12 (1960), 415–426

work page 1960
[8]

Greenberg, Maximal Fuchsian groups, Bull

L. Greenberg, Maximal Fuchsian groups, Bull. Amer. Math. Soc. 69 (1963), 569–573

work page 1963

Show all 22 references

[9]

Greenberg, Maximal groups and signatures, in Discontinuous Groups and Riemann Surfaces, Proc

L. Greenberg, Maximal groups and signatures, in Discontinuous Groups and Riemann Surfaces, Proc. Conf. Maryland 1973, ed. L. Greenberg, Princeton University Press, Princeton N J, 1974, 207–226. A SHORT PROOF OF GREENBERG’S THEOREM 5

1973
[10]

Grothendieck, Esquisse d’un Programme (1984), in Geometric Galois Actions 1 , ed

A. Grothendieck, Esquisse d’un Programme (1984), in Geometric Galois Actions 1 , ed. L. Schneps and P . Lochak, London Math. Soc. Lecture Note Ser. 242, Cambridge U niversity Press, Cambridge, 1997, 5–47

1984
[11]

Hidalgo, Automorphism groups of dessins d’enfants, Arch

R. Hidalgo, Automorphism groups of dessins d’enfants, Arch. Math. (Basel) 112 (2019), 13–18

2019
[12]

G. A. Jones, Realisation of groups as automorphism grou ps in categories, arXiv:1807.00547v3 [math.GR]

arXiv
[13]

G. A. Jones, Primitive permutation groups containing a cycle, Bull Aust. Math. Soc. 89 (2014), 159–165

2014
[14]

G. A. Jones and D. Singerman, Complex Function Theory , Cambridge University Press, Cambridge, 1987

1987
[15]

G. A. Jones and M. Streit, Galois groups, monodromy grou ps and cartographic groups, in Geometric Galois Actions 2 , ed. P . Lochak and L. Schneps, London Math. Soc. Lecture Note Ser. 243, Cambridge University Press, Cambridge, 25–65
[16]

G. A. Jones and J. Wolfart, Dessins d’Enfants on Riemann Surfaces , Springer, Cham, 2016

2016
[17]

A. M. Macbeath, Generators of the linear fractional gro ups, in Number Theory (Houston 1967), ed. W . J. Leveque and E. G. Straus, Proc. Sympos. Pure Math. 12, Amer. Math. Soc ., Providence RI, 1969, 14–32

1967
[18]

Margulis, Discrete Subgroups of Semisimple Lie Groups , Springer-V erlag, Berlin, 1991

G. Margulis, Discrete Subgroups of Semisimple Lie Groups , Springer-V erlag, Berlin, 1991

1991
[19]

Serre, Topics in Galois Theory, Jones and Bartlett, Boston Mass., 1992

J-P . Serre, Topics in Galois Theory, Jones and Bartlett, Boston Mass., 1992

1992
[20]

Singerman, Finitely maximal Fuchsian groups, J

D. Singerman, Finitely maximal Fuchsian groups, J. London Math. Soc. (2) 6 (1972), 29–38

1972
[21]

Singerman and R

D. Singerman and R. I. Syddall, The Riemann surface of a u niform dessin, Beitr¨ age Algebra Geom.44 (2003), 413–430

2003
[22]

Takeuchi, Arithmetic triangle groups, J

K. Takeuchi, Arithmetic triangle groups, J. Math. Soc. Japan 29 (1977), 91–106. School of MathematicalSciences, University of Southampton, Southampton SO17 1BJ, UK E-mail address: G.A.Jones@maths.soton.ac.uk

1977

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

Allcock, Hyperbolic surfaces with prescribed inﬁnit e symmetry groups, Proc

D. Allcock, Hyperbolic surfaces with prescribed inﬁnit e symmetry groups, Proc. Amer. Math. Soc. 134 (2006), 3057–3057

work page 2006

[2] [2]

G. V . Bely˘ ı, On Galois extensions of a maximal cyclotomi c ﬁeld, Izv. Akad. Nauk SSSR Ser. Mat. 43 (1979), 267–276, 479

work page 1979

[3] [3]

L. E. Dickson, Linear Groups, Dover, New Y ork, 1958

work page 1958

[4] [4]

Garion, On Beauville structures for PSL2(q), J

S. Garion, On Beauville structures for PSL2(q), J. Group Theory 18 (2015), 981–1019

work page 2015

[5] [5]

Girondo, D

E. Girondo, D. Torres-Teigell and J. Wolfart, Shimura cu rves with many uniform dessins, Math. Z. 271 (2012), 757–779

work page 2012

[6] [6]

Greenberg, Conformal transformations of Riemann sur faces, Amer

L. Greenberg, Conformal transformations of Riemann sur faces, Amer. J. Math. 82 (1960), 749–760

work page 1960

[7] [7]

Greenberg, Discrete groups of motions, Canad

L. Greenberg, Discrete groups of motions, Canad. J. Math. 12 (1960), 415–426

work page 1960

[8] [8]

Greenberg, Maximal Fuchsian groups, Bull

L. Greenberg, Maximal Fuchsian groups, Bull. Amer. Math. Soc. 69 (1963), 569–573

work page 1963