Combinatorial Problems Related to Automorphism Groups of Compact - - PowerPoint PPT Presentation

Combinatorial Problems Related to Automorphism Groups of Compact Riemann Surfaces

Charles Camacho Oregon State University

Rainwater Seminar University of Washington

February 26, 2019 camachoc@math.oregonstate.edu

Charles Camacho Oregon State University 1

Introduction

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Inspiration from the Platonic Solids

The Platonic solids can be described by ramified coverings of the sphere branching over three points, 0, 1, ∞, represented as white and black vertices, and face centers, respectively. The values below are the branch orders above 0, 1, ∞. ∆(2, 3, 5) ∆(2, 3, 4) ∆(2, 3, 3) ∆(2, 4, 3) ∆(2, 5, 3)

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Klein Quartic of Genus Three (1879)

Glue edges 1 ↔ 10, 2 ↔ 7, 3 ↔ 12, 4 ↔ 9, 5 ↔ 14, 6 ↔ 11, 8 ↔ 13. Rotation about the center of the hyperbolic 14-gon by 2π/7 fixes three points.

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References for an Introduction to the Subject

Broughton, S. A. (1990). The equisymmetric stratification of the moduli space and the Krull dimension of mapping class groups. Topology and its Applications, 37(2), 101-113. Broughton, S. A. (1991). Classifying finite group actions on surfaces

• f low genus. Journal of Pure and Applied Algebra, 69(3), 233-270.

Girondo, E., Gonz´ alez-Diez, G. Introduction to Compact Riemann Surfaces and Dessins d’Enfants. Cambridge: Cambridge UP, 2012. Print. Jones, G. A., Wolfart, J. (2016). Dessins D’enfants on Riemann

• Surfaces. Springer.

O’Sullivan, C., Weaver, A. A diophantine frobenius problem related to Riemann surfaces. Glasgow Mathematical Journal, 53(3), 501-522. (2011). Weaver, A. Genus spectra for split metacyclic groups. Glasgow Mathematical Journal, 43(2), 209-218. 2001.

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Main Talk

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Results - The Number of Quasiplatonic Cyclic Group Actions

Theorem (C. 2018): Let QC(n) denote the number of distinct quasiplatonic topological Cn-actions on surfaces of genus at least two. Write n = r

i=1 pai i . Then

QC(n) = 1 6n

• p|n
• 1 + 1

p

• − 1 + a · 2r

where a =                    1/4 p1 = 2, a1 = 1 1/2 p1 = 2, a1 = 2 1 p1 = 2, a1 ≥ 3 5/6 pi ≡ 1 mod 6 for 1 ≤ i ≤ r 2/3 p1 = 3, a1 = 1, pi ≡ 1 mod 6 for 2 ≤ i ≤ r 1/2 p1 = 3, a1 ≥ 2, pi ≡ 1 mod 6 for 2 ≤ i ≤ r 1/2 pi ≡ 5 mod 6 for some 1 ≤ i ≤ r .

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Graph

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Preliminary Results - The Quasiplatonic Cyclic Genus Spectrum

Let QCGS(n) be the set of genera two or greater admitting a quasiplatonic Cn-action, called the quasiplatonic cyclic genus spectrum. Write QCGS(n) = {σn,1, σn,2, . . . , σn,sn}, where 2 ≤ σn,1 < σn,2 < · · · < σn,sn.

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Preliminary Results - The Quasiplatonic Cyclic Genus Spectrum

Let QCGS(n) be the set of genera two or greater admitting a quasiplatonic Cn-action, called the quasiplatonic cyclic genus spectrum. Write QCGS(n) = {σn,1, σn,2, . . . , σn,sn}, where 2 ≤ σn,1 < σn,2 < · · · < σn,sn. Theorem (C. 2019): If n = 2pa, then σn,2 = (p − 1)pa−1. If n = 2p2pa3

3 · · · par r , then σn,2 =

• p2−1

2

• ·
• n

p2 − 1

• .

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Conjecture for σn,2

σn,2 =            a − 1 2

• · n

a b − 1 2

• ·

n b − 1

• where

a =                                            p1 p1 ≥ 3, a1 = 1 p2 p1 = 2 and either a1 = 1, a2 ≥ 2;

• r a1 = 1, r = 2;
• r a1 = 2, p2 = 3, a2 = 1, r = 2;
• r a1 ≥ 3, p2 = 3, a2 ≥ 2.

