Discrete group actions on orientable surfaces

Transcription

1 Discrete group actions on orientable surfaces Ján Karabáš and Roman Nedela Matej Bel University Mathematical Institute, Slovak Academy of Sciences Banská Bystrica, Slovakia Geometry Days, Novosibirsk, August 2013

2 Maps Map is a drawing of a finite connected graph on a surface without edge-crossings. More formally: (topological) Map is a cell-decomposition of a surface. Surfaces: closed, connected, orientable. Combinatorial map: M = (D; R, L), D - finite set, R, L Sym(D), R, L is transitive on D, L 2 = id. Underlying graph: V orbits of R, E orbits of L, faces orbits RL on D. Incidency by intersections. (Orientation-preserving) automorphism: ϕ C Sym(D) ( R, L ) = Aut + (M). Reflections: S = {ϱ : ϱ Sym(D), R ϱ = R 1 }; Automorphism group of a map M: Aut(M) = Aut + (M), S, [Aut(M) : Aut + (M)] 2; An automorphism of a map on a surface S extends to a self-homeomorphism of S. Group actions... (J. Karabáš, UMB) 2 / 20

3 Highly symmetrical maps (a) (33 ) (f) (3.6.6) (b) (43 ) (g) ( ) (h) (3.8.8) (m) ( ) (c) (34 ) (i) (4.6.6) (e) (35 ) (j) ( ) (k) (4.6.10) (n) ( ) (o) (5.6.6) (p) ( ) (q) (4.6.10) (s) (4.4.5) Group actions... (J. Karabáš, UMB) (d) (53 ) (l) (34.5)± (r) (34.5)± (t) ( ) 3 / 20

4 Example: A toroidal map of local type ( ) A map defined from the universal cover, the parameters determining the fundamental region are a = 2, b = 0 Group actions... (J. Karabáš, UMB) 4 / 20

5 Example: The same map defined as a lift of its quotient Aut + (M) = x, y x 3 = y 2 = (xy) 6 = 1, (x 1 yxy) 2 (xy 1 xy) 0 = 1 Group actions... (J. Karabáš, UMB) 5 / 20

6 Maps and group actions 1 Map Action of a discrete group on an underlying surface of the map. 2 Every finite group of automorphisms of a surface S is a group of automorphisms of a (Cayley) vertex-transitive map on S; 3 Every finite group appears as a discrete group of automorphisms of S (compact, closed); 4 Not all actions of finite groups can be seen; 5 Cyclic point stabilisers orientable surfaces; 6 Dihedral point stabilisers non-orientable surfaces; Group actions... (J. Karabáš, UMB) 6 / 20

7 Main problem Problem Given genus g classify maps with high degree of symmetry (few orbits of Aut + (M) or of Aut(M) on vertices, edges, faces, darts, flags,... ). Subproblem A. Classify Archimedean maps of given genus. Subproblem B. Classify non-degenerate (polytopal, polyhedral) edge-transitive maps of given genus Remark 1. Each highly symmetrical map defines a discrete group of automorphisms, Remark 2. A discrete group determines a map completely!!! Remark 3. Turn attention to the classification of discrete groups, an advantage: much more applications Group actions... (J. Karabáš, UMB) 7 / 20

8 Group/covering duality and universal covers Smooth coverings - classical topic in homotopy theory; Regular branched coverings between orientable surfaces, Riemann-Hurwitz equation an action of a discrete group G is orientation preserving, 2 2g = G ( 2 2γ r i=1 ( 1 1 m i ) ) ; i : m i 2 Z; m i G ; Fundamental group π 1 (S): elements (eq. classes of) closed curves is S, contractible (closed) curves are identities, operation composition of curves; Subgroup/covering correspondence: a covering determines a subgroup of π 1 (S), a subgroup G π 1 (S) determines a regular covering S S with CT p = G; Universal cover over S: a simply connected surface S (1 π 1 (S)) covering all the covers of S; Fundamental group of a surface of genus g: g π 1 (S g) = a 1, b 1,..., a g, b g [a i, b i ]. i=1 Group actions... (J. Karabáš, UMB) 8 / 20

9 Uniformization Koebe Theorem ( S, π 1 (O)) U K (S g, G) (O(γ; {m 1,..., m r}), I) G A discrete group G is an epimorphic image G = π 1 (O)/K for a quotient orbifold S g/g = O(γ; {m 1, m 2,..., m r}). Fuchsian group F = π 1 (O) x 1,..., x r, a 1, b 1,..., a γ, b γ x m 1 1 = x m 2 2 =... = x mr r = 1, is the orbifold fundamental group π 1 (O). γ r [a i, b i ] x j = 1. i=1 j=1 Group actions... (J. Karabáš, UMB) 9 / 20

