Problem Classify regular dessins. ◮ Classify regular dessins on a surface of a given genus;
Problem Classify regular dessins. ◮ Classify regular dessins on a surface of a given genus; ◮ Classify regular dessins with a given underlying graph;
Problem Classify regular dessins. ◮ Classify regular dessins on a surface of a given genus; ◮ Classify regular dessins with a given underlying graph; ◮ Classify regular dessins with a given automorphism group.
Algebraic description Each dessin D can be regarded as a transitive permutation representation θ : F 2 → A = Mon ( D ) , X �→ ρ, Y �→ λ, where F 2 = � X , Y | −� is the free group of rank two.
Algebraic description Each dessin D can be regarded as a transitive permutation representation θ : F 2 → A = Mon ( D ) , X �→ ρ, Y �→ λ, where F 2 = � X , Y | −� is the free group of rank two. Define N = θ − 1 ( A e ) , e ∈ E . Then the subgroup N is unique up to conjugacy, and is called the dessin subgroup of D .
Algebraic description Each dessin D can be regarded as a transitive permutation representation θ : F 2 → A = Mon ( D ) , X �→ ρ, Y �→ λ, where F 2 = � X , Y | −� is the free group of rank two. Define N = θ − 1 ( A e ) , e ∈ E . Then the subgroup N is unique up to conjugacy, and is called the dessin subgroup of D . Theorem A dessin D is regular iff the associated dessin subgroup N is normal in F 2 , in which case Aut ( D ) ∼ = F 2 / N.
Theorem The set R ( G ) of isomorphism classes of regular dessins with a given automorphism group G corresponds bijectively ◮ to the set N ( G ) of normal subgroups N of F 2 such that G = F 2 / N,
Theorem The set R ( G ) of isomorphism classes of regular dessins with a given automorphism group G corresponds bijectively ◮ to the set N ( G ) of normal subgroups N of F 2 such that G = F 2 / N, ◮ or to the orbits of Aut ( G ) on the set Ω( G ) of generating pairs ( x , y ) of G, i.e., Ω( G ) = { ( x , y ) | G = � x , y �} .
Theorem The set R ( G ) of isomorphism classes of regular dessins with a given automorphism group G corresponds bijectively ◮ to the set N ( G ) of normal subgroups N of F 2 such that G = F 2 / N, ◮ or to the orbits of Aut ( G ) on the set Ω( G ) of generating pairs ( x , y ) of G, i.e., Ω( G ) = { ( x , y ) | G = � x , y �} . Corollary Let G be a finite group. Then the number r ( G ) of isomorphism classes of regular dessin D with Aut ( D ) ∼ = G is | Ω( G ) | r ( G ) = | Aut ( G ) | .
Example ◮ Let G = C n be the cyclic group of order n , then r ( G ) = ψ ( n ) where (1 + 1 � ψ ( n ) = n p ) p | n is the Dedekind’s totient function.
Example ◮ Let G = C n be the cyclic group of order n , then r ( G ) = ψ ( n ) where (1 + 1 � ψ ( n ) = n p ) p | n is the Dedekind’s totient function. ◮ Let G = C n × C m where m | n , then r ( G ) = ψ ( n / m ).
Example ◮ Let G = C n be the cyclic group of order n , then r ( G ) = ψ ( n ) where (1 + 1 � ψ ( n ) = n p ) p | n is the Dedekind’s totient function. ◮ Let G = C n × C m where m | n , then r ( G ) = ψ ( n / m ). ◮ Let G = D 2 n , n ≥ 3, then r ( G ) = 3.
Example ◮ Let G = C n be the cyclic group of order n , then r ( G ) = ψ ( n ) where (1 + 1 � ψ ( n ) = n p ) p | n is the Dedekind’s totient function. ◮ Let G = C n × C m where m | n , then r ( G ) = ψ ( n / m ). ◮ Let G = D 2 n , n ≥ 3, then r ( G ) = 3. ◮ Let G = A 5 , then r ( G ) = 19.