Or p1 ≥ 3, a1 ≥ 3, a2 ≥ 2, p2 < p2

1

4 p1 = 2, a1 ≥ 3, p2 > 3 p2

1

p1 ≥ 3, a1 ≥ 3, p2 > p2

1.

b =                        p2 p1 = 2 and either a1 = 1, a2 = 1, r ≥ 3;

• r a1 = 2, p2 = 3, a2 = 1, r ≥ 3;
• r a1 ≥ 3, p2 = 3, a2 ≥ 2.

Or p1 ≥ 3, a1 ≥ 3, a2 = 1, p2 < p2

1

4 p1 = 2, a1 = 2, p2 > 3. .

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Graph

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Part One: Counting Quasiplatonic Cyclic Actions

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∆(2, 3, 7)

A tessellation of H by hyperbolic triangles, each with angles π/2, π/3 and π/7.

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Example - C7 Actions

Consider the permutation ρ = (1, 2, 3, 4, 5, 6, 7) ∈ S7.

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Example - C7 Actions

Consider the permutation ρ = (1, 2, 3, 4, 5, 6, 7) ∈ S7. In terms of rotations about the white and black vertices, and the face, respectively:

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Example - C7 Actions

Consider the permutation ρ = (1, 2, 3, 4, 5, 6, 7) ∈ S7. In terms of rotations about the white and black vertices, and the face, respectively:

1 2 3 4 5 6 7 5 6 7 1 2 3 4

(3, 3, 1) is described by ρ3, ρ3, ρ...

1 2 3 4 5 6 7 6 7 1 2 3 4 5

...and (2, 4, 1) is described by ρ2, ρ4, ρ.

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Counting Cn-Actions with a Given Signature (n1, n2, n3)

Theorem (Benim, Wootton 2014)

Write n = r

i=1 pai i . Suppose Cn acts on X of signature (n1, n2, n3).

Let w ≥ 0 be the number of primes shared in common among n1, n2, n3 which also share the same exponent. Relabel the primes to start with p1, . . . , pw. Then the number T of quasiplatonic topological actions of Cn on X is given in the three possibilities below.

• 1. If each ni is distinct and w = 0, then

T = φ(gcd(n1, n2, n3))

w

• i=1

pi − 2 pi − 1. If w = 0, then T = φ(gcd(n1, n2, n3)).

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Counting Cn-Actions with a Given Signature (n1, n2, n3)

Theorem (Benim, Wootton 2014)

• 2. If two ni are equal so that the signature is (n1, n2, n3) = (n1, n, n) with

n1 = n and w = 0, then T = 1 2

• τ1(n, n1) + φ(n1)

w

• i=1

pi − 2 pi − 1

• .

If w = 0, then T = 1

2 (τ1(n, n1) + φ(n1)).

• 3. If all ni are equal so that the signature is (n1, n2, n3) = (n, n, n) and

w = 0, then T = 1 6

• 3 + 2τ2(n) + φ(n)

w

• i=1

pi − 2 pi − 1

• .

If w = 0, then T = 1

6 (3 + 2τ2(n) + φ(n)).

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Example: Deriving QC(n) when p1 = 2, a1 = 1

Claim: QC(n) = 1 2

r

• i=2
• pai−1

i

(pi + 1)

• + 2r−2 − 1.

Proof Sketch.

First prove the following recursive formula by induction on r: QC(n · par+1

r+1 ) = (QC(n) + 1 − 2r−2)par+1−1 r+1

(pr+1 + 1) − 1 + 2r−1. Then prove the main claim by using induction again on r, with the base case being QC(2pa) = 1 2pa−1(p + 1).

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Further Work

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Further Work

Analyze the growth rate of QC(n). Find a geometric interpretation of the constant term in the formula for QC(n). Derive analogous formulas for quasiplatonic actions of other groups, e.g., abelian, elementary abelian C n

p , semidirect products Cm ⋊ Cn.

Does a count on the number of group actions have a physical interpretations (e.g., in string theory)? Visualization of these group actions?

