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Regular maps with a given automorphism group, and on a given surface Jozef Sir a n Open University and Slovak University of Technology AGT 2016 Pilsen Jozef Sir a n Open University and Slovak University of Technology


  1. �� �� � � ���� ���� �������� �������� �������� �������� �� �� � � ���� ���� �������� �������� �������� �������� � � ���� ���� �������� �������� �������� �������� � � �������� �������� �������� �������� ���� ���� � � �������� �������� �������� �������� ���� ���� �� �� �������� �������� �������� �������� �� �� ���� ���� � � � �� �� �� �� � �������� �������� �������� �������� �� �� �� �� ���� ���� � � ���� ���� �������� �������� �������� �������� ����������������� ����������������� ���� ���� �������� �������� ����������������� ����������������� � � ���� ���� �������� �������� ����������������� ����������������� � � ���� ���� �������� �������� ����������������� ����������������� � � ���� ���� �������� �������� �� �� ����������������� ����������������� � � �������� �������� �� �� ���� ���� ����������������� ����������������� � � �������� �������� ���� ���� ����������������� ����������������� �������� �������� ���� ���� � � ����������������� ����������������� �������� �������� ���� ���� � � ����� ����� ����������������� ����������������� �������� �������� � � ����������������� ����������������� ����� ����� �������� �������� � � � � ����������������� ����������������� ����� ����� �������� �������� � � � � �� �� ����������������� ����������������� ����� ����� �������� �������� � � ����������������� ����������������� ����� ����� �������� �������� � � ����������������� ����������������� ����� ����� �������� �������� � � ����������������� ����������������� ����� ����� �������� �������� � � ����������������� ����������������� �������� �������� ����� ����� �������� �������� � � �������� �������� ����� ����� �������� �������� � � �������� �������� ����� ����� �������� �������� � � �������� �������� ����� ����� �������� �������� �������� �������� ����� ����� �������� �������� �������� �������� � � ����� ����� �������� �������� �������� �������� � � ����� ����� �������� �������� �������� �������� ����� ����� �������� �������� �� �� �� �� �������� �������� ����� ����� �������� �������� �� �� �� �� Example of an orientably-regular map: K 5 on a torus Jozef ˇ Sir´ aˇ n Open University and Slovak University of Technology Regular maps with a given automorphism group, and on a given surface AGT 2016 Pilsen 4 / 18

  2. Example of an orientably-regular map: K 5 on a torus �� �� � � ���� ���� �������� �������� �������� �������� �� �� � � ���� ���� �������� �������� �������� �������� � � �������� �������� �������� �������� ���� ���� � � �������� �������� �������� �������� ���� ���� � � �������� �������� �������� �������� ���� ���� �� �� �������� �������� �������� �������� �� �� ���� ���� � � � �� �� �� �� � �� �� �� �� ���� ���� �������� �������� �������� �������� � � ���� ���� �������� �������� �������� �������� ����������������� ����������������� ���� ���� �������� �������� ����������������� ����������������� � � ���� ���� �������� �������� ����������������� ����������������� � � ���� ���� �������� �������� ����������������� ����������������� � � ���� ���� �������� �������� �� �� ����������������� ����������������� � � �������� �������� �� �� ���� ���� ����������������� ����������������� � � �������� �������� ���� ���� ����������������� ����������������� �������� �������� ���� ���� � � ����������������� ����������������� �������� �������� ���� ���� � � ����� ����� ����������������� ����������������� �������� �������� � � ����������������� ����������������� ����� ����� �������� �������� � � � � ����������������� ����������������� ����� ����� �������� �������� � � � � �� �� ����������������� ����������������� ����� ����� �������� �������� � � ����������������� ����������������� ����� ����� �������� �������� � � ����������������� ����������������� ����� ����� �������� �������� � � ����������������� ����������������� ����� ����� �������� �������� ����������������� �������� �������� ����� ����� � � ����������������� �������� �������� � � �������� �������� ����� ����� �������� �������� � � �������� �������� ����� ����� �������� �������� � � �������� �������� ����� ����� �������� �������� �������� �������� ����� ����� �������� �������� �������� �������� � � ����� ����� �������� �������� �������� �������� � � ����� ����� �������� �������� �������� �������� ����� ����� �������� �������� �� �� �� �� �������� �������� ����� ����� �������� �������� �� �� �� �� • Presentation: Aut + ( M ) = � r, s | r 4 = s 4 = ( rs ) 2 = r 2 s 2 rs − 1 = 1 � Jozef ˇ Sir´ aˇ n Open University and Slovak University of Technology Regular maps with a given automorphism group, and on a given surface AGT 2016 Pilsen 4 / 18

  3. Example of an orientably-regular map: K 5 on a torus �� �� � � ���� ���� �������� �������� �������� �������� �� �� � � ���� ���� �������� �������� �������� �������� � � �������� �������� �������� �������� ���� ���� � � �������� �������� �������� �������� ���� ���� � � �������� �������� �������� �������� ���� ���� �� �� �������� �������� �������� �������� �� �� ���� ���� � � � �� �� �� �� � �� �� �� �� ���� ���� �������� �������� �������� �������� � � ���� ���� �������� �������� �������� �������� ����������������� ����������������� ���� ���� �������� �������� ����������������� ����������������� � � ���� ���� �������� �������� ����������������� ����������������� � � ���� ���� �������� �������� