Dessins d’Enfants and Counting Quasiplatonic Surfaces
Charles Camacho
Oregon State University
USTARS at Reed College, Portland, Oregon April 8, 2018
1 The Compact Surface X ...
2 ...with a Bipartite Map...
4 2 6 1 7 5 1 3 7 2 6 3 5 4
3 ...as an Algebraic Curve over Q...
4 2 6 1 7 5 1 3 7 2 6 3 5 4
y 2 = x7 − x
4 ∼ ...with a Uniformizing Fuchsian Group Γ = π1(X )...
6 1
7 7
5 2
6 1
4 3
5 2
3 4 ∼ X = H/Γ
5 ...Contained in a Hyperbolic Triangle Group...
6 1
7 7
5 2
6 1
4 3
5 2
3 4 ∼ X = H/Γ Γ C ∆(n1, n2, n3)
6 ...with the Cyclic Group as Quotient
6 1
7 7
5 2
6 1
4 3
5 2
3 4 ∼ X = H/Γ Γ C ∆(n1, n2, n3) ∼ ∆(n1, n2, n3)/Γ = Cn, the cyclic group of order n
7 Method: Enumerate all quasiplatonic actions of Cn on surfaces!
Main Question
How many quasiplatonic surfaces have regular dessins d’enfants with automorphism group Cn?
8 Main Question
How many quasiplatonic surfaces have regular dessins d’enfants with automorphism group Cn?
Method: Enumerate all quasiplatonic actions of Cn on surfaces!
8 Example Three dessins with order-seven symmetry on the Riemann sphere.
Dessins d’Enfants
A dessin d’enfant is a pair (X , D), where X is an orientable, compact surface and D ⊂ X is a finite graph such that 1 D is connected, 2 D is bicolored (i.e., bipartite), 3 X \ D is the union of finitely many topological discs, called the faces of D.
9 Dessins d’Enfants
A dessin d’enfant is a pair (X , D), where X is an orientable, compact surface and D ⊂ X is a finite graph such that 1 D is connected, 2 D is bicolored (i.e., bipartite), 3 X \ D is the union of finitely many topological discs, called the faces of D. Example Three dessins with order-seven symmetry on the Riemann sphere.
9 The automorphism group Aut(D) of a dessin is the set of orientation-preserving homeomorphisms which preserve the dessin. A regular dessin is a dessin whose automorphism group acts transitively on the edges. In other words, regular dessins are the most symmetric of all dessins.
Regular Dessins
10 A regular dessin is a dessin whose automorphism group acts transitively on the edges. In other words, regular dessins are the most symmetric of all dessins.
Regular Dessins
The automorphism group Aut(D) of a dessin is the set of orientation-preserving homeomorphisms which preserve the dessin.
10 In other words, regular dessins are the most symmetric of all dessins.
Regular Dessins
The automorphism group Aut(D) of a dessin is the set of orientation-preserving homeomorphisms which preserve the dessin. A regular dessin is a dessin whose automorphism group acts transitively on the edges.
10 Regular Dessins
The automorphism group Aut(D) of a dessin is the set of orientation-preserving homeomorphisms which preserve the dessin. A regular dessin is a dessin whose automorphism group acts transitively on the edges. In other words, regular dessins are the most symmetric of all dessins.
10 In other words, quasiplatonic surfaces are the most symmetric of all compact Riemann surfaces.
Figure: The Klein quartic of genus three with a regular dessin having 168 edges. (Credit: Girondo, Gonz´alez-Diez)
Quasiplatonic Surfaces
A quasiplatonic surface is a compact Riemann surface of genus at least two that admits a regular dessin.
11 Figure: The Klein quartic of genus three with a regular dessin having 168 edges. (Credit: Girondo, Gonz´alez-Diez)
Quasiplatonic Surfaces
A quasiplatonic surface is a compact Riemann surface of genus at least two that admits a regular dessin. In other words, quasiplatonic surfaces are the most symmetric of all compact Riemann surfaces.
11 Quasiplatonic Surfaces
A quasiplatonic surface is a compact Riemann surface of genus at least two that admits a regular dessin. In other words, quasiplatonic surfaces are the most symmetric of all compact Riemann surfaces.