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Part Two: The Quasiplatonic Cyclic Genus Spectrum

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Set-Up

Write n =

r

• i=1

pai

i . Let

Aj = {1 ≤ i ≤ r : ki = j, ℓi = hi = ai}, Bj = {1 ≤ i ≤ r : ℓi = j, ki = hi = ai}, Cj = {1 ≤ i ≤ r : hi = j, ki = ℓi = ai}. Let tn = max1≤i≤r{ai}. Then S(n1, n2, n3) = 1 pk1

1 · · · pkr r

+ 1 pℓ1

1 · · · pℓr r

+ 1 ph1

1 · · · phr r

= 1 n    

tn

• m=0
• i∈Am

pai−m

i

  +  

tn

• m=0
• i∈Bm

pai−m

i

  +  

tn

• m=0
• i∈Cm

pai−m

i

    .

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Key Lemma

Lemma (C. 2019)

All signatures that achieve ˆ S1 satisfy Aj = Bj = Cj = ∅ for j ≥ 2 and |A1 ∪ B1 ∪ C1| ≤ 1. If d ≥ 2 and n = pa for prime p and a ≥ 1, then all signatures that achieve ˆ Sd satisfy Aj = Bj = Cj = ∅ for j ≥ d and |Ad−1 ∪ Bd−1 ∪ Cd−1| ≤ 1.

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Example: C12-action with Signature (2, 12, 12)

1 2 3 4 5 6 7 8 9 10 11 12 1 2 3 4 5 6 7 8 9 10 11 12 (2, 12, 12)

1 7 2 8 9 3 10 4 115 6 12 8 1 6 11 4 9 2 7 12 5 10 3

σ = 3

A0 = {2} A1 = {1} A2 = ∅ B0 = ∅ B1 = ∅ B2 = ∅ C0 = ∅ C1 = ∅ C2 = ∅

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Example: C12-action with Signature (3, 12, 12)

1 2 3 4 5 6 7 8 9 10 11 12 (3, 12, 12) 1 2 3 4 5 6 7 8 9 10 11 12

8 8 1 1 3 3 10 10 5 5 12 12 7 7 2 2 9 9 4 4 11 11 6 6

σ = 4

A0 = {1} A1 = {2} A2 = ∅ B0 = ∅ B1 = ∅ B2 = ∅ C0 = ∅ C1 = ∅ C2 = ∅

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Example: C12-action with Signature (6, 12, 12)

1 2 3 4 5 6 7 8 9 10 11 12 (6, 12, 12) 7 8 9 10 11 12 1 2 3 4 5 6

1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 10 10 11 11 12 12

σ = 5

A0 = ∅ A1 = {1, 2} A2 = ∅ B0 = ∅ B1 = ∅ B2 = ∅ C0 = ∅ C1 = ∅ C2 = ∅

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Preliminary Results

Theorem (C. 2019)

When n = 2pa, we have σn,2 = (p − 1)pa−1.

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Preliminary Results

Theorem (C. 2019)

When n = 2pa, we have σn,2 = (p − 1)pa−1.

Proof Sketch.

The admissible signatures are of the form (2pk, pℓ, 2ph) (so assume WLOG that 1 ∈ B0).

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Preliminary Results

Theorem (C. 2019)

When n = 2pa, we have σn,2 = (p − 1)pa−1.

Proof Sketch.

The admissible signatures are of the form (2pk, pℓ, 2ph) (so assume WLOG that 1 ∈ B0). If |A1 ∪ B1 ∪ C1| = 0, these signatures achieve ˆ S1 = 3/(2pa) + 1/2.

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Preliminary Results

Theorem (C. 2019)

When n = 2pa, we have σn,2 = (p − 1)pa−1.

Proof Sketch.

The admissible signatures are of the form (2pk, pℓ, 2ph) (so assume WLOG that 1 ∈ B0). If |A1 ∪ B1 ∪ C1| = 0, these signatures achieve ˆ S1 = 3/(2pa) + 1/2. When |A1 ∪ B1 ∪ C1| = 1, there are two cases: either A1 = {2} or B1 = {2} (omit C1 by symmetry). The value of S is verified to be larger in the case when B1 = {2}. This gives ˆ S2 = 1/pa + 1/p.

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Preliminary Results

Theorem (C. 2019)

When n = 2pa, we have σn,2 = (p − 1)pa−1.

Proof Sketch.