Example ◮ Let G = C n be the cyclic group of order n , then r ( G ) = ψ ( n ) where (1 + 1 � ψ ( n ) = n p ) p | n is the Dedekind’s totient function. ◮ Let G = C n × C m where m | n , then r ( G ) = ψ ( n / m ). ◮ Let G = D 2 n , n ≥ 3, then r ( G ) = 3. ◮ Let G = A 5 , then r ( G ) = 19. ◮ Let G = Q 8 , then r ( G ) = 1.
Dessin operations Let D be a dessin, and N ≤ F 2 be the associated dessin subgroup. Then ◮ each τ ∈ Aut ( F 2 ) transforms N to N τ , and hence transforms D to a dessin D τ .
Dessin operations Let D be a dessin, and N ≤ F 2 be the associated dessin subgroup. Then ◮ each τ ∈ Aut ( F 2 ) transforms N to N τ , and hence transforms D to a dessin D τ . ◮ if τ ∈ Inn ( F 2 ), then N is conjugate to N τ , and hence D ∼ = D τ .
Dessin operations Let D be a dessin, and N ≤ F 2 be the associated dessin subgroup. Then ◮ each τ ∈ Aut ( F 2 ) transforms N to N τ , and hence transforms D to a dessin D τ . ◮ if τ ∈ Inn ( F 2 ), then N is conjugate to N τ , and hence D ∼ = D τ . ◮ so the outer automorphism Out ( F 2 ) = Aut ( F 2 ) Inn ( F 2 ) acts on the isomorphism classes of dessins, and it is called the group of dessin operations.
Invariants of Out ( F 2 ) Up to group isomorphism, ◮ the monodromy group of a dessin D is invariant under Out ( F 2 ); ◮ the automorphism group of D is invariant under Out ( F 2 ).
External symmetry External symmetry D possesses an external symmetry σ if = D σ where σ ∈ Aut ( F 2 ) \ Inn ( F 2 ) . D ∼
External symmetry External symmetry D possesses an external symmetry σ if = D σ where σ ∈ Aut ( F 2 ) \ Inn ( F 2 ) . D ∼ Symmetric dessin D ∼ = D σ where σ : X �→ Y , Y �→ X .
External symmetry External symmetry D possesses an external symmetry σ if = D σ where σ ∈ Aut ( F 2 ) \ Inn ( F 2 ) . D ∼ Symmetric dessin D ∼ = D σ where σ : X �→ Y , Y �→ X . = D σ where σ : X �→ X − 1 , Y �→ Y − 1 . Reflexible dessin D ∼
External symmetry External symmetry D possesses an external symmetry σ if = D σ where σ ∈ Aut ( F 2 ) \ Inn ( F 2 ) . D ∼ Symmetric dessin D ∼ = D σ where σ : X �→ Y , Y �→ X . = D σ where σ : X �→ X − 1 , Y �→ Y − 1 . Reflexible dessin D ∼ Self-Petrie-dual dessin D ∼ = D τ where σ : X �→ X − 1 , Y �→ Y .
External symmetry External symmetry D possesses an external symmetry σ if = D σ where σ ∈ Aut ( F 2 ) \ Inn ( F 2 ) . D ∼ Symmetric dessin D ∼ = D σ where σ : X �→ Y , Y �→ X . = D σ where σ : X �→ X − 1 , Y �→ Y − 1 . Reflexible dessin D ∼ Self-Petrie-dual dessin D ∼ = D τ where σ : X �→ X − 1 , Y �→ Y . Totally symmetric dessin a regular dessin which is invariant under all dessin operations.
Example ◮ The cube on the sphere is symmetric and reflexible, but not self-Petrie-dual.
Example ◮ The cube on the sphere is symmetric and reflexible, but not self-Petrie-dual. ◮ The cube on the torus is symmetric and reflexible, but not self-Petrie-dual.