Figure: The Klein quartic of genus three with a regular dessin having 168 edges. (Credit: Girondo, Gonz´alez-Diez) 11 For any compact Riemann surface X of genus g ≥ 2, we have |Aut(X )| ≤ 84(g − 1). This is the Hurwitz Automorphism Theorem. Some consequences are... 1 For any fixed genus g, there are finitely many regular dessins on genus g surfaces. 2 Thus, there are finitely many quasiplatonic surfaces of genus g. 3 For a given finite group G, there are only finitely many quasiplatonic surfaces X with a regular dessin D such that Aut(D) =∼ G. How do we enumerate such surfaces for a given G?
Key Facts
12 Some consequences are... 1 For any fixed genus g, there are finitely many regular dessins on genus g surfaces. 2 Thus, there are finitely many quasiplatonic surfaces of genus g. 3 For a given finite group G, there are only finitely many quasiplatonic surfaces X with a regular dessin D such that Aut(D) =∼ G. How do we enumerate such surfaces for a given G?
Key Facts
For any compact Riemann surface X of genus g ≥ 2, we have |Aut(X )| ≤ 84(g − 1). This is the Hurwitz Automorphism Theorem.
12 1 For any fixed genus g, there are finitely many regular dessins on genus g surfaces. 2 Thus, there are finitely many quasiplatonic surfaces of genus g. 3 For a given finite group G, there are only finitely many quasiplatonic surfaces X with a regular dessin D such that Aut(D) =∼ G. How do we enumerate such surfaces for a given G?
Key Facts
For any compact Riemann surface X of genus g ≥ 2, we have |Aut(X )| ≤ 84(g − 1). This is the Hurwitz Automorphism Theorem. Some consequences are...
12 2 Thus, there are finitely many quasiplatonic surfaces of genus g. 3 For a given finite group G, there are only finitely many quasiplatonic surfaces X with a regular dessin D such that Aut(D) =∼ G. How do we enumerate such surfaces for a given G?
Key Facts
For any compact Riemann surface X of genus g ≥ 2, we have |Aut(X )| ≤ 84(g − 1). This is the Hurwitz Automorphism Theorem. Some consequences are... 1 For any fixed genus g, there are finitely many regular dessins on genus g surfaces.
12 3 For a given finite group G, there are only finitely many quasiplatonic surfaces X with a regular dessin D such that Aut(D) =∼ G. How do we enumerate such surfaces for a given G?
Key Facts
For any compact Riemann surface X of genus g ≥ 2, we have |Aut(X )| ≤ 84(g − 1). This is the Hurwitz Automorphism Theorem. Some consequences are... 1 For any fixed genus g, there are finitely many regular dessins on genus g surfaces. 2 Thus, there are finitely many quasiplatonic surfaces of genus g.
12 Key Facts
For any compact Riemann surface X of genus g ≥ 2, we have |Aut(X )| ≤ 84(g − 1). This is the Hurwitz Automorphism Theorem. Some consequences are... 1 For any fixed genus g, there are finitely many regular dessins on genus g surfaces. 2 Thus, there are finitely many quasiplatonic surfaces of genus g. 3 For a given finite group G, there are only finitely many quasiplatonic surfaces X with a regular dessin D such that Aut(D) =∼ G. How do we enumerate such surfaces for a given G?
12 Two actions 1 and 2 are equivalent if 1(G) and 2(G) are conjugate in Homeo+(X ).
For a Riemann surface X of genus g ≥ 2, a group G acts conformally X if there is a monomorphism : G → Aut(X ).
Group Acting on a Surface
A group G acts topologically on a surface X of genus g ≥ 2 if there is a monomorphism : G → Homeo+(X ).
13 For a Riemann surface X of genus g ≥ 2, a group G acts conformally X if there is a monomorphism : G → Aut(X ).
Group Acting on a Surface
A group G acts topologically on a surface X of genus g ≥ 2 if there is a monomorphism : G → Homeo+(X ).
Two actions 1 and 2 are equivalent if 1(G) and 2(G) are conjugate in Homeo+(X ).
13 Group Acting on a Surface
A group G acts topologically on a surface X of genus g ≥ 2 if there is a monomorphism : G → Homeo+(X ).
Two actions 1 and 2 are equivalent if 1(G) and 2(G) are conjugate in Homeo+(X ).