The admissible signatures are of the form (2pk, pℓ, 2ph) (so assume WLOG that 1 ∈ B0). If |A1 ∪ B1 ∪ C1| = 0, these signatures achieve ˆ S1 = 3/(2pa) + 1/2. When |A1 ∪ B1 ∪ C1| = 1, there are two cases: either A1 = {2} or B1 = {2} (omit C1 by symmetry). The value of S is verified to be larger in the case when B1 = {2}. This gives ˆ S2 = 1/pa + 1/p. Therefore, B0 = {1} and B1 = {2}. The associated signature is (2pa, p, 2pa), so the Riemann-Hurwitz formula gives the result.

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Preliminary Results (cont.)

Theorem (C. 2019)

When n = 2p2pa3

3 · · · par r

for r ≥ 3, σn,2 = p2 − 1 2

• ·

n p2 − 1

• ,
• ccurring with associated signature (p2, n/p2, n).

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Proof Sketch.

Assume WLOG 1 ∈ B0.

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Proof Sketch.

Assume WLOG 1 ∈ B0. If |A1 ∪ B1 ∪ C1| = 0, then S is of the form S = 1 n  

i∈A0

pai

i + 2

• i∈B0

pai

i +

• i∈C0

pai

i

  .

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Proof Sketch.

Assume WLOG 1 ∈ B0. If |A1 ∪ B1 ∪ C1| = 0, then S is of the form S = 1 n  

i∈A0

pai

i + 2

• i∈B0

pai

i +

• i∈C0

pai

i

  . Write u =

i∈A0 pai i , v = i∈B0 pai i , w = i∈C0 pai i

and note that uvw = n/2.

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Proof Sketch.

Assume WLOG 1 ∈ B0. If |A1 ∪ B1 ∪ C1| = 0, then S is of the form S = 1 n  

i∈A0

pai

i + 2

• i∈B0

pai

i +

• i∈C0

pai

i

  . Write u =

i∈A0 pai i , v = i∈B0 pai i , w = i∈C0 pai i

and note that uvw = n/2. Goal: find the second-largest maximum of the function f (u, v) = u + 2v + n/(2uv).

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Proof Sketch.

Assume WLOG 1 ∈ B0. If |A1 ∪ B1 ∪ C1| = 0, then S is of the form S = 1 n  

i∈A0

pai

i + 2

• i∈B0

pai

i +

• i∈C0

pai

i

  . Write u =

i∈A0 pai i , v = i∈B0 pai i , w = i∈C0 pai i

and note that uvw = n/2. Goal: find the second-largest maximum of the function f (u, v) = u + 2v + n/(2uv). This turns out to occur when B0 = {1, 3, 4, . . . , r} and A ∪ C = {2}. The associated signature is (p2, n/p2, n).

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Proof Sketch.

Assume WLOG 1 ∈ B0. If |A1 ∪ B1 ∪ C1| = 0, then S is of the form S = 1 n  

i∈A0

pai

i + 2

• i∈B0

pai

i +

• i∈C0

pai

i

  . Write u =

i∈A0 pai i , v = i∈B0 pai i , w = i∈C0 pai i

and note that uvw = n/2. Goal: find the second-largest maximum of the function f (u, v) = u + 2v + n/(2uv). This turns out to occur when B0 = {1, 3, 4, . . . , r} and A ∪ C = {2}. The associated signature is (p2, n/p2, n). If |A1 ∪ B1 ∪ C1| = 1, S turns out to be smaller than in the previous case.

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Further Work

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Further Work

Determine all the genera σn,d. Analyze the sequence of gaps in the genus spectrum. Extend to the non-quasiplatonic case.

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References

John Baez’s webpage: http://math.ucr.edu/home/baez/klein.html Benim, R., Wootton, A. (2013). Enumerating quasiplatonic cyclic group actions. Rocky Mountain Journal of Mathematics, 43(5), 1459-1480. Greg Egan’s webpage: http://www.gregegan.net/SCIENCE/KleinQuartic/KleinQuartic.html Jones, G. A. (2014). Regular dessins with a given automorphism group. Contemporary Mathematics, 629, 245-260. Jones, G. A., Singerman, D. (1987). Complex functions: an algebraic and geometric viewpoint. Cambridge university press. Kulkarni, R. S. (1991). Isolated Points in the Branch Locus of the Moduli Space

• f Compact Riemann Surfaces.

