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Vertex-quasiprimitivity in regular maps Jozef Sir a n OU and STU 27th May 2015 Jozef Sir a n OU and STU Vertex-quasiprimitivity in regular maps 27th May 2015 1 / 15 The five Platonic maps M Jozef Sir a n OU

  1. Is this embedding of K 5 in a torus orientably-regular? �� �� � � �������� �������� �������� �������� ���� ���� �� �� � � ���� ���� �������� �������� �������� �������� � � �������� �������� �������� �������� ���� ���� � � �������� �������� �������� �������� ���� ���� � � ���� ���� �������� �������� �������� �������� �� �� �������� �������� �������� �������� �� �� ���� ���� � � � �� �� �� �� � �������� �������� �������� �������� �� �� �� �� ���� ���� � � ���� ���� �������� �������� �������� �������� ����������������� ����������������� ���� ���� �������� �������� ����������������� ����������������� � � �������� �������� ���� ���� ����������������� ����������������� � � ���� ���� �������� �������� ����������������� ����������������� � � ���� ���� �������� �������� �� �� ����������������� ����������������� �������� �������� �� �� � � ���� ���� ����������������� ����������������� � � ���� ���� �������� �������� ����������������� ����������������� ���� ���� �������� �������� � � ����������������� ����������������� �������� �������� ���� ���� � � ����� ����� ����������������� ����������������� �������� �������� � � ����������������� ����������������� ����� ����� �������� �������� � � � � ����������������� ����������������� ����� ����� �������� �������� � � �� �� � � ����� ����� ����������������� ����������������� �������� �������� � � ����������������� ����������������� ����� ����� �������� �������� � � ����������������� ����������������� ����� ����� �������� �������� ����� ����� � � ����������������� ����������������� �������� �������� �������� �������� � � ����������������� ����������������� ����� ����� �������� �������� � � �������� �������� ����� ����� �������� �������� � � �������� �������� ����� ����� �������� �������� � � �������� �������� ����� ����� �������� �������� �������� �������� ����� ����� �������� �������� � � �������� �������� ����� ����� �������� �������� �������� �������� � � ����� ����� �������� �������� �������� �������� ����� ����� �������� �������� �� �� �� �� �������� �������� ����� ����� �������� �������� �� �� �� �� Jozef ˇ Sir´ aˇ n OU and STU Vertex-quasiprimitivity in regular maps 27th May 2015 3 / 15

  2. Is this embedding of K 5 in a torus orientably-regular? �� �� � � �������� �������� �������� �������� ���� ���� �� �� � � ���� ���� �������� �������� �������� �������� � � �������� �������� �������� �������� ���� ���� � � �������� �������� �������� �������� ���� ���� � � ���� ���� �������� �������� �������� �������� �� �� �������� �������� �������� �������� �� �� ���� ���� � � � �� �� �� �� � �������� �������� �������� �������� �� �� �� �� ���� ���� � � ���� ���� �������� �������� �������� �������� ����������������� ����������������� ���� ���� �������� �������� ����������������� ����������������� � � �������� �������� ���� ���� ����������������� ����������������� � � ���� ���� �������� �������� ����������������� ����������������� � � ���� ���� �������� �������� �� �� ����������������� ����������������� �������� �������� �� �� � � ���� ���� ����������������� ����������������� � � ���� ���� �������� �������� ����������������� ����������������� ���� ���� �������� �������� � � ����������������� ����������������� �������� �������� ���� ���� � � ����� ����� ����������������� ����������������� �������� �������� � � ����������������� ����������������� ����� ����� �������� �������� � � � � ����������������� ����������������� ����� ����� �������� �������� � � �� �� � � ����� ����� ����������������� ����������������� �������� �������� � � ����������������� ����������������� ����� ����� �������� �������� � � ����������������� ����������������� ����� ����� �������� �������� ����� ����� � � ����������������� ����������������� �������� �������� �������� �������� � � ����������������� ����������������� ����� ����� �������� �������� � � �������� �������� ����� ����� �������� �������� � � �������� �������� ����� ����� �������� �������� � � �������� �������� ����� ����� �������� �������� �������� �������� ����� ����� �������� �������� � � �������� �������� ����� ����� �������� �������� �������� �������� � � ����� ����� �������� �������� �������� �������� ����� ����� �������� �������� �� �� �� �� �������� �������� ����� ����� �������� �������� �� �� �� �� • Presentation: Aut + ( M ) = � r , s ; r 4 = s 4 = ( rs ) 2 = ... = 1 � Jozef ˇ Sir´ aˇ n OU and STU Vertex-quasiprimitivity in regular maps 27th May 2015 3 / 15

  3. � � �� �� �� �� � � � � �� �� �� �� � � � � � � �� �� �� �� � � � �� �� �� �� � � � �� �� �� �� � � � � � � �� �� � � �� �� � � � � � � �� �� � � � � �� �� � � �� �� � � �� �� � � �� �� � � �� �� � � �� �� � � � � �� �� �� �� � � � � �� �� �� �� � � Is this embedding of K 5 regular? Chirality Jozef ˇ Sir´ aˇ n OU and STU Vertex-quasiprimitivity in regular maps 27th May 2015 4 / 15

  4. Is this embedding of K 5 regular? Chirality � � �� �� �� �� � � � � �� �� �� �� � � � � � � �� �� �� �� � � � �� �� �� �� � � � �� �� �� �� � � � � � � �� �� � � �� �� � � � � � � �� �� � � � � �� �� � � � � �� �� �� �� � � �� �� � � �� �� � � �� �� � � � � �� �� �� �� � � � � �� �� �� �� � � Jozef ˇ Sir´ aˇ n OU and STU Vertex-quasiprimitivity in regular maps 27th May 2015 4 / 15

  5. Regular and orientably-regular maps Jozef ˇ Sir´ aˇ n OU and STU Vertex-quasiprimitivity in regular maps 27th May 2015 5 / 15

  6. Regular and orientably-regular maps A map is regular if its automorphism group is regular on flags. Jozef ˇ Sir´ aˇ n OU and STU Vertex-quasiprimitivity in regular maps 27th May 2015 5 / 15

  7. Regular and orientably-regular maps A map is regular if its automorphism group is regular on flags. Jozef ˇ Sir´ aˇ n OU and STU Vertex-quasiprimitivity in regular maps 27th May 2015 5 / 15

  8. Regular and orientably-regular maps A map is regular if its automorphism group is regular on flags. Aut ( M ) = � x , y , z | x 2 = y 2 = z 2 = ( yz ) k = ( zx ) m = ( xy ) 2 = . . . = 1 � Jozef ˇ Sir´ aˇ n OU and STU Vertex-quasiprimitivity in regular maps 27th May 2015 5 / 15

  9. Regular and orientably-regular maps A map is regular if its automorphism group is regular on flags. Aut ( M ) = � x , y , z | x 2 = y 2 = z 2 = ( yz ) k = ( zx ) m = ( xy ) 2 = . . . = 1 � Orientable regularity - the orientation-preserving map automorphism group Aut + ( M ) = � r , s | r k = s m = ( rs ) 2 = . . . = 1 � is regular on arcs, or darts: Jozef ˇ Sir´ aˇ n OU and STU Vertex-quasiprimitivity in regular maps 27th May 2015 5 / 15