For a Riemann surface X of genus g ≥ 2, a group G acts conformally X if there is a monomorphism : G → Aut(X ).
13 Proposition (Wootton, 2007) ∼ The number of topological group actions of G on X = H/Γ where ∼ G = ∆(n1, n2, n3)/Γ equals the number of PSL2(R)-conjugacy classes of Γ.
The triple (n1, n2, n3) is called a signature of X . Each ni is called a period.
Group Actions and Riemann Surfaces
Theorem (Wootton, 2007) A topological group action of G on a surface X extends to a unique conformal action if and only if G is a quotient of a hyperbolic triangle group ∆(n1, n2, n3).
14 The triple (n1, n2, n3) is called a signature of X . Each ni is called a period.
Group Actions and Riemann Surfaces
Theorem (Wootton, 2007) A topological group action of G on a surface X extends to a unique conformal action if and only if G is a quotient of a hyperbolic triangle group ∆(n1, n2, n3).
Proposition (Wootton, 2007) ∼ The number of topological group actions of G on X = H/Γ where ∼ G = ∆(n1, n2, n3)/Γ equals the number of PSL2(R)-conjugacy classes of Γ.
14 Group Actions and Riemann Surfaces
Theorem (Wootton, 2007) A topological group action of G on a surface X extends to a unique conformal action if and only if G is a quotient of a hyperbolic triangle group ∆(n1, n2, n3).
Proposition (Wootton, 2007) ∼ The number of topological group actions of G on X = H/Γ where ∼ G = ∆(n1, n2, n3)/Γ equals the number of PSL2(R)-conjugacy classes of Γ.
The triple (n1, n2, n3) is called a signature of X . Each ni is called a period.
14 Qr ai Let n = i=1 pi be the prime factorization of n. Signature T = number of distinct topological actions w Y pi − 2 (n1, n2, n3) T = φ(gcd(n1, n2, n3)) · pi − 1 i=1 w ! 1 Y pi − 2 (n1, n, n) T = 2 τ1(n, n1) + φ(n) · pi − 1 i=1 r ! 1 Y pi − 2 (n, n, n) T = 6 3 + 2τ2(n) + φ(n) · pi − 1 i=1 τ1(n1, n) = number of noncongruent, nonzero solutions to 2 x + 2x ≡ 0 mod n where gcd(x, n) = n/n1; 2 τ2(n) = number of noncongruent solutions to x + x + 1 ≡ 0 mod n; w ≥ 0 is an integer representing the number of primes (including multiplicity) shared in common among n1, n2, n3.
Enumerating Topological Cyclic Actions - Benim, Wootton 2013
15 τ1(n1, n) = number of noncongruent, nonzero solutions to 2 x + 2x ≡ 0 mod n where gcd(x, n) = n/n1; 2 τ2(n) = number of noncongruent solutions to x + x + 1 ≡ 0 mod n; w ≥ 0 is an integer representing the number of primes (including multiplicity) shared in common among n1, n2, n3.
Enumerating Topological Cyclic Actions - Benim, Wootton 2013
Qr ai Let n = i=1 pi be the prime factorization of n. Signature T = number of distinct topological actions w Y pi − 2 (n1, n2, n3) T = φ(gcd(n1, n2, n3)) · pi − 1 i=1 w ! 1 Y pi − 2 (n1, n, n) T = 2 τ1(n, n1) + φ(n) · pi − 1 i=1 r ! 1 Y pi − 2 (n, n, n) T = 6 3 + 2τ2(n) + φ(n) · pi − 1 i=1
15 Enumerating Topological Cyclic Actions - Benim, Wootton 2013
Qr ai Let n = i=1 pi be the prime factorization of n. Signature T = number of distinct topological actions w Y pi − 2 (n1, n2, n3) T = φ(gcd(n1, n2, n3)) · pi − 1 i=1 w ! 1 Y pi − 2 (n1, n, n) T = 2 τ1(n, n1) + φ(n) · pi − 1 i=1 r ! 1 Y pi − 2 (n, n, n) T = 6 3 + 2τ2(n) + φ(n) · pi − 1 i=1 τ1(n1, n) = number of noncongruent, nonzero solutions to 2 x + 2x ≡ 0 mod n where gcd(x, n) = n/n1; 2 τ2(n) = number of noncongruent solutions to x + x + 1 ≡ 0 mod n; w ≥ 0 is an integer representing the number of primes (including multiplicity) shared in common among n1, n2, n3. 15 Let QC(n) be the number of distinct topological actions of Cn on quasiplatonic surfaces. This also counts the number of quasiplatonic surfaces. Compute QC(n) via the following procedure:
1 find all admissible signatures( n1, n2, n3) for a given n; 2 for each signature, use one of three different formulas giving the number of nonequivalent quasiplatonic cyclic actions on surfaces of that signature; 3 add up all values given by the formulas from all possible signatures for n. This number will be QC(n).