By Tamfang - Own work, Public Domain, https://commons.wikimedia.org/w/index.php?curid=12806647 Wootton, A. (2007). Extending topological group actions to conformal group

• actions. Albanian Journal of Mathematics (ISNN: 1930-1235), 1(3), 133-143.

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Thank you!

1 2 3 4 5 6 7 1 6 4 2 7 5 3 1 3 5 7 2 4 6 1 5 2 6 3 7 4

camachoc@math.oregonstate.edu

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Appendix

τ1(m, n) denotes the number of nonzero, noncongruent solutions x to the equation x2 + 2x ≡ 0 mod m where gcd(x, m) = m/n. τ2(n) is the number of nonzero, noncongruent solutions x to the equation x2 + x + 1 ≡ 0 mod n.

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Appendix

Fuchsian groups ∆ having compact quotient space H/∆ of genus g can be shown to have a presentation ∆ ∼ =

• a1, b1, . . . , ag, bg, c1, . . . , ck | cm1

1

= · · · = cmr

r

= 1,

g

• i=1

[ai, bi] ·

r

• i=1

ci = 1

• ,

for mi positive integers, and [x, y] := xyx−1y−1. We say ∆ has signature (g; m1, . . . , mr) with periods m1, . . . , mr. Theorem (e.g., Broughton (1991) and Wootton (2007)): Two (0; m1, m2, m3)-generating vectors of the finite group G define the same equivalence class of G-actions if and only if the generating vector lie in the same Aut(G) × Aut(Γ)-class. (There is an analogous statement for general group actions, not necessarily quasiplatonic.)

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RAINWATER SEMINAR NOTES

CHARLES CAMACHO

• 1. Introduction

1.1. Illustrative Example. [Genus-three surface with three-fold rotational sym- metry having genus one quotient, sphere with three-fold rotation related to dessins and Bely˘ ı maps.] We focus on two combinatorial problems inspired by the theory of automorphism groups of compact Riemann surfaces: (i) How many different ways are there to rotate a surface n times? (ii) What are the genera of surfaces having such an n-fold rotation? The first problem is to enumerate the distinct topological actions of the cyclic group Cn on a surface, while the second problem is to determine the genus spectrum of

• Cn. General results are challenging, so we discuss the first nontrivial case: the

quasiplatonic actions. 1.2. The General Set-up. Ramified covering map of degree d = |G| given as X − → X/G where

• X: compact, connected surface of genus σ
• G: finite group
• X/G: quotient surface of genus g with k branch points of branch orders

n1, . . . , nk. If such a map exists, we say X has signature (g; n1, . . . , nk). We generally assume σ ≥ 2 so that ni ≥ 2 for each i. A quasiplatonic action is when g = 0 and k = 3. These quantities are related by the Riemann-Hurwitz formula: σ = 1 + |G| 2

• 2g − 2 +

k

• i=1
• 1 − 1

ni

• .
• 2. Topological Group Actions

G acts topologically on X if there exists monomorphism ǫ : G ֒ → Homeo+(X). Another action ǫ′ : G′ ֒ − → Homeo+(Y ) will be equivalent to ǫ if there exists a group isomorphism ω : G → G′ and a homeomorphism h : X → Y such that the following diagram commutes: G × X X G′ × Y Y

ǫG ω h h ǫG′ 1

2 CHARLES CAMACHO

That is, for any g ∈ G, x ∈ X, h ◦ ǫ(g) ◦ h−1(x) = ǫ′(w(g))(x). 2.1. Facts.

• G acts topologically on a surface X of genus σ ≥ 2 iff G-action on X

is topologically equivalent to ∆/Γ-action on H/Γ, where ∆ is a Fuchsian group and Γ ⊳ ∆ is torsion-free and Γ ∼ = π1(X).

• G acts topologically on X iff there exists a surface kernel epimorphism

ϕ : ∆ ։ G with ker ϕ = Γ.

• ∆ has presentation

∆ ∼ =

• a1, b1, . . . , ag, bg, c1, . . . , ck |

g

• i=1

[ai, bi]

k

• i=1

ci = 1, cn1

1

= · · · = cnk

k

= 1

• .
• The images of the generators a1, b1, . . . , ag, bg, c1, . . . , ck of ∆ under ϕ forms

a (g; n1, . . . , nk)-generating vector for G. Thus, G acts topologically on X with signature (g; n1, . . . , nk) iff G has a (g; n1, . . . , nk)-generating vector and the Riemann-Hurwitz formula is satisfied.