  10. Regular and orientably-regular maps A map is regular if its automorphism group is regular on flags. Aut ( M ) = � x , y , z | x 2 = y 2 = z 2 = ( yz ) k = ( zx ) m = ( xy ) 2 = . . . = 1 � Orientable regularity - the orientation-preserving map automorphism group Aut + ( M ) = � r , s | r k = s m = ( rs ) 2 = . . . = 1 � is regular on arcs, or darts: Conversely, such group presentations determine (orientably-) regular maps. Jozef ˇ Sir´ aˇ n OU and STU Vertex-quasiprimitivity in regular maps 27th May 2015 5 / 15

  11. The famous Klein map of genus 3 Jozef ˇ Sir´ aˇ n OU and STU Vertex-quasiprimitivity in regular maps 27th May 2015 6 / 15

  12. The famous Klein map of genus 3 Jozef ˇ Sir´ aˇ n OU and STU Vertex-quasiprimitivity in regular maps 27th May 2015 6 / 15

  13. The Klein map of genus 3 – algebraically Jozef ˇ Sir´ aˇ n OU and STU Vertex-quasiprimitivity in regular maps 27th May 2015 7 / 15

  14. The Klein map of genus 3 – algebraically • Regular, of type { 7 , 3 } Jozef ˇ Sir´ aˇ n OU and STU Vertex-quasiprimitivity in regular maps 27th May 2015 7 / 15

  15. The Klein map of genus 3 – algebraically • Regular, of type { 7 , 3 } • Aut ( M ) = � x , y , z ; x 2 = y 2 = z 2 = ( yz ) 3 = ( zx ) 7 = ( xy ) 2 = . . . = 1 � Jozef ˇ Sir´ aˇ n OU and STU Vertex-quasiprimitivity in regular maps 27th May 2015 7 / 15

  16. The Klein map of genus 3 – algebraically • Regular, of type { 7 , 3 } • Aut ( M ) = � x , y , z ; x 2 = y 2 = z 2 = ( yz ) 3 = ( zx ) 7 = ( xy ) 2 = . . . = 1 � • 336 flags Jozef ˇ Sir´ aˇ n OU and STU Vertex-quasiprimitivity in regular maps 27th May 2015 7 / 15

  17. The Klein map of genus 3 – algebraically • Regular, of type { 7 , 3 } • Aut ( M ) = � x , y , z ; x 2 = y 2 = z 2 = ( yz ) 3 = ( zx ) 7 = ( xy ) 2 = . . . = 1 � • 336 flags • Aut ( M ) ≃ PGL (2 , 7) Jozef ˇ Sir´ aˇ n OU and STU Vertex-quasiprimitivity in regular maps 27th May 2015 7 / 15

  18. Example of a non-orientable regular map Jozef ˇ Sir´ aˇ n OU and STU Vertex-quasiprimitivity in regular maps 27th May 2015 8 / 15

  19. Example of a non-orientable regular map The Petersen Graph on the projective plane, with its dual – K 6 : Jozef ˇ Sir´ aˇ n OU and STU Vertex-quasiprimitivity in regular maps 27th May 2015 8 / 15

  20. Example of a non-orientable regular map The Petersen Graph on the projective plane, with its dual – K 6 : Jozef ˇ Sir´ aˇ n OU and STU Vertex-quasiprimitivity in regular maps 27th May 2015 8 / 15

  21. Example of a non-orientable regular map The Petersen Graph on the projective plane, with its dual – K 6 : • Regular, of type { 5 , 3 } Jozef ˇ Sir´ aˇ n OU and STU Vertex-quasiprimitivity in regular maps 27th May 2015 8 / 15

  22. Example of a non-orientable regular map The Petersen Graph on the projective plane, with its dual – K 6 : • Regular, of type { 5 , 3 } • Aut ( M ) = � x , y , z ; x 2 = y 2 = z 2 = ( yz ) 3 = ( zx ) 5 = ( xy ) 2 = . . . = 1 � Jozef ˇ Sir´ aˇ n OU and STU Vertex-quasiprimitivity in regular maps 27th May 2015 8 / 15

  23. Example of a non-orientable regular map The Petersen Graph on the projective plane, with its dual – K 6 : • Regular, of type { 5 , 3 } • Aut ( M ) = � x , y , z ; x 2 = y 2 = z 2 = ( yz ) 3 = ( zx ) 5 = ( xy ) 2 = . . . = 1 � • 60 flags Jozef ˇ Sir´ aˇ n OU and STU Vertex-quasiprimitivity in regular maps 27th May 2015 8 / 15

  24. Example of a non-orientable regular map The Petersen Graph on the projective plane, with its dual – K 6 : • Regular, of type { 5 , 3 } • Aut ( M ) = � x , y , z ; x 2 = y 2 = z 2 = ( yz ) 3 = ( zx ) 5 = ( xy ) 2 = . . . = 1 � • 60 flags • Aut ( M ) ≃ A 5 Jozef ˇ Sir´ aˇ n OU and STU Vertex-quasiprimitivity in regular maps 27th May 2015 8 / 15

  25. Classification by automorphism groups Jozef ˇ Sir´ aˇ n OU and STU Vertex-quasiprimitivity in regular maps 27th May 2015 9 / 15

  26. Classification by automorphism groups Available results: Jozef ˇ Sir´ aˇ n OU and STU Vertex-quasiprimitivity in regular maps 27th May 2015 9 / 15

  27. Classification by automorphism groups Available results: • Abelian, dihedral: Folklore Jozef ˇ Sir´ aˇ n OU and STU Vertex-quasiprimitivity in regular maps 27th May 2015 9 / 15

  28. Classification by automorphism groups Available results: • Abelian, dihedral: Folklore • Sylow-cyclic: Classification of groups - H¨ older 1895, Burnside 1905, cnik, ˇ Zassenhaus 1936; maps - Conder, Potoˇ S 2010 Jozef ˇ Sir´ aˇ n OU and STU Vertex-quasiprimitivity in regular maps 27th May 2015 9 / 15

  29. Classification by automorphism groups Available results: • Abelian, dihedral: Folklore • Sylow-cyclic: Classification of groups - H¨ older 1895, Burnside 1905, cnik, ˇ Zassenhaus 1936; maps - Conder, Potoˇ S 2010 • Almost Sylow-cyclic: Solvable groups - Zassenhaus 1936; unsolvable - cnik, ˇ Suzuki 1955, Wong 1966; maps - Conder, Potoˇ S 2010 Jozef ˇ Sir´ aˇ n OU and STU Vertex-quasiprimitivity in regular maps 27th May 2015 9 / 15

  30. Classification by automorphism groups Available results: • Abelian, dihedral: Folklore • Sylow-cyclic: Classification of groups - H¨ older 1895, Burnside 1905, cnik, ˇ Zassenhaus 1936; maps - Conder, Potoˇ S 2010 • Almost Sylow-cyclic: Solvable groups - Zassenhaus 1936; unsolvable - cnik, ˇ Suzuki 1955, Wong 1966; maps - Conder, Potoˇ S 2010 c, Nedela, ˇ • Nilpotent maps - class 2 Malniˇ Skoviera 2012 Jozef ˇ Sir´ aˇ n OU and STU Vertex-quasiprimitivity in regular maps 27th May 2015 9 / 15