10 Canonical presentation of orbifold fundamental group 1 canonical quotient map M is a bouquet of r loops on the surface of genus γ, 2 every loop is the boundary of a face containing exactly one branch-point with respective branch-index m i, 3 outer face of the map is an (r + 2γ)-gon; contains no branch-point. O(0; {2, 2, 3, 3}) O(1; {3, 3}) x, y, z, w x 2 = y 2 = z 3 = w 3 = xyzw = 1 x, y, a, b x 3 = y 3 = [a, b]xy = 1 Group actions... (J. Karabáš, UMB) 10 / 20

11 Classification: The procedure Requirements: g > 1 - genus of a surface Output: list of all actions of finite groups acting on S g. (1) solve Riemann-Hurwitz equation numerically; (2) construct Fuchsian groups F(γ, {m 1,..., m r}) given by solutions of (1); low-index subgroups approach (3 ) search for all low-index normal subgroups of index G ; (4 ) for every K F test whether ε : F F/K is order-preserving on elliptic generators of F; STOP. or examining epimorphisms F G (3 ) given F and G construct all epimorphisms F G; (4 ) test whether contructed epimorphisms are order-preserving on elliptic generators of F; (5 ) choose epimorphisms which will represent actions i.e. one epimorphism for a particular K F, s.t. G = F/K; STOP. Group actions... (J. Karabáš, UMB) 11 / 20

12 Equivalence of actions If F is a Fuchsian group, epimorphisms ϕ: F G and ψ : F G give equivalent actions, provided that for α Aut(G) and the generators x i, a j, b j of F we have α(ϕ(x i )) = ψ(x i ), α(ϕ(a j )) = ψ(a j ), α(ϕ(b j )) = ψ(b j ), i {1,..., r}, j {0,..., γ}, j {0,..., γ}. Example Symetric group S 3 possesses two non-equivalent actions on genus 2 surface, described by Cayley vectors (x, x, y 1, y, ) (xy, x, y 1, y 1 ) where S 3 = x, y x 2, y 3, (y 1 x) 2 and orbifold has signature (0; {2, 2, 3, 3}). Group actions... (J. Karabáš, UMB) 12 / 20

13 Riemann-Hurwitz equation: Numeric solutions Rieman-Hurwitz equation ( 2 2g = G 2 2γ Criteria for a solution: 1 γ g, 2 r 2g + 2, r ) (1 ) 1mi 3 for any i, m i is a non-trivial divisor of G, 4 G 84(g 1). We obtain a set of pairs (signature of an orbifold, order of respective group) G, (γ; {m 1,..., m r}). i=1 Not every signature is g-admissible: RH numerically holds, but no action of G does exist. (0; {2, 3, 7}) is not 2-admissible no group of order 84 acts on S 2 ; (0; {2, 4, 5}) is not 3-admissible no group of order 20 acts on S 3, etc... Group actions... (J. Karabáš, UMB) 13 / 20

14 Genus 2 actions: Arithmetics vs. group theory G Orbifold Action G Orbifold Action 1 (2; {}) 1 12 (0; {2, 2, 2, 3}) D 12 2 (1; {2, 2}) C 2 12 (0; {3, 4, 4}) C 3 : C 4 2 (0; {2, 2, 2, 2, 2, 2}) C 2 12 (0; {3, 3, 6}) 3 (1; {3}) 12 (0; {2, 6, 6}) C 6 C 2 3 (0; {3, 3, 3, 3}) C 3 12 (0; {2, 4, 12}) 4 (1; {2}) 15 (0; {3, 3, 5}) 4 (0; {2, 2, 2, 2, 2}) C 2 C 2 16 (0; {2, 4, 8}) QD 16 4 (0; {2, 2, 4, 4}) C 4 18 (0; {2, 3, 18}) 5 (0; {5, 5, 5}) C 5 20 (0; {2, 5, 5}) 6 (0; {2, 2, 3, 3}) C 6, S 3 24 (0; {3, 3, 3}) SL(2, 3) 6 (0; {2, 2, 2, 6}) 24 (0; {2, 4, 6}) (C 6 C 2 ) : C 2 6 (0; {3, 6, 6}) C 6 24 (0; {2, 3, 12}) 8 (0; {2, 2, 2, 4}) D 8 30 (0; {2, 3, 10}) 8 (0; {4, 4, 4}) Q 8 36 (0; {2, 3, 9}) 8 (0; {2, 8, 8}) C 8 40 (0; {2, 4, 5}) 9 (0; {3, 3, 9}) 48 (0; {2, 3, 8}) GL(2, 3) 10 (0; {2, 5, 10}) C (0; {2, 3, 7}) Group actions... (J. Karabáš, UMB) 14 / 20

15 Main result Previous results Broughton 89 classification for genera 2 and 3 Bogopolski 91 classification for genus 4 Kuribayashi and Kimura 90 s classification for genus 5 State-of-art Abstract structure of groups and g-admissible orbifold types were dermined up to genus 21 (JK), for large groups much further (see Conder s web page, G 4(g 1)) (almost) All actions determined by g-admissible groups are known for 2 g 21 (JK); At present we have completed the list of actions up to genus 11, see Small 11-admissible troublemakers with more than kernels: C 2 C 2 of types (2; {2 6 }), (1; {2 10 }) or (0; {2 14 }), S 3 of type (0; {2 8, 3 2 }), and several other abelian and dihedral groups with certain types. Group actions... (J. Karabáš, UMB) 15 / 20