• If G = Cn, Harvey’s Theorem gives necessary and sufficient conditions
• n the signatures (n1, . . . , nk) guaranteeing a Cn-action. The signatures

satisfying Harvey’s Theorem are called admissible signatures. 2.2. History. (i) 1969: (Lloyd) Generating function for prime power cyclic group actions. (ii) 1979: (Bely˘ ı) Compact Riemann surfaces admitting a ramified covering of the sphere over three points are in one-to-one correspondence with alge- braic curves over number fields. (iii) 1983: (Kerckhoff) Groups of automorphisms of a surface of genus σ ≥ 2 are subgroups of Γσ, and conversely, all subgroups of Γσ arise in this way. (iv) 1999: (Jones) Enumerating subgroups of non-Euclidean crystallographic groups. (v) 2007: (Broughton, Wootton) Enumeration formulas for finite abelian group actions. (vi) 2007: (Wootton) Enumeration formulas for triangle group actions. (vii) 2013: (Jones) Enumerating regular dessins with a given automorphism group. (viii) 2014: (Benim, Wootton) Explicit formulae enumerating the quasiplatonic topological cyclic group actions with a given signature. 2.3. Motivations.

• Determining number of topological actions amounts to finding the conju-

gacy classes of finite subgroups of the mapping class group Γσ for a fixed genus σ ≥ 2. The formula for QC(n) explicitly enumerates the conjugacy classes of cyclic subgroups of Γσ as σ ranges over the quasiplatonic genus spectrum.

• The moduli space Mσ of conformal equivalence classes of Riemann surfaces
• f genus σ is decomposed into finite, disjoint equisymmetric strata, each
• f which corresponds to a unique equivalence class of actions of some finite
• group. See Broughton, 1990.

RAINWATER SEMINAR NOTES 3

• Many quasiplatonic groups are maximal in Γσ (e.g., the Hurwitz groups).

Enumerating the number of quasiplatonic topological actions of the cyclic group of a given admissible signature yields a lower bound on the number

• f conjugacy classes of maximal finite cyclic subgroups of Γσ. The formula

for QC(n) determines an explicit lower bound on the number of conjugacy classes of maximal finite cyclic subgroups of Γσ for σ ∈ QCGS(n).

• Quasiplatonic G-actions are related to regular dessins d’enfants D with

Aut(D) ∼ = G. 2.4. Enumerating Actions. Restrict to the quasiplatonic case. Let T(n1, n2, n3) denote the number of distinct topological Cn-actions with signature (n1, n2, n3). We may also refer to the number of topological actions of Cn of signature (n1, n2, n3) as the T-value of (n1, n2, n3). Let τ1(m, n) denote the number of nonzero, noncon- gruent solutions x to the equation x2 + 2x ≡ 0 mod m where gcd(x, m) = m/n. Also, let τ2(n) be the number of nonzero, noncongruent solutions x to the equation x2 + x + 1 ≡ 0 mod n. Theorem 2.1 (Benim and Wootton 2014). Let n be a positive integer and let p1, . . . , pr be the distinct primes dividing n. Suppose Cn acts on X of signature (n1, n2, n3). Let w ≥ 0 be the number of primes shared in common among n1, n2, n3 which also share the same exponent and relabel these primes as p1, . . . , pw. Then the number T of distinct topological actions of Cn on X is given in the three possibilities below. (i) If each ni is distinct and w = 0, then T = φ(gcd(n1, n2, n3))

w

• i=1

pi − 2 pi − 1. If w = 0, then T = φ(gcd(n1, n2, n3)). (ii) If two ni are equal so that the signature is (n1, n2, n3) = (n1, n, n) with n1 = n and w = 0, then T = 1 2

• τ1(n, n1) + φ(n1)

w

• i=1

pi − 2 pi − 1

• .

If w = 0, then T = 1 2 (τ1(n, n1) + φ(n1)) . (iii) If all ni are equal so that the signature is (n1, n2, n3) = (n, n, n) and w = 0, then T = 1 6

• 3 + 2τ2(n) + φ(n)

w

• i=1

pi − 2 pi − 1

• .