  31. Classification by automorphism groups Available results: • Abelian, dihedral: Folklore • Sylow-cyclic: Classification of groups - H¨ older 1895, Burnside 1905, cnik, ˇ Zassenhaus 1936; maps - Conder, Potoˇ S 2010 • Almost Sylow-cyclic: Solvable groups - Zassenhaus 1936; unsolvable - cnik, ˇ Suzuki 1955, Wong 1966; maps - Conder, Potoˇ S 2010 c, Nedela, ˇ • Nilpotent maps - class 2 Malniˇ Skoviera 2012 • Nilpotent maps - class 3 Du, Nedela, ˇ Skoviera et al 201? Jozef ˇ Sir´ aˇ n OU and STU Vertex-quasiprimitivity in regular maps 27th May 2015 9 / 15

  32. Classification by automorphism groups Available results: • Abelian, dihedral: Folklore • Sylow-cyclic: Classification of groups - H¨ older 1895, Burnside 1905, cnik, ˇ Zassenhaus 1936; maps - Conder, Potoˇ S 2010 • Almost Sylow-cyclic: Solvable groups - Zassenhaus 1936; unsolvable - cnik, ˇ Suzuki 1955, Wong 1966; maps - Conder, Potoˇ S 2010 c, Nedela, ˇ • Nilpotent maps - class 2 Malniˇ Skoviera 2012 • Nilpotent maps - class 3 Du, Nedela, ˇ Skoviera et al 201? cnik, ˇ • Maps on PSL , PGL (2 , q ) - Sah 1969, Conder, Potoˇ S 2010 Jozef ˇ Sir´ aˇ n OU and STU Vertex-quasiprimitivity in regular maps 27th May 2015 9 / 15

  33. Classification by automorphism groups Available results: • Abelian, dihedral: Folklore • Sylow-cyclic: Classification of groups - H¨ older 1895, Burnside 1905, cnik, ˇ Zassenhaus 1936; maps - Conder, Potoˇ S 2010 • Almost Sylow-cyclic: Solvable groups - Zassenhaus 1936; unsolvable - cnik, ˇ Suzuki 1955, Wong 1966; maps - Conder, Potoˇ S 2010 c, Nedela, ˇ • Nilpotent maps - class 2 Malniˇ Skoviera 2012 • Nilpotent maps - class 3 Du, Nedela, ˇ Skoviera et al 201? cnik, ˇ • Maps on PSL , PGL (2 , q ) - Sah 1969, Conder, Potoˇ S 2010 • Maps of type { 4 , 5 } on Suzuki groups - Jones, Silver 1993 Jozef ˇ Sir´ aˇ n OU and STU Vertex-quasiprimitivity in regular maps 27th May 2015 9 / 15

  34. Classification by automorphism groups Available results: • Abelian, dihedral: Folklore • Sylow-cyclic: Classification of groups - H¨ older 1895, Burnside 1905, cnik, ˇ Zassenhaus 1936; maps - Conder, Potoˇ S 2010 • Almost Sylow-cyclic: Solvable groups - Zassenhaus 1936; unsolvable - cnik, ˇ Suzuki 1955, Wong 1966; maps - Conder, Potoˇ S 2010 c, Nedela, ˇ • Nilpotent maps - class 2 Malniˇ Skoviera 2012 • Nilpotent maps - class 3 Du, Nedela, ˇ Skoviera et al 201? cnik, ˇ • Maps on PSL , PGL (2 , q ) - Sah 1969, Conder, Potoˇ S 2010 • Maps of type { 4 , 5 } on Suzuki groups - Jones, Silver 1993 • Maps of type { 3 , p } , p ≡ − 1 mod 12, on Ree groups - Jones 1994 Jozef ˇ Sir´ aˇ n OU and STU Vertex-quasiprimitivity in regular maps 27th May 2015 9 / 15

  35. Classification by automorphism groups Available results: • Abelian, dihedral: Folklore • Sylow-cyclic: Classification of groups - H¨ older 1895, Burnside 1905, cnik, ˇ Zassenhaus 1936; maps - Conder, Potoˇ S 2010 • Almost Sylow-cyclic: Solvable groups - Zassenhaus 1936; unsolvable - cnik, ˇ Suzuki 1955, Wong 1966; maps - Conder, Potoˇ S 2010 c, Nedela, ˇ • Nilpotent maps - class 2 Malniˇ Skoviera 2012 • Nilpotent maps - class 3 Du, Nedela, ˇ Skoviera et al 201? cnik, ˇ • Maps on PSL , PGL (2 , q ) - Sah 1969, Conder, Potoˇ S 2010 • Maps of type { 4 , 5 } on Suzuki groups - Jones, Silver 1993 • Maps of type { 3 , p } , p ≡ − 1 mod 12, on Ree groups - Jones 1994 Classification also considered by underlying graphs and supporting surfaces. Jozef ˇ Sir´ aˇ n OU and STU Vertex-quasiprimitivity in regular maps 27th May 2015 9 / 15

  36. Large regular maps from small ones by lifting Jozef ˇ Sir´ aˇ n OU and STU Vertex-quasiprimitivity in regular maps 27th May 2015 10 / 15

  37. Large regular maps from small ones by lifting M : orientably-regular, G = Aut + ( M ) = � r , s | r k = s m = ( rs ) 2 = . . . = 1 � Jozef ˇ Sir´ aˇ n OU and STU Vertex-quasiprimitivity in regular maps 27th May 2015 10 / 15

  38. Large regular maps from small ones by lifting M : orientably-regular, G = Aut + ( M ) = � r , s | r k = s m = ( rs ) 2 = . . . = 1 � Taking K ⊳ G we may form the orientably-regular quotient map M / K with G / K = Aut + ( M / K ) = � rK , sK | ( rK ) κ = ( sK ) µ = ( rsK ) 2 = . . . = 1 � Jozef ˇ Sir´ aˇ n OU and STU Vertex-quasiprimitivity in regular maps 27th May 2015 10 / 15

  39. Large regular maps from small ones by lifting M : orientably-regular, G = Aut + ( M ) = � r , s | r k = s m = ( rs ) 2 = . . . = 1 � Taking K ⊳ G we may form the orientably-regular quotient map M / K with G / K = Aut + ( M / K ) = � rK , sK | ( rK ) κ = ( sK ) µ = ( rsK ) 2 = . . . = 1 � The projection G → G / K induces a branched covering M → M / K of maps (and supporting surfaces); Jozef ˇ Sir´ aˇ n OU and STU Vertex-quasiprimitivity in regular maps 27th May 2015 10 / 15

  40. Large regular maps from small ones by lifting M : orientably-regular, G = Aut + ( M ) = � r , s | r k = s m = ( rs ) 2 = . . . = 1 � Taking K ⊳ G we may form the orientably-regular quotient map M / K with G / K = Aut + ( M / K ) = � rK , sK | ( rK ) κ = ( sK ) µ = ( rsK ) 2 = . . . = 1 � The projection G → G / K induces a branched covering M → M / K of maps (and supporting surfaces); smooth iff κ = k and µ = m . Jozef ˇ Sir´ aˇ n OU and STU Vertex-quasiprimitivity in regular maps 27th May 2015 10 / 15

  41. Large regular maps from small ones by lifting M : orientably-regular, G = Aut + ( M ) = � r , s | r k = s m = ( rs ) 2 = . . . = 1 � Taking K ⊳ G we may form the orientably-regular quotient map M / K with G / K = Aut + ( M / K ) = � rK , sK | ( rK ) κ = ( sK ) µ = ( rsK ) 2 = . . . = 1 � The projection G → G / K induces a branched covering M → M / K of maps (and supporting surfaces); smooth iff κ = k and µ = m . Knowing M ′ = M / K and K , we may reverse the process and lift M ′ to M along K ; the lift may not be unique. Jozef ˇ Sir´ aˇ n OU and STU Vertex-quasiprimitivity in regular maps 27th May 2015 10 / 15