Example ◮ The cube on the sphere is symmetric and reflexible, but not self-Petrie-dual. ◮ The cube on the torus is symmetric and reflexible, but not self-Petrie-dual. In fact, each of them has the other as its Petrie-dual.
Petrie-dual of cube Example (Cube on the sphere revisited) A zig-zag path in the cube on the sphere. Cut the cube along all zig-zag pathes and glue them together along the new face boundaries we get the cube embedded into the torus.
Petrie-dual of cube Example (Cube on the sphere revisited) A zig-zag path in the cube on the sphere. Cut the cube along all zig-zag pathes and glue them together along the new face boundaries we get the cube embedded into the torus.
Petrie-dual of cube Example (Cube on the sphere revisited) A zig-zag path in the cube on the sphere. Cut the cube along all zig-zag pathes and glue them together along the new face boundaries we get the cube embedded into the torus.
Petrie-dual of cube Example (Cube on the sphere revisited) A zig-zag path in the cube on the sphere. Cut the cube along all zig-zag pathes and glue them together along the new face boundaries we get the cube embedded into the torus.
Petrie-dual of cube Example (Cube on the sphere revisited) A zig-zag path in the cube on the sphere. Cut the cube along all zig-zag pathes and glue them together along the new face boundaries we get the cube embedded into the torus.
Petrie-dual of cube Example (Cube on the sphere revisited) A zig-zag path in the cube on the sphere. Cut the cube along all zig-zag pathes and glue them together along the new face boundaries we get the cube embedded into the torus.
Petrie-dual of cube Example (Cube on the sphere revisited) A zig-zag path in the cube on the sphere. Cut the cube along all zig-zag pathes and glue them together along the new face boundaries we get the cube embedded into the torus.
The universal dessin Let G be a finite 2-generator group, define � K ( G ) = N . N ∈N ( G )
The universal dessin Let G be a finite 2-generator group, define � K ( G ) = N . N ∈N ( G ) Then K ( G ) is the intersection of finitely many normal subgroups of finite index in F 2 , and hence it is normal of finite index in F 2 as well.
The universal dessin Let G be a finite 2-generator group, define � K ( G ) = N . N ∈N ( G ) Then K ( G ) is the intersection of finitely many normal subgroups of finite index in F 2 , and hence it is normal of finite index in F 2 as well. Define U ( G ) to be the regular dessin correponding to K ( G ), and ˆ G = F 2 / K ( G ) .
Theorem (Jones, 2013) ◮ U ( G ) is the smallest regular dessin which covers all regular dessins in R ( G ) .
Theorem (Jones, 2013) ◮ U ( G ) is the smallest regular dessin which covers all regular dessins in R ( G ) . ◮ The group ˆ G underlies a unique regular dessin, i.e., the dessin U ( G ) .
Theorem (Jones, 2013) ◮ U ( G ) is the smallest regular dessin which covers all regular dessins in R ( G ) . ◮ The group ˆ G underlies a unique regular dessin, i.e., the dessin U ( G ) . ◮ The dessin U ( G ) is totally symmetric.
Theorem (Jones, 2013) ◮ U ( G ) is the smallest regular dessin which covers all regular dessins in R ( G ) . ◮ The group ˆ G underlies a unique regular dessin, i.e., the dessin U ( G ) . ◮ The dessin U ( G ) is totally symmetric. ◮ The dessin U ( G ) is defined over the field Q of rational numbers.
Theorem (Jones, 2013) ◮ U ( G ) is the smallest regular dessin which covers all regular dessins in R ( G ) . ◮ The group ˆ G underlies a unique regular dessin, i.e., the dessin U ( G ) . ◮ The dessin U ( G ) is totally symmetric. ◮ The dessin U ( G ) is defined over the field Q of rational numbers. ◮ If G is non-abelian simple, then ˆ G = G r , where r = r ( G ) and G r is the rth direct product of G.