If w = 0, then T = 1 6 (3 + 2τ2(n) + φ(n)) .

4 CHARLES CAMACHO

Let QC(n) denote the total number of quasiplatonic topological Cn-actions on surfaces of genus two or greater. It follows that QC(n) =

• (n1,n2,n3)

T(n1, n2, n3), the sum being taken over all admissible signature triples. Theorem 2.2 (C. 2018). QC(n) = 1 6n

• p|n
• 1 + 1

p

• − 1 + a · 2r

where a =                                           

1 4

p1 = 2, a1 = 1

1 2

p1 = 2, a1 = 2 1 p1 = 2, a1 ≥ 3

5 6

pi ≡ 1 mod 6 for 1 ≤ i ≤ r

2 3

p1 = 3, a1 = 1, pi ≡ 1 mod 6 for 2 ≤ i ≤ r

1 2

p1 = 3, a1 ≥ 2, pi ≡ 1 mod 6 for 2 ≤ i ≤ r

1 2

pi ≡ 5 mod 6 for some 1 ≤ i ≤ r .

• 3. Genus Spectrum

The Riemann-Hurwitz formula applied to all admissible signatures (guaranteed by Harvey’s Theorem) yields a set QCGS(n) of genera admitting a quasiplatonic Cn-action, called the (quasiplatonic) genus spectrum of Cn: QCGS(n) = {σn,1, . . . , σn,sn}, where 2 ≤ σn,1 < σn,2 < · · · < σn,sn. 3.1. Facts.

• Harvey determined σn,1: The minimum genus σn,1 of a surface admitting

an automorphism of order n = pa1

1 · · · par r

is σn,1 =    max

• 2,

p1−1

2

• · n

p1

• a1 > 1 or n is prime

max

• 2,

p1−1

2

• ·
• n

p1 − 1

• a1 = 1

. Moreover, the corresponding Cn-action on the surface of genus σn,1 is quasiplatonic.

• Without restricting the signature type, Kulkarni proved that all finite

groups will eventually act in all genera. That is, there exists a constant NG ∈ N called the stable genus increment such that if g is in the genus spectrum of G, then g ≡ 1 mod NG and, expect for finitely many g where g ≡ 1 mod NG, g is in the genus spectrum.

RAINWATER SEMINAR NOTES 5

• The minimum genus µ(G) is known for cyclic groups, non-cyclic abelian

groups, metacyclic groups, alternating and symmetric groups, all sporadic simple groups.

• The maximum genus of a surface admitting a quasiplatonic Cn-action is

quickly computed from the Riemann-Hurwitz formula. This occurs for Cn-actions resulting from Fuchsian triangle groups of signature (n, n, n) for n odd or (n/2, n, n) for n even. That is, σn,sn =        n − 1 2 n is odd n − 2 2 n is even . 3.2. History. (i) 1965: (Maclachlan) Minimum genus for noncyclic abelian. (ii) 1966: (Harvey) Minimum genus for cyclic group. (iii) 1985: (Conder) Minimum genus for alternating and symmetric groups. (iv) 1985: (Glover, Sjerve) Minimum genus for PSL2(p). (v) 1987: (Kulkarni) Stable genus and genus spectra for cyclic p-groups. (vi) 1987: (Kulkarni) Determination of genus increment NG. (vii) 1998: (Machlachlan, Talu) Genus spectra for elementary abelian p-groups, p-groups of cyclic p-deficiency at most 2, and some other p-groups. (viii) 1990s, 2001: (Conder, Wilson, Woldar) Minimum genus for all 26 sporadic groups. (ix) 2001: (Weaver) Genus spectra for split metacyclic groups. 3.3. Motivations.

• The largest non-genus action is related to a Diophantine Frobenius prob-

lem: the Frobenius number of a set {a1, . . . , ak} of relatively prime integers strictly greater than 1 is the largest number which is not a linear combi- nation of the ai, which is NP-hard for k ≥ 3.

• There is a connection between gaps in the genus spectrum and “unstable”

torsion in the cohomology of the mapping class group, i.e., no p-torsion in the mapping class of the mapping class group of a particular genus κ(p), but there is p-torsion in H2(p−1)(Γκ(p); Z).

• There is interest in determining the stable genus increment NG.