  42. Large regular maps from small ones by lifting M : orientably-regular, G = Aut + ( M ) = � r , s | r k = s m = ( rs ) 2 = . . . = 1 � Taking K ⊳ G we may form the orientably-regular quotient map M / K with G / K = Aut + ( M / K ) = � rK , sK | ( rK ) κ = ( sK ) µ = ( rsK ) 2 = . . . = 1 � The projection G → G / K induces a branched covering M → M / K of maps (and supporting surfaces); smooth iff κ = k and µ = m . Knowing M ′ = M / K and K , we may reverse the process and lift M ′ to M along K ; the lift may not be unique. Theory of covers and lifts: A. Malniˇ c; Jozef ˇ Sir´ aˇ n OU and STU Vertex-quasiprimitivity in regular maps 27th May 2015 10 / 15

  43. Large regular maps from small ones by lifting M : orientably-regular, G = Aut + ( M ) = � r , s | r k = s m = ( rs ) 2 = . . . = 1 � Taking K ⊳ G we may form the orientably-regular quotient map M / K with G / K = Aut + ( M / K ) = � rK , sK | ( rK ) κ = ( sK ) µ = ( rsK ) 2 = . . . = 1 � The projection G → G / K induces a branched covering M → M / K of maps (and supporting surfaces); smooth iff κ = k and µ = m . Knowing M ′ = M / K and K , we may reverse the process and lift M ′ to M along K ; the lift may not be unique. Theory of covers and lifts: A. Malniˇ c; elementary Abelian case well understood: Malniˇ c, Maruˇ siˇ c, Potoˇ cnik 2004. Jozef ˇ Sir´ aˇ n OU and STU Vertex-quasiprimitivity in regular maps 27th May 2015 10 / 15

  44. Large regular maps from small ones by lifting M : orientably-regular, G = Aut + ( M ) = � r , s | r k = s m = ( rs ) 2 = . . . = 1 � Taking K ⊳ G we may form the orientably-regular quotient map M / K with G / K = Aut + ( M / K ) = � rK , sK | ( rK ) κ = ( sK ) µ = ( rsK ) 2 = . . . = 1 � The projection G → G / K induces a branched covering M → M / K of maps (and supporting surfaces); smooth iff κ = k and µ = m . Knowing M ′ = M / K and K , we may reverse the process and lift M ′ to M along K ; the lift may not be unique. Theory of covers and lifts: A. Malniˇ c; elementary Abelian case well understood: Malniˇ c, Maruˇ siˇ c, Potoˇ cnik 2004. Regular cyclic lifts of Platonic maps: Jones and Surowski 2000, generalised by Hu, Nedela and Wang 2014. Jozef ˇ Sir´ aˇ n OU and STU Vertex-quasiprimitivity in regular maps 27th May 2015 10 / 15

  45. Large regular maps from small ones by lifting M : orientably-regular, G = Aut + ( M ) = � r , s | r k = s m = ( rs ) 2 = . . . = 1 � Taking K ⊳ G we may form the orientably-regular quotient map M / K with G / K = Aut + ( M / K ) = � rK , sK | ( rK ) κ = ( sK ) µ = ( rsK ) 2 = . . . = 1 � The projection G → G / K induces a branched covering M → M / K of maps (and supporting surfaces); smooth iff κ = k and µ = m . Knowing M ′ = M / K and K , we may reverse the process and lift M ′ to M along K ; the lift may not be unique. Theory of covers and lifts: A. Malniˇ c; elementary Abelian case well understood: Malniˇ c, Maruˇ siˇ c, Potoˇ cnik 2004. Regular cyclic lifts of Platonic maps: Jones and Surowski 2000, generalised by Hu, Nedela and Wang 2014. Examples of lifts of embeddings of the θ -graph: Jozef ˇ Sir´ aˇ n OU and STU Vertex-quasiprimitivity in regular maps 27th May 2015 10 / 15

  46. Quasiprimitivity Jozef ˇ Sir´ aˇ n OU and STU Vertex-quasiprimitivity in regular maps 27th May 2015 11 / 15

  47. Quasiprimitivity Take an orientably-regular map M ; view G = Aut + ( M ) as a permutation group on the set Ω of vertices of M . Jozef ˇ Sir´ aˇ n OU and STU Vertex-quasiprimitivity in regular maps 27th May 2015 11 / 15

  48. Quasiprimitivity Take an orientably-regular map M ; view G = Aut + ( M ) as a permutation group on the set Ω of vertices of M . This works fine if G has trivial core on Ω, Jozef ˇ Sir´ aˇ n OU and STU Vertex-quasiprimitivity in regular maps 27th May 2015 11 / 15

  49. Quasiprimitivity Take an orientably-regular map M ; view G = Aut + ( M ) as a permutation group on the set Ω of vertices of M . This works fine if G has trivial core on Ω, where, for G transitive on Ω (which is our case), core ( G ) = { h ∈ G ; h ( v ) = v for each v ∈ Ω } ( g ∈ G ) g − 1 Hg ; H = Stab G ( u ) for some u ∈ Ω. = � Jozef ˇ Sir´ aˇ n OU and STU Vertex-quasiprimitivity in regular maps 27th May 2015 11 / 15

  50. Quasiprimitivity Take an orientably-regular map M ; view G = Aut + ( M ) as a permutation group on the set Ω of vertices of M . This works fine if G has trivial core on Ω, where, for G transitive on Ω (which is our case), core ( G ) = { h ∈ G ; h ( v ) = v for each v ∈ Ω } ( g ∈ G ) g − 1 Hg ; H = Stab G ( u ) for some u ∈ Ω. = � This is always the case when the underlying graph of M is simple. Jozef ˇ Sir´ aˇ n OU and STU Vertex-quasiprimitivity in regular maps 27th May 2015 11 / 15

  51. Quasiprimitivity Take an orientably-regular map M ; view G = Aut + ( M ) as a permutation group on the set Ω of vertices of M . This works fine if G has trivial core on Ω, where, for G transitive on Ω (which is our case), core ( G ) = { h ∈ G ; h ( v ) = v for each v ∈ Ω } ( g ∈ G ) g − 1 Hg ; H = Stab G ( u ) for some u ∈ Ω. = � This is always the case when the underlying graph of M is simple. From now on: G = Aut + ( M ) acts faithfully as a permutation group on Ω. Jozef ˇ Sir´ aˇ n OU and STU Vertex-quasiprimitivity in regular maps 27th May 2015 11 / 15

  52. Quasiprimitivity Take an orientably-regular map M ; view G = Aut + ( M ) as a permutation group on the set Ω of vertices of M . This works fine if G has trivial core on Ω, where, for G transitive on Ω (which is our case), core ( G ) = { h ∈ G ; h ( v ) = v for each v ∈ Ω } ( g ∈ G ) g − 1 Hg ; H = Stab G ( u ) for some u ∈ Ω. = � This is always the case when the underlying graph of M is simple. From now on: G = Aut + ( M ) acts faithfully as a permutation group on Ω. If there is a K ⊳ G such that M / K has at least two vertices, try lifts. Jozef ˇ Sir´ aˇ n OU and STU Vertex-quasiprimitivity in regular maps 27th May 2015 11 / 15