Theorem (Jones, 2013) ◮ U ( G ) is the smallest regular dessin which covers all regular dessins in R ( G ) . ◮ The group ˆ G underlies a unique regular dessin, i.e., the dessin U ( G ) . ◮ The dessin U ( G ) is totally symmetric. ◮ The dessin U ( G ) is defined over the field Q of rational numbers. ◮ If G is non-abelian simple, then ˆ G = G r , where r = r ( G ) and G r is the rth direct product of G. ◮ If G is solvable of derived length d, then so is ˆ G.
Theorem (Jones, 2013) ◮ U ( G ) is the smallest regular dessin which covers all regular dessins in R ( G ) . ◮ The group ˆ G underlies a unique regular dessin, i.e., the dessin U ( G ) . ◮ The dessin U ( G ) is totally symmetric. ◮ The dessin U ( G ) is defined over the field Q of rational numbers. ◮ If G is non-abelian simple, then ˆ G = G r , where r = r ( G ) and G r is the rth direct product of G. ◮ If G is solvable of derived length d, then so is ˆ G. ◮ If G is nilpotent of nilpotence class c, then so is ˆ G.
Example (Jones, 2013) ◮ If G = C n , then ˆ G = C n × C n . In fact, if G = C n × C m , m | n , then ˆ G = C n × C n .
Example (Jones, 2013) ◮ If G = C n , then ˆ G = C n × C n . In fact, if G = C n × C m , m | n , then ˆ G = C n × C n . ◮ If G = A 5 , then ˆ G = G 19 .
Problem Classify finite groups which underlie a unique regular dessin.
Problem Classify finite groups which underlie a unique regular dessin. A a subproblem, we are interested in Problem Classify finite nilpotent groups which underlie a unique regular dessin.
Problem Classify finite groups which underlie a unique regular dessin. A a subproblem, we are interested in Problem Classify finite nilpotent groups which underlie a unique regular dessin. Since every finite nilpotent group is the direct product of its Sylow subgroups, the problem is reduced to Problem Classify finite p-groups which underly a unique regular dessin.
Example Let p be an odd prime and G be the non-abelian non-metacyclic p -group of order p 3 , that is, G = � x , y | x p = y p = z p = [ x , z ] = [ y , z ] = 1 , z := [ x , y ] � .
Example Let p be an odd prime and G be the non-abelian non-metacyclic p -group of order p 3 , that is, G = � x , y | x p = y p = z p = [ x , z ] = [ y , z ] = 1 , z := [ x , y ] � . Then G underlies a unique regular dessin.
Example Let p be an odd prime and G be the non-abelian non-metacyclic p -group of order p 3 , that is, G = � x , y | x p = y p = z p = [ x , z ] = [ y , z ] = 1 , z := [ x , y ] � . Then G underlies a unique regular dessin. For instance when p = 3, the dessin is a regular embedding of the Pappus graph into the torus.
Pappus graph on the torus Example
Main results Theorem (H., Roman Nedela, Na-Er Wang, 2014) A finite p-group G of class at most three which underlies a unique regular dessin is isomorphic to one of the following groups: (A) A single family of class c ( G ) = 1 : G = � x , y | x p a = y p a = [ x , y ] = 1 � ∼ = C p a × C p a , a ≥ 0 .