16 Samples Discrete groups with actions on S 0 G Epi O Sign. G Epi O Sign. I 1 (0; ) A 4 24 (0; {2, 3, 3}) C m ϕ(m) (0; {m, m}) S 4 24 (0; {2, 3, 4}) D 2m mϕ(m) (0; {2, 2, m}) A (0; {2, 3, 5}) By RH bound, for g > 1 there are finitely many g-admissible orbifolds. see for 2 g 21 Discrete groups acting on S 2 G Epi O #act. Sign. G Epi O #act Sign. I 1 1 (2; ) Q (0; {4, 4, 4}) C (1; {2, 2}) D (0; {2 3, 4}) C (0; {2 6 }) C (0; {2, 5, 10}) C (0; {3 4 }) C 2 C (0; {2, 6, 6}) C (0; {2 2, 4 2 }) C 3 : C (0; {3, 4, 4}) C (0; {2 5 }) D (0; {2 3, 3}) C (0; {5, 5, 5}) C 8 : C (0; {2, 4, 8}) C (0; {3, 6, 6}) (C 2 C 6 ) : C (0; {2, 4, 6}) C (0; {2 2, 3 2 }) SL 2 (3) 24 1 (0; {3, 3, 4}) S (0; {2 2, 3 2 }) GL 2 (3) 48 1 (0; {2, 3, 8}) C (0; {2, 8, 8}) Group actions... (J. Karabáš, UMB) 16 / 20

17 Maximal actions Accola & MacLachlan - the maximal order of a discrete group acting on S g is 8(g + 1). Hurwitz action of genus g - action of a group of order 84(g 1) with orbifold type (0; {2, 3, 7}) - attained for g = 3, 7, 14, 17, 118, 129, 146, Conder, and others Local maximum for a genus g - actions of group of order > 8(g + 1) with orbifold type (0; {k, m, n}) - a typical situation. From Accola & MacLachlan - the respective orbifold type of a maximal action is either (0; {k, m, n}), or (0; {2, 2, 2, 3})! Problem Does there exist an orientable surface S g such that the respective maximal action is of orbifold type (0; {2, 2, 2, 3})? YES: g = 126, G = 1500 is the first case - Conder Open Problem Does the series of surfaces S g, such that maximal action on S g is of type (0; {2, 2, 2, 3}) exist? Compute its members. Is the series infinite? Group actions... (J. Karabáš, UMB) 17 / 20

18 Equivalence of actions (again) Having ϕ: F G and ψ : F G and α Aut(G), two actions are equivalent if α(ϕ(x i )) = ψ(x i ), α(ϕ(a j )) = ψ(a j ), α(ϕ(b j )) = ψ(b j ), i {1,..., r}, j {0,..., γ}, j {0,..., γ}. Actions are equivariant (Malnič, Nedela, and Škoviera) - equivalent actions define the same maps by voltage assignments. Other possibilities: Take α Inn(G), preserving the prescribed order of generators; Take α A Aut(G), preserving the prescribed order of generators; Define equivalence α ϕ = ψ, i.e. α({ϕ(x i ), ϕ(a j ), ϕ(b j )}) = {ψ(x i ), ψ(a j ), ψ(b j )}, i = 0,..., r, j = 1,..., γ;??? Open Problem Find (convenient) group-theoretic expressions of common equivalences known to topologists. Group actions... (J. Karabáš, UMB) 18 / 20

19 Classification of discrete group actions on arbitrary surfaces? Klein equation relating a surface with its quotients r ) µ(s) = G α γ + k 2 + (1 1mi i=1 k h i i=1 j=1 ) (1 1nij gives the signature (γ; ±; [m 1,..., m r]; {(n 1,1,..., n 1,h1 ),..., (n k,1,..., n k,h k )}). Further direct attack is (surely) intractable. Backdoor attack (Gromadzki, JK, and Kozłowska-Walania) 1 Take an orientable action of G and the its (+)signature σ = (γ, {m 1,..., m r}); 2 Fuchsian group F σ is an index 2 subgroup of several NEC groups Λ i ; 3 We have an algorithm giving all possible signatures (and presentations of Λ i ); 4 Koebe Theorem gives Λ i /K = C 2.G, where K Λ i, torsion free; 5 Parameters of a surface where C 2.G is acting are given by µ(s) = αg + k 2; DONE. Backdoor attack does not give complete census it is done by algebraic genus. We are able to check previous results and go much further... Group actions... (J. Karabáš, UMB) 19 / 20

20 Done. Thank you. Group actions... (J. Karabáš, UMB) 20 / 20