The Number of Quasiplatonic Cyclic Surfaces
16 Compute QC(n) via the following procedure:
1 find all admissible signatures( n1, n2, n3) for a given n; 2 for each signature, use one of three different formulas giving the number of nonequivalent quasiplatonic cyclic actions on surfaces of that signature; 3 add up all values given by the formulas from all possible signatures for n. This number will be QC(n).
The Number of Quasiplatonic Cyclic Surfaces
Let QC(n) be the number of distinct topological actions of Cn on quasiplatonic surfaces. This also counts the number of quasiplatonic surfaces.
16 1 find all admissible signatures( n1, n2, n3) for a given n; 2 for each signature, use one of three different formulas giving the number of nonequivalent quasiplatonic cyclic actions on surfaces of that signature; 3 add up all values given by the formulas from all possible signatures for n. This number will be QC(n).
The Number of Quasiplatonic Cyclic Surfaces
Let QC(n) be the number of distinct topological actions of Cn on quasiplatonic surfaces. This also counts the number of quasiplatonic surfaces. Compute QC(n) via the following procedure:
16 2 for each signature, use one of three different formulas giving the number of nonequivalent quasiplatonic cyclic actions on surfaces of that signature; 3 add up all values given by the formulas from all possible signatures for n. This number will be QC(n).
The Number of Quasiplatonic Cyclic Surfaces
Let QC(n) be the number of distinct topological actions of Cn on quasiplatonic surfaces. This also counts the number of quasiplatonic surfaces. Compute QC(n) via the following procedure:
1 find all admissible signatures( n1, n2, n3) for a given n;
16 3 add up all values given by the formulas from all possible signatures for n. This number will be QC(n).
The Number of Quasiplatonic Cyclic Surfaces
Let QC(n) be the number of distinct topological actions of Cn on quasiplatonic surfaces. This also counts the number of quasiplatonic surfaces. Compute QC(n) via the following procedure:
1 find all admissible signatures( n1, n2, n3) for a given n; 2 for each signature, use one of three different formulas giving the number of nonequivalent quasiplatonic cyclic actions on surfaces of that signature;
16 The Number of Quasiplatonic Cyclic Surfaces
Let QC(n) be the number of distinct topological actions of Cn on quasiplatonic surfaces. This also counts the number of quasiplatonic surfaces. Compute QC(n) via the following procedure:
1 find all admissible signatures( n1, n2, n3) for a given n; 2 for each signature, use one of three different formulas giving the number of nonequivalent quasiplatonic cyclic actions on surfaces of that signature; 3 add up all values given by the formulas from all possible signatures for n. This number will be QC(n).
16 Signature T (n1, n2, n3) (4, 5, 20) T = 1 (4, 10, 20) T = 1 (2, 20, 20) T = 1 (5, 20, 20) T = 2 (10, 20, 20) T = 2
Then QC(20) = 1 + 1 + 1 + 2 + 2 = 7.
Example
Let n = 20. Let T (n1, n2, n3) denote the number of topological actions of Cn on a surface with signature (n1, n2, n3).
17 Then QC(20) = 1 + 1 + 1 + 2 + 2 = 7.
Example
Let n = 20. Let T (n1, n2, n3) denote the number of topological actions of Cn on a surface with signature (n1, n2, n3).
Signature T (n1, n2, n3) (4, 5, 20) T = 1 (4, 10, 20) T = 1 (2, 20, 20) T = 1 (5, 20, 20) T = 2 (10, 20, 20) T = 2
17 Example
Let n = 20. Let T (n1, n2, n3) denote the number of topological actions of Cn on a surface with signature (n1, n2, n3).