  53. Quasiprimitivity Take an orientably-regular map M ; view G = Aut + ( M ) as a permutation group on the set Ω of vertices of M . This works fine if G has trivial core on Ω, where, for G transitive on Ω (which is our case), core ( G ) = { h ∈ G ; h ( v ) = v for each v ∈ Ω } ( g ∈ G ) g − 1 Hg ; H = Stab G ( u ) for some u ∈ Ω. = � This is always the case when the underlying graph of M is simple. From now on: G = Aut + ( M ) acts faithfully as a permutation group on Ω. If there is a K ⊳ G such that M / K has at least two vertices, try lifts. But what if there is no such K ⊳ G ? Jozef ˇ Sir´ aˇ n OU and STU Vertex-quasiprimitivity in regular maps 27th May 2015 11 / 15

  54. Quasiprimitivity Take an orientably-regular map M ; view G = Aut + ( M ) as a permutation group on the set Ω of vertices of M . This works fine if G has trivial core on Ω, where, for G transitive on Ω (which is our case), core ( G ) = { h ∈ G ; h ( v ) = v for each v ∈ Ω } ( g ∈ G ) g − 1 Hg ; H = Stab G ( u ) for some u ∈ Ω. = � This is always the case when the underlying graph of M is simple. From now on: G = Aut + ( M ) acts faithfully as a permutation group on Ω. If there is a K ⊳ G such that M / K has at least two vertices, try lifts. But what if there is no such K ⊳ G ? Equivalently, what if every normal subgroup of our permutation group G is transitive on Ω ? Jozef ˇ Sir´ aˇ n OU and STU Vertex-quasiprimitivity in regular maps 27th May 2015 11 / 15

  55. Quasiprimitivity Take an orientably-regular map M ; view G = Aut + ( M ) as a permutation group on the set Ω of vertices of M . This works fine if G has trivial core on Ω, where, for G transitive on Ω (which is our case), core ( G ) = { h ∈ G ; h ( v ) = v for each v ∈ Ω } ( g ∈ G ) g − 1 Hg ; H = Stab G ( u ) for some u ∈ Ω. = � This is always the case when the underlying graph of M is simple. From now on: G = Aut + ( M ) acts faithfully as a permutation group on Ω. If there is a K ⊳ G such that M / K has at least two vertices, try lifts. But what if there is no such K ⊳ G ? Equivalently, what if every normal subgroup of our permutation group G is transitive on Ω ? Such permutation groups are known as quasiprimitive. Jozef ˇ Sir´ aˇ n OU and STU Vertex-quasiprimitivity in regular maps 27th May 2015 11 / 15

  56. Quasiprimitivity Take an orientably-regular map M ; view G = Aut + ( M ) as a permutation group on the set Ω of vertices of M . This works fine if G has trivial core on Ω, where, for G transitive on Ω (which is our case), core ( G ) = { h ∈ G ; h ( v ) = v for each v ∈ Ω } ( g ∈ G ) g − 1 Hg ; H = Stab G ( u ) for some u ∈ Ω. = � This is always the case when the underlying graph of M is simple. From now on: G = Aut + ( M ) acts faithfully as a permutation group on Ω. If there is a K ⊳ G such that M / K has at least two vertices, try lifts. But what if there is no such K ⊳ G ? Equivalently, what if every normal subgroup of our permutation group G is transitive on Ω ? Such permutation groups are known as quasiprimitive. The corresponding orientably-regular and regular maps may be thought of as ‘irreducible’. Jozef ˇ Sir´ aˇ n OU and STU Vertex-quasiprimitivity in regular maps 27th May 2015 11 / 15

  57. The O’Nan-Scott-Praeger Theorem (1979,1993) Jozef ˇ Sir´ aˇ n OU and STU Vertex-quasiprimitivity in regular maps 27th May 2015 12 / 15

  58. The O’Nan-Scott-Praeger Theorem (1979,1993) Quasiprimitive permutation groups G on Ω (with all normal subgroups transitive) have N = soc ( G ) = T k for some simple group T , and: Jozef ˇ Sir´ aˇ n OU and STU Vertex-quasiprimitivity in regular maps 27th May 2015 12 / 15

  59. The O’Nan-Scott-Praeger Theorem (1979,1993) Quasiprimitive permutation groups G on Ω (with all normal subgroups transitive) have N = soc ( G ) = T k for some simple group T , and: • HA : Ω = Z k p , G < AGL ( k , p ), Stab G ( u ) irreducible, N = Z k p Jozef ˇ Sir´ aˇ n OU and STU Vertex-quasiprimitivity in regular maps 27th May 2015 12 / 15

  60. The O’Nan-Scott-Praeger Theorem (1979,1993) Quasiprimitive permutation groups G on Ω (with all normal subgroups transitive) have N = soc ( G ) = T k for some simple group T , and: • HA : Ω = Z k p , G < AGL ( k , p ), Stab G ( u ) irreducible, N = Z k p • HS : Ω = T , k = 2, N = T . Inn ( T ) < G < T . Aut ( T ) Jozef ˇ Sir´ aˇ n OU and STU Vertex-quasiprimitivity in regular maps 27th May 2015 12 / 15

  61. The O’Nan-Scott-Praeger Theorem (1979,1993) Quasiprimitive permutation groups G on Ω (with all normal subgroups transitive) have N = soc ( G ) = T k for some simple group T , and: • HA : Ω = Z k p , G < AGL ( k , p ), Stab G ( u ) irreducible, N = Z k p • HS : Ω = T , k = 2, N = T . Inn ( T ) < G < T . Aut ( T ) • HC : Ω = T ℓ , k = 2 ℓ > 2, N = T ℓ . Inn ( T ℓ ) < G < T ℓ . Aut ( T ℓ ) Jozef ˇ Sir´ aˇ n OU and STU Vertex-quasiprimitivity in regular maps 27th May 2015 12 / 15

  62. The O’Nan-Scott-Praeger Theorem (1979,1993) Quasiprimitive permutation groups G on Ω (with all normal subgroups transitive) have N = soc ( G ) = T k for some simple group T , and: • HA : Ω = Z k p , G < AGL ( k , p ), Stab G ( u ) irreducible, N = Z k p • HS : Ω = T , k = 2, N = T . Inn ( T ) < G < T . Aut ( T ) • HC : Ω = T ℓ , k = 2 ℓ > 2, N = T ℓ . Inn ( T ℓ ) < G < T ℓ . Aut ( T ℓ ) • AS : k = 1, N = T < G < Aut ( T ), T transitive on some set Ω Jozef ˇ Sir´ aˇ n OU and STU Vertex-quasiprimitivity in regular maps 27th May 2015 12 / 15

  63. The O’Nan-Scott-Praeger Theorem (1979,1993) Quasiprimitive permutation groups G on Ω (with all normal subgroups transitive) have N = soc ( G ) = T k for some simple group T , and: • HA : Ω = Z k p , G < AGL ( k , p ), Stab G ( u ) irreducible, N = Z k p • HS : Ω = T , k = 2, N = T . Inn ( T ) < G < T . Aut ( T ) • HC : Ω = T ℓ , k = 2 ℓ > 2, N = T ℓ . Inn ( T ℓ ) < G < T ℓ . Aut ( T ℓ ) • AS : k = 1, N = T < G < Aut ( T ), T transitive on some set Ω • SD : Ω = N / H , H < dia N , N ⊳ G < N . Out ( T ) × S k < S Ω , k ≥ 2 Jozef ˇ Sir´ aˇ n OU and STU Vertex-quasiprimitivity in regular maps 27th May 2015 12 / 15