Continued (B) Three families of class c ( G ) = 2: (1) p > 2 and 1 ≤ b ≤ a , G = � x , y | x p a = y p a = z p b = [ x , z ] = [ y , z ] = 1 , z := [ x , y ] � . (2) p = 2 and 1 ≤ b ≤ a − 1, G = � x , y | x 2 a = y 2 a = z 2 b = [ x , z ] = [ y , z ] = 1 , z := [ x , y ] � . (3) p = 2 and a ≥ 2, G = � x , y | x 2 a = [ x , z ] = [ y , z ] = 1 , x 2 a − 1 = y 2 a − 1 = z 2 a − 2 , z := [ x , y ]
Continued (C) Six families of class c ( G ) = 3: (1) p = 3 and 1 ≤ c < b = a or 1 ≤ c ≤ b ≤ a − 1, G = � x , y | x 3 a = y 3 a = z 3 b = u 3 c = v 3 c = [ x , u ] = [ x , v ] = [ y , u ] = [ y , v ] = 1 , z := [ x , y ] , u := [ z , x ] , v := [ z , y ] � . (2) p > 3 and 1 ≤ c ≤ b ≤ a , G = � x , y | x p a = y p a = z p b = u p c = v p c = [ x , u ] = [ x , v ] = [ y , u ] = [ y , v ] = 1 , z := [ x , y ] , u := [ z , x ] , v := [ z , y ] � . (3) p = 2 and 1 ≤ c ≤ b ≤ a − 1 , G = � x , y | x 2 a = y 2 a = z 2 b = u 2 c = v 2 c = [ x , u ] = [ x , v ] = [ y , u ] = [ y , v ] = 1 , z := [ x , y ] , u := [ z , x ] , v := [ z , y ] � .
Continued (4) p = 2 and 1 ≤ c ≤ b ≤ a − 2 , G = � x , y | x 2 a = y 2 a = z 2 b = u 2 c = v 2 c = [ x , u ] = [ x , v ] = [ y , u ] = [ y , v ] = 1 , x 2 a − 1 = u 2 c − 1 , y 2 a − 1 = v 2 c − 1 , z := [ x , y ] , u := [ z , x ] , v := [ z , y ] � . (5) p = 2 and 1 ≤ c ≤ b ≤ a − 1 , G = � x , y | x 2 a = y 2 a = z 2 b = u 2 c = v 2 c = [ x , u ] = [ x , v ] = [ y , u ] = [ y , v ] = 1 , x 2 a − 1 = z 2 b − 1 , y 2 a − 1 = z 2 b − 1 , z := [ x , y ] , u := [ z , x ] , v := [ z , y ] � .
Continued (6) p = 2 and 1 ≤ c ≤ a − 2 , G = � x , y | x 2 a = y 2 a = z 2 a − 1 = u 2 c = v 2 c = [ x , u ] = [ x , v ] = [ y , u ] = [ y , v ] = 1 , x 2 a − 1 = z 2 a − 2 u 2 c − 1 , y 2 a − 1 = z 2 a − 2 v 2 c − 1 , z := [ x , y ] , u := [ z , x ] , v := [ z , y ] � . Moreover, the groups from distinct families, or from the same family but with distinct parameters, are pairwise non-isomorphic.
Remark In “Groups of Prime Power Order, 2008”, Y. Berkovich and Z. Janko posed a problem of studying p -group G such that | G : Φ( G ) | = p d and | Aut ( G ) | = ( p d − 1) . . . ( p d − p d − 1 ) | Φ( G ) | d , that is, G is a d -generator p -group and its automorphism group Aut ( G ) acts transitively on its generating d -tuples; see Research Problems and Themes I 35(a). Our main result solves this problem when d = 2 and c ( G ) ≤ 3.
G. Gonz´ alez-Diez, A. Jaikin-Zapirain, The absolute Galois group acts faithfully on regular dessins and on Beauville surfaces, preprint, 2013. A. Grothendieck, Esquisse d’un programme, preprint, 1984. K. Hu, R. Nedela, N.-E. Wang, Nilpotent groups of class two which underly a unique regular dessin, Geometriae Dedicate, DOI. 10.1007/s10711-015-0074-8 K. Hu, R. Nedela, N.-E. Wang, Nilpotent groups of class three which underly a unique regular dessin, submitted. G.A. Jones, Regular dessins with a given automorphism group, arXiv:1309.5219 [math.GR], 2013. G.A. Jones, D. Singerman, Belyˇ ı functions, hypermaps and Galois groups, Bull. London Math. Soc. 28 (1996) 561–590.
Recommend
More recommend