Signature T (n1, n2, n3) (4, 5, 20) T = 1 (4, 10, 20) T = 1 (2, 20, 20) T = 1 (5, 20, 20) T = 2 (10, 20, 20) T = 2
Then QC(20) = 1 + 1 + 1 + 2 + 2 = 7.
17 Then
QC(p) = T (p, p, p) 1 (p + 1) p ≡ 5 mod 6 6 = . 1 2 6 (p + 1) + 3 p ≡ 1 mod 6
Example
For n = p ≥ 5 a prime, there is only one admissible signature: (p, p, p).
18 Example
For n = p ≥ 5 a prime, there is only one admissible signature: (p, p, p). Then
QC(p) = T (p, p, p) 1 (p + 1) p ≡ 5 mod 6 6 = . 1 2 6 (p + 1) + 3 p ≡ 1 mod 6
18 Theorem (C, 2018)
a1 Qr ai Suppose n is even and n ≥ 8, so that n = 2 i=2 pi . Then the number of distinct topological actions of Cn on quasiplatonic surfaces is
! r−2 r 2 a1 = 1 a1 a2 ar a1−2 Y ai −1 r−1 QC(2 p2 ··· pr ) = 2 pi (pi + 1) − 1 + 2 a1 = 2 . r i=2 2 a1 ≥ 3
Corollary (C, 2018)
Let r(Cn) denote the number of regular dessins with Cn as their automorphism group. Then for even n ≥ 8,
r−2 2 a1 = 1 1 r−1 QC(n) − r(Cn) = −1 + 2 a1 = 2 . 6 r 2 a1 ≥ 3
Preliminary Results
19 Corollary (C, 2018)
Let r(Cn) denote the number of regular dessins with Cn as their automorphism group. Then for even n ≥ 8,
r−2 2 a1 = 1 1 r−1 QC(n) − r(Cn) = −1 + 2 a1 = 2 . 6 r 2 a1 ≥ 3
Preliminary Results
Theorem (C, 2018)
a1 Qr ai Suppose n is even and n ≥ 8, so that n = 2 i=2 pi . Then the number of distinct topological actions of Cn on quasiplatonic surfaces is
! r−2 r 2 a1 = 1 a1 a2 ar a1−2 Y ai −1 r−1 QC(2 p2 ··· pr ) = 2 pi (pi + 1) − 1 + 2 a1 = 2 . r i=2 2 a1 ≥ 3
19 Preliminary Results
Theorem (C, 2018)
a1 Qr ai Suppose n is even and n ≥ 8, so that n = 2 i=2 pi . Then the number of distinct topological actions of Cn on quasiplatonic surfaces is
! r−2 r 2 a1 = 1 a1 a2 ar a1−2 Y ai −1 r−1 QC(2 p2 ··· pr ) = 2 pi (pi + 1) − 1 + 2 a1 = 2 . r i=2 2 a1 ≥ 3
Corollary (C, 2018)
Let r(Cn) denote the number of regular dessins with Cn as their automorphism group. Then for even n ≥ 8,
r−2 2 a1 = 1 1 r−1 QC(n) − r(Cn) = −1 + 2 a1 = 2 . 6 r 2 a1 ≥ 3
19 QC(n) Graph
20 Derive QC(n) formula for positive, odd integers n. Explore the darker lines of the QC(n) graph. Generalize methods to any quasiplatonic group; i.e., find all ∼ topological actions of G = ∆/Γ on surfaces X = H/Γ.
Current Work and Future Directions
21 Explore the darker lines of the QC(n) graph. Generalize methods to any quasiplatonic group; i.e., find all ∼ topological actions of G = ∆/Γ on surfaces X = H/Γ.
Current Work and Future Directions
Derive QC(n) formula for positive, odd integers n.
21 Generalize methods to any quasiplatonic group; i.e., find all ∼ topological actions of G = ∆/Γ on surfaces X = H/Γ.
Current Work and Future Directions
Derive QC(n) formula for positive, odd integers n. Explore the darker lines of the QC(n) graph.
21 Current Work and Future Directions
Derive QC(n) formula for positive, odd integers n. Explore the darker lines of the QC(n) graph. Generalize methods to any quasiplatonic group; i.e., find all ∼ topological actions of G = ∆/Γ on surfaces X = H/Γ.