  64. The O’Nan-Scott-Praeger Theorem (1979,1993) Quasiprimitive permutation groups G on Ω (with all normal subgroups transitive) have N = soc ( G ) = T k for some simple group T , and: • HA : Ω = Z k p , G < AGL ( k , p ), Stab G ( u ) irreducible, N = Z k p • HS : Ω = T , k = 2, N = T . Inn ( T ) < G < T . Aut ( T ) • HC : Ω = T ℓ , k = 2 ℓ > 2, N = T ℓ . Inn ( T ℓ ) < G < T ℓ . Aut ( T ℓ ) • AS : k = 1, N = T < G < Aut ( T ), T transitive on some set Ω • SD : Ω = N / H , H < dia N , N ⊳ G < N . Out ( T ) × S k < S Ω , k ≥ 2 • CD : Ω = N , G < H wr S k , H quasiprimitive on T of SD-type, k ≥ 4 Jozef ˇ Sir´ aˇ n OU and STU Vertex-quasiprimitivity in regular maps 27th May 2015 12 / 15

  65. The O’Nan-Scott-Praeger Theorem (1979,1993) Quasiprimitive permutation groups G on Ω (with all normal subgroups transitive) have N = soc ( G ) = T k for some simple group T , and: • HA : Ω = Z k p , G < AGL ( k , p ), Stab G ( u ) irreducible, N = Z k p • HS : Ω = T , k = 2, N = T . Inn ( T ) < G < T . Aut ( T ) • HC : Ω = T ℓ , k = 2 ℓ > 2, N = T ℓ . Inn ( T ℓ ) < G < T ℓ . Aut ( T ℓ ) • AS : k = 1, N = T < G < Aut ( T ), T transitive on some set Ω • SD : Ω = N / H , H < dia N , N ⊳ G < N . Out ( T ) × S k < S Ω , k ≥ 2 • CD : Ω = N , G < H wr S k , H quasiprimitive on T of SD-type, k ≥ 4 • TW : Ω = N , T non-Abelian, G is a ‘twisted wreath product’ Jozef ˇ Sir´ aˇ n OU and STU Vertex-quasiprimitivity in regular maps 27th May 2015 12 / 15

  66. The O’Nan-Scott-Praeger Theorem (1979,1993) Quasiprimitive permutation groups G on Ω (with all normal subgroups transitive) have N = soc ( G ) = T k for some simple group T , and: • HA : Ω = Z k p , G < AGL ( k , p ), Stab G ( u ) irreducible, N = Z k p • HS : Ω = T , k = 2, N = T . Inn ( T ) < G < T . Aut ( T ) • HC : Ω = T ℓ , k = 2 ℓ > 2, N = T ℓ . Inn ( T ℓ ) < G < T ℓ . Aut ( T ℓ ) • AS : k = 1, N = T < G < Aut ( T ), T transitive on some set Ω • SD : Ω = N / H , H < dia N , N ⊳ G < N . Out ( T ) × S k < S Ω , k ≥ 2 • CD : Ω = N , G < H wr S k , H quasiprimitive on T of SD-type, k ≥ 4 • TW : Ω = N , T non-Abelian, G is a ‘twisted wreath product’ • PA : Ω = N ⊳ G < H wr S k , T < H < Aut ( T ), ‘product action’ of G on Ω. Jozef ˇ Sir´ aˇ n OU and STU Vertex-quasiprimitivity in regular maps 27th May 2015 12 / 15

  67. The O’Nan-Scott-Praeger Theorem (1979,1993) Quasiprimitive permutation groups G on Ω (with all normal subgroups transitive) have N = soc ( G ) = T k for some simple group T , and: • HA : Ω = Z k p , G < AGL ( k , p ), Stab G ( u ) irreducible, N = Z k p • HS : Ω = T , k = 2, N = T . Inn ( T ) < G < T . Aut ( T ) • HC : Ω = T ℓ , k = 2 ℓ > 2, N = T ℓ . Inn ( T ℓ ) < G < T ℓ . Aut ( T ℓ ) • AS : k = 1, N = T < G < Aut ( T ), T transitive on some set Ω • SD : Ω = N / H , H < dia N , N ⊳ G < N . Out ( T ) × S k < S Ω , k ≥ 2 • CD : Ω = N , G < H wr S k , H quasiprimitive on T of SD-type, k ≥ 4 • TW : Ω = N , T non-Abelian, G is a ‘twisted wreath product’ • PA : Ω = N ⊳ G < H wr S k , T < H < Aut ( T ), ‘product action’ of G on Ω. Lemma [Li, ˇ S, Wang] If G is the automorphism group of a regular or an orientably-regular map, quasiprimitive on V , then G is HA, AS, TW or PA. Jozef ˇ Sir´ aˇ n OU and STU Vertex-quasiprimitivity in regular maps 27th May 2015 12 / 15

  68. The ‘Holomorph-Abelian’ case: A sample result Jozef ˇ Sir´ aˇ n OU and STU Vertex-quasiprimitivity in regular maps 27th May 2015 13 / 15

  69. The ‘Holomorph-Abelian’ case: A sample result This is the only case in which we have a characterisation: Jozef ˇ Sir´ aˇ n OU and STU Vertex-quasiprimitivity in regular maps 27th May 2015 13 / 15

  70. The ‘Holomorph-Abelian’ case: A sample result This is the only case in which we have a characterisation: Theorem [Li, ˇ S, Wang, w. i. p.] Let N = Z d p for an odd prime p and let G = N ⋊ H, where H = � h � ≃ Z k is an irreducible subgroup of AGL ( d , p ) Jozef ˇ Sir´ aˇ n OU and STU Vertex-quasiprimitivity in regular maps 27th May 2015 13 / 15

  71. The ‘Holomorph-Abelian’ case: A sample result This is the only case in which we have a characterisation: Theorem [Li, ˇ S, Wang, w. i. p.] Let N = Z d p for an odd prime p and let G = N ⋊ H, where H = � h � ≃ Z k is an irreducible subgroup of AGL ( d , p ) (that is, k is an even primitive divisor of p d − 1 ) Jozef ˇ Sir´ aˇ n OU and STU Vertex-quasiprimitivity in regular maps 27th May 2015 13 / 15

  72. The ‘Holomorph-Abelian’ case: A sample result This is the only case in which we have a characterisation: Theorem [Li, ˇ S, Wang, w. i. p.] Let N = Z d p for an odd prime p and let G = N ⋊ H, where H = � h � ≃ Z k is an irreducible subgroup of AGL ( d , p ) (that is, k is an even primitive divisor of p d − 1 ) such that h k / 2 inverts N. Let 1 � = g ∈ N and let S = { g h i ; i ∈ Z k } , with natural cyclic ordering. Jozef ˇ Sir´ aˇ n OU and STU Vertex-quasiprimitivity in regular maps 27th May 2015 13 / 15