3.4. How to Determine σn,d. Recall the Riemann-Hurwitz formula in the quasi- platonic case for an admissible signature (a, b, c) for Cn: σ = 1 + |G| 2

• 1 −

1 pk1

1 · · · pkr r

− 1 pℓ1

1 · · · pℓr r

− 1 ph1

1 · · · phr r

• .

(Here, we wrote a, b and c as factors of n = r

i=1 pai i .) Let

S(a, b, c) = 1 pk1

1 · · · pkr r

+ 1 pℓ1

1 · · · pℓr r

+ 1 ph1

1 · · · phr r

6 CHARLES CAMACHO

so that σ = 1 + (n/2)(1 − S(a, b, c)). We write S(a, b, c) = S for clarity. For each 0 ≤ j ≤ max1≤i≤r{ai}, define the following subsets of the index set {1, . . . , r}: Aj = {1 ≤ i ≤ r : ki = j, ℓi = hi = ai}, Bj = {1 ≤ i ≤ r : ℓi = j, ki = hi = ai}, Cj = {1 ≤ i ≤ r : hi = j, ki = ℓi = ai}. Let tn = max1≤i≤r{ai}. It follows that S = 1 n tn

• m=0
• i∈Am

pai−m

i

• +

tn

• m=0
• i∈Bm

pai−m

i

• +

tn

• m=0
• i∈Cm

pai−m

i

• .

Define the following: F = {(a, b, c) ∈ Z3 : (a, b, c) is an admissible triple for Cn}, ˆ Sd = the d-th largest maximum of S over F, Fd = {(a, b, c) ∈ F : S(a, b, c) = ˆ Sd}. Then ˆ S1 = max{S(a, b, c) : (a, b, c) ∈ F}, and for d ≥ 1, ˆ Sd+1 = max   S(a, b, c) : (a, b, c) ∈ F \  

1≤t≤d

Ft      . The following lemma describes the sufficient conditions on the sets Ak, Bk and Ck guaranteeing when a signature attains ˆ Sd. Lemma 3.1. All signatures that achieve ˆ S1 satisfy Aj = Bj = Cj = ∅ for j ≥ 2 and |A1 ∪ B1 ∪ C1| ≤ 1. If d ≥ 2 and n = pa for prime p and a ≥ 1, then all signatures that achieve ˆ Sd satisfy Aj = Bj = Cj = ∅ for j ≥ d and |Ad−1 ∪ Bd−1 ∪ Cd−1| ≤ 1. 3.5. Preliminary Results on σn,2. Theorem 3.2. When n = 2pa, we have σn,2 = (p − 1)pa−1. Theorem 3.3. When n = 2p2pa3

3 · · · par r

for r ≥ 3, σn,2 = p2 − 1 2

• ·

n p2 − 1

• ,
• ccurring with associated signature (p2, n/p2, n).

Theorem 3.4 (CONJECTURE). The two types of signatures giving σn,2 are of the form (a, n, n) for a ∈ {p1, p2, 4, p2

1} and (b, n/b, n) for b ∈ {p2, 4}. Thus, the

value of σn,2 is given by σn,2 =            a − 1 2

• · n

a b − 1 2

• ·

n b − 1

RAINWATER SEMINAR NOTES 7

where a =                                        p1 p1 ≥ 3, a1 = 1 p2 p1 = 2 and either a1 = 1, a2 ≥ 2;

• r a1 = 1, r = 2;
• r a1 = 2, p2 = 3, a2 = 1, r = 2;
• r a1 ≥ 3, p2 = 3, a2 ≥ 2.

Or p1 ≥ 3, a1 ≥ 3, a2 ≥ 2, p2 < p2

1

4 p1 = 2, a1 ≥ 3, p2 > 3 p2

1

p1 ≥ 3, a1 ≥ 3, p2 > p2

1.

and b =                    p2 p1 = 2 and either a1 = 1, a2 = 1, r ≥ 3;

• r a1 = 2, p2 = 3, a2 = 1, r ≥ 3;
• r a1 ≥ 3, p2 = 3, a2 ≥ 2.

Or p1 ≥ 3, a1 ≥ 3, a2 = 1, p2 < p2

1

4 p1 = 2, a1 = 2, p2 > 3. .

Oregon State University E-mail address: camachoc@math.oregonstate.edu