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 n 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 n 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 jAut(X )j ≤ 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 jAut(X )j ≤ 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 jAut(X )j ≤ 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 jAut(X )j ≤ 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 jAut(X )j ≤ 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 jAut(X )j ≤ 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).