  73. The ‘Holomorph-Abelian’ case: A sample result This is the only case in which we have a characterisation: Theorem [Li, ˇ S, Wang, w. i. p.] Let N = Z d p for an odd prime p and let G = N ⋊ H, where H = � h � ≃ Z k is an irreducible subgroup of AGL ( d , p ) (that is, k is an even primitive divisor of p d − 1 ) such that h k / 2 inverts N. Let 1 � = g ∈ N and let S = { g h i ; i ∈ Z k } , with natural cyclic ordering. Then, the Cayley graph Cay ( N , S ) embeds as an orientably-regular Cayley map; Jozef ˇ Sir´ aˇ n OU and STU Vertex-quasiprimitivity in regular maps 27th May 2015 13 / 15

  74. The ‘Holomorph-Abelian’ case: A sample result This is the only case in which we have a characterisation: Theorem [Li, ˇ S, Wang, w. i. p.] Let N = Z d p for an odd prime p and let G = N ⋊ H, where H = � h � ≃ Z k is an irreducible subgroup of AGL ( d , p ) (that is, k is an even primitive divisor of p d − 1 ) such that h k / 2 inverts N. Let 1 � = g ∈ N and let S = { g h i ; i ∈ Z k } , with natural cyclic ordering. Then, the Cayley graph Cay ( N , S ) embeds as an orientably-regular Cayley map; the map is regular if and only if d = 2 e and k divides p e + 1 . Jozef ˇ Sir´ aˇ n OU and STU Vertex-quasiprimitivity in regular maps 27th May 2015 13 / 15

  75. The ‘Holomorph-Abelian’ case: A sample result This is the only case in which we have a characterisation: Theorem [Li, ˇ S, Wang, w. i. p.] Let N = Z d p for an odd prime p and let G = N ⋊ H, where H = � h � ≃ Z k is an irreducible subgroup of AGL ( d , p ) (that is, k is an even primitive divisor of p d − 1 ) such that h k / 2 inverts N. Let 1 � = g ∈ N and let S = { g h i ; i ∈ Z k } , with natural cyclic ordering. Then, the Cayley graph Cay ( N , S ) embeds as an orientably-regular Cayley map; the map is regular if and only if d = 2 e and k divides p e + 1 . Moreover, all (orientably) regular maps with a vertex-quasiprimitive automorphism group of type HA for odd p arise this way. Jozef ˇ Sir´ aˇ n OU and STU Vertex-quasiprimitivity in regular maps 27th May 2015 13 / 15

  76. The ‘Holomorph-Abelian’ case: A sample result This is the only case in which we have a characterisation: Theorem [Li, ˇ S, Wang, w. i. p.] Let N = Z d p for an odd prime p and let G = N ⋊ H, where H = � h � ≃ Z k is an irreducible subgroup of AGL ( d , p ) (that is, k is an even primitive divisor of p d − 1 ) such that h k / 2 inverts N. Let 1 � = g ∈ N and let S = { g h i ; i ∈ Z k } , with natural cyclic ordering. Then, the Cayley graph Cay ( N , S ) embeds as an orientably-regular Cayley map; the map is regular if and only if d = 2 e and k divides p e + 1 . Moreover, all (orientably) regular maps with a vertex-quasiprimitive automorphism group of type HA for odd p arise this way. Similar results for non-orientable regular maps; modifications for p = 2. Jozef ˇ Sir´ aˇ n OU and STU Vertex-quasiprimitivity in regular maps 27th May 2015 13 / 15

  77. The ‘Twisted Wreath Product’ case Jozef ˇ Sir´ aˇ n OU and STU Vertex-quasiprimitivity in regular maps 27th May 2015 14 / 15

  78. The ‘Twisted Wreath Product’ case Let a group B have a transitive action on { 1 , . . . , k } , with S = Stab B (1). Jozef ˇ Sir´ aˇ n OU and STU Vertex-quasiprimitivity in regular maps 27th May 2015 14 / 15

  79. The ‘Twisted Wreath Product’ case Let a group B have a transitive action on { 1 , . . . , k } , with S = Stab B (1). Let ϕ : S �→ Aut ( T ) be such that core B ( ϕ − 1 ( Inn ( T )) = { 1 B } . Jozef ˇ Sir´ aˇ n OU and STU Vertex-quasiprimitivity in regular maps 27th May 2015 14 / 15

  80. The ‘Twisted Wreath Product’ case Let a group B have a transitive action on { 1 , . . . , k } , with S = Stab B (1). Let ϕ : S �→ Aut ( T ) be such that core B ( ϕ − 1 ( Inn ( T )) = { 1 B } . Define A := { f : B �→ T : f ( bs ) = f ( b ) ϕ ( s ) for all b ∈ B , s ∈ S } . Jozef ˇ Sir´ aˇ n OU and STU Vertex-quasiprimitivity in regular maps 27th May 2015 14 / 15

  81. The ‘Twisted Wreath Product’ case Let a group B have a transitive action on { 1 , . . . , k } , with S = Stab B (1). Let ϕ : S �→ Aut ( T ) be such that core B ( ϕ − 1 ( Inn ( T )) = { 1 B } . Define A := { f : B �→ T : f ( bs ) = f ( b ) ϕ ( s ) for all b ∈ B , s ∈ S } . Then A is a group under pointwise multiplication and A ∼ = T k . Jozef ˇ Sir´ aˇ n OU and STU Vertex-quasiprimitivity in regular maps 27th May 2015 14 / 15

  82. The ‘Twisted Wreath Product’ case Let a group B have a transitive action on { 1 , . . . , k } , with S = Stab B (1). Let ϕ : S �→ Aut ( T ) be such that core B ( ϕ − 1 ( Inn ( T )) = { 1 B } . Define A := { f : B �→ T : f ( bs ) = f ( b ) ϕ ( s ) for all b ∈ B , s ∈ S } . Then A is a group under pointwise multiplication and A ∼ = T k . Further, let B act on A by ψ : f b ( x ) := f ( bx ) for all b , x ∈ B . Jozef ˇ Sir´ aˇ n OU and STU Vertex-quasiprimitivity in regular maps 27th May 2015 14 / 15

  83. The ‘Twisted Wreath Product’ case Let a group B have a transitive action on { 1 , . . . , k } , with S = Stab B (1). Let ϕ : S �→ Aut ( T ) be such that core B ( ϕ − 1 ( Inn ( T )) = { 1 B } . Define A := { f : B �→ T : f ( bs ) = f ( b ) ϕ ( s ) for all b ∈ B , s ∈ S } . Then A is a group under pointwise multiplication and A ∼ = T k . Further, let B act on A by ψ : f b ( x ) := f ( bx ) for all b , x ∈ B . The twisted wreath product of T and B is T twr ϕ B := A ⋊ ψ B ; Jozef ˇ Sir´ aˇ n OU and STU Vertex-quasiprimitivity in regular maps 27th May 2015 14 / 15

  84. The ‘Twisted Wreath Product’ case Let a group B have a transitive action on { 1 , . . . , k } , with S = Stab B (1). Let ϕ : S �→ Aut ( T ) be such that core B ( ϕ − 1 ( Inn ( T )) = { 1 B } . Define A := { f : B �→ T : f ( bs ) = f ( b ) ϕ ( s ) for all b ∈ B , s ∈ S } . Then A is a group under pointwise multiplication and A ∼ = T k . Further, let B act on A by ψ : f b ( x ) := f ( bx ) for all b , x ∈ B . The twisted wreath product of T and B is T twr ϕ B := A ⋊ ψ B ; it has a transitive action on Ω= A given by f ( g , b ) = gf b for f , g ∈ A and b ∈ B . Jozef ˇ Sir´ aˇ n OU and STU Vertex-quasiprimitivity in regular maps 27th May 2015 14 / 15