21 References
Benim, R., Wootton, A. Enumerating Quasiplatonic Cyclic Group Actions. Journal of Mathematics, 43(5), 2013. Girondo, E., Gonz´alez-Diez, G. Introduction to Compact Riemann Surfaces and Dessins d’Enfants. Cambridge: Cambridge UP, 2012. Print. Harvey, W. J. Cyclic groups of automorphisms of a compact Riemann surface. The Quarterly Journal of Mathematics, 17(1), 86-97. 1966. Jones, G. A. Regular dessins with a given automorphism group. Contemporary Mathematics, 629, 245-260. 2014. Jones, G. A., Wolfart, J. Dessins d’Enfants on Riemann Surfaces. Switzerland: Springer International Publishing, 2016. Print. Wootton, A. Extending topological group actions to conformal group actions. Albanian Journal of Mathematics (ISNN: 1930-1235), 1(3), 133-143. 2007.
Questions? Thank you!
22 Theorem (Harvey, 1966)
Let n = lcm(n1, n2, n3).Then Cn acts on X of genus g ≥ 2 with signature (n1, n2, n3) if and only if
1 n = lcm(n1, n2) = lcm(n1, n3) = lcm(n2, n3);
2 for n even, exactly two of n1, n2, n3 must be divisible by the maximum power of two dividing n; 3 the Riemann-Hurwitz formula holds: n 1 1 1 g = 1 + 1 − − − . 2 n1 n2 n3
Harvey’s Theorem for the Quasiplatonic Case
23 Then Cn acts on X of genus g ≥ 2 with signature (n1, n2, n3) if and only if
1 n = lcm(n1, n2) = lcm(n1, n3) = lcm(n2, n3);
2 for n even, exactly two of n1, n2, n3 must be divisible by the maximum power of two dividing n; 3 the Riemann-Hurwitz formula holds: n 1 1 1 g = 1 + 1 − − − . 2 n1 n2 n3
Harvey’s Theorem for the Quasiplatonic Case
Theorem (Harvey, 1966)
Let n = lcm(n1, n2, n3).
23 1 n = lcm(n1, n2) = lcm(n1, n3) = lcm(n2, n3);
2 for n even, exactly two of n1, n2, n3 must be divisible by the maximum power of two dividing n; 3 the Riemann-Hurwitz formula holds: n 1 1 1 g = 1 + 1 − − − . 2 n1 n2 n3
Harvey’s Theorem for the Quasiplatonic Case
Theorem (Harvey, 1966)
Let n = lcm(n1, n2, n3).Then Cn acts on X of genus g ≥ 2 with signature (n1, n2, n3) if and only if
23 2 for n even, exactly two of n1, n2, n3 must be divisible by the maximum power of two dividing n; 3 the Riemann-Hurwitz formula holds: n 1 1 1 g = 1 + 1 − − − . 2 n1 n2 n3
Harvey’s Theorem for the Quasiplatonic Case
Theorem (Harvey, 1966)
Let n = lcm(n1, n2, n3).Then Cn acts on X of genus g ≥ 2 with signature (n1, n2, n3) if and only if
1 n = lcm(n1, n2) = lcm(n1, n3) = lcm(n2, n3);
23 3 the Riemann-Hurwitz formula holds: n 1 1 1 g = 1 + 1 − − − . 2 n1 n2 n3
Harvey’s Theorem for the Quasiplatonic Case
Theorem (Harvey, 1966)
Let n = lcm(n1, n2, n3).Then Cn acts on X of genus g ≥ 2 with signature (n1, n2, n3) if and only if
1 n = lcm(n1, n2) = lcm(n1, n3) = lcm(n2, n3);
2 for n even, exactly two of n1, n2, n3 must be divisible by the maximum power of two dividing n;
23 Harvey’s Theorem for the Quasiplatonic Case
Theorem (Harvey, 1966)
Let n = lcm(n1, n2, n3).Then Cn acts on X of genus g ≥ 2 with signature (n1, n2, n3) if and only if
1 n = lcm(n1, n2) = lcm(n1, n3) = lcm(n2, n3);
2 for n even, exactly two of n1, n2, n3 must be divisible by the maximum power of two dividing n; 3 the Riemann-Hurwitz formula holds: n 1 1 1 g = 1 + 1 − − − . 2 n1 n2 n3
23