  85. The ‘Twisted Wreath Product’ case Let a group B have a transitive action on { 1 , . . . , k } , with S = Stab B (1). Let ϕ : S �→ Aut ( T ) be such that core B ( ϕ − 1 ( Inn ( T )) = { 1 B } . Define A := { f : B �→ T : f ( bs ) = f ( b ) ϕ ( s ) for all b ∈ B , s ∈ S } . Then A is a group under pointwise multiplication and A ∼ = T k . Further, let B act on A by ψ : f b ( x ) := f ( bx ) for all b , x ∈ B . The twisted wreath product of T and B is T twr ϕ B := A ⋊ ψ B ; it has a transitive action on Ω= A given by f ( g , b ) = gf b for f , g ∈ A and b ∈ B . A sample of results: Jozef ˇ Sir´ aˇ n OU and STU Vertex-quasiprimitivity in regular maps 27th May 2015 14 / 15

  86. The ‘Twisted Wreath Product’ case Let a group B have a transitive action on { 1 , . . . , k } , with S = Stab B (1). Let ϕ : S �→ Aut ( T ) be such that core B ( ϕ − 1 ( Inn ( T )) = { 1 B } . Define A := { f : B �→ T : f ( bs ) = f ( b ) ϕ ( s ) for all b ∈ B , s ∈ S } . Then A is a group under pointwise multiplication and A ∼ = T k . Further, let B act on A by ψ : f b ( x ) := f ( bx ) for all b , x ∈ B . The twisted wreath product of T and B is T twr ϕ B := A ⋊ ψ B ; it has a transitive action on Ω= A given by f ( g , b ) = gf b for f , g ∈ A and b ∈ B . A sample of results: • Every quasiprimitive regular map of type TW is orientable. Jozef ˇ Sir´ aˇ n OU and STU Vertex-quasiprimitivity in regular maps 27th May 2015 14 / 15

  87. The ‘Twisted Wreath Product’ case Let a group B have a transitive action on { 1 , . . . , k } , with S = Stab B (1). Let ϕ : S �→ Aut ( T ) be such that core B ( ϕ − 1 ( Inn ( T )) = { 1 B } . Define A := { f : B �→ T : f ( bs ) = f ( b ) ϕ ( s ) for all b ∈ B , s ∈ S } . Then A is a group under pointwise multiplication and A ∼ = T k . Further, let B act on A by ψ : f b ( x ) := f ( bx ) for all b , x ∈ B . The twisted wreath product of T and B is T twr ϕ B := A ⋊ ψ B ; it has a transitive action on Ω= A given by f ( g , b ) = gf b for f , g ∈ A and b ∈ B . A sample of results: • Every quasiprimitive regular map of type TW is orientable. • Infinite families of orientably-regular (chiral, if desired) map M with Aut + ( M ) ∼ = T twr ϕ B , where T = PSL (2 , p ), B = Z 2 k , ϕ ( z ) = z mod k . Jozef ˇ Sir´ aˇ n OU and STU Vertex-quasiprimitivity in regular maps 27th May 2015 14 / 15

  88. The ‘Product Action’ and ‘Almost Simple’ cases Jozef ˇ Sir´ aˇ n OU and STU Vertex-quasiprimitivity in regular maps 27th May 2015 15 / 15

  89. The ‘Product Action’ and ‘Almost Simple’ cases • The ‘Product Action’ type (description omitted) splits into the ‘Straight Diagonal’, ‘Twisted Diagonal’ and ‘Non-Diagonal’ subcases. Jozef ˇ Sir´ aˇ n OU and STU Vertex-quasiprimitivity in regular maps 27th May 2015 15 / 15

  90. The ‘Product Action’ and ‘Almost Simple’ cases • The ‘Product Action’ type (description omitted) splits into the ‘Straight Diagonal’, ‘Twisted Diagonal’ and ‘Non-Diagonal’ subcases. We have infinite families of examples of (orientably-) regular maps in the first two categories (for chiral examples we need to invoke the family of Suzuki simple groups), and a proof that no example exists in the third category. Jozef ˇ Sir´ aˇ n OU and STU Vertex-quasiprimitivity in regular maps 27th May 2015 15 / 15

  91. The ‘Product Action’ and ‘Almost Simple’ cases • The ‘Product Action’ type (description omitted) splits into the ‘Straight Diagonal’, ‘Twisted Diagonal’ and ‘Non-Diagonal’ subcases. We have infinite families of examples of (orientably-) regular maps in the first two categories (for chiral examples we need to invoke the family of Suzuki simple groups), and a proof that no example exists in the third category. • The ‘Almost Simple’ type: k =1, N = T < G < Aut ( T ), T transitive on Ω Jozef ˇ Sir´ aˇ n OU and STU Vertex-quasiprimitivity in regular maps 27th May 2015 15 / 15

  92. The ‘Product Action’ and ‘Almost Simple’ cases • The ‘Product Action’ type (description omitted) splits into the ‘Straight Diagonal’, ‘Twisted Diagonal’ and ‘Non-Diagonal’ subcases. We have infinite families of examples of (orientably-) regular maps in the first two categories (for chiral examples we need to invoke the family of Suzuki simple groups), and a proof that no example exists in the third category. • The ‘Almost Simple’ type: k =1, N = T < G < Aut ( T ), T transitive on Ω A consequence of a deep result of Malle, Saxl, and Weigel 1991: Every finite simple group is isomorphic to the automorphism group of an orientably regular map. Jozef ˇ Sir´ aˇ n OU and STU Vertex-quasiprimitivity in regular maps 27th May 2015 15 / 15

  93. The ‘Product Action’ and ‘Almost Simple’ cases • The ‘Product Action’ type (description omitted) splits into the ‘Straight Diagonal’, ‘Twisted Diagonal’ and ‘Non-Diagonal’ subcases. We have infinite families of examples of (orientably-) regular maps in the first two categories (for chiral examples we need to invoke the family of Suzuki simple groups), and a proof that no example exists in the third category. • The ‘Almost Simple’ type: k =1, N = T < G < Aut ( T ), T transitive on Ω A consequence of a deep result of Malle, Saxl, and Weigel 1991: Every finite simple group is isomorphic to the automorphism group of an orientably regular map. With some exceptions, the map can be assumed to be trivalent; Jozef ˇ Sir´ aˇ n OU and STU Vertex-quasiprimitivity in regular maps 27th May 2015 15 / 15

  94. The ‘Product Action’ and ‘Almost Simple’ cases • The ‘Product Action’ type (description omitted) splits into the ‘Straight Diagonal’, ‘Twisted Diagonal’ and ‘Non-Diagonal’ subcases. We have infinite families of examples of (orientably-) regular maps in the first two categories (for chiral examples we need to invoke the family of Suzuki simple groups), and a proof that no example exists in the third category. • The ‘Almost Simple’ type: k =1, N = T < G < Aut ( T ), T transitive on Ω A consequence of a deep result of Malle, Saxl, and Weigel 1991: Every finite simple group is isomorphic to the automorphism group of an orientably regular map. With some exceptions, the map can be assumed to be trivalent; with a larger set of exceptions the result extends to regular maps. Jozef ˇ Sir´ aˇ n OU and STU Vertex-quasiprimitivity in regular maps 27th May 2015 15 / 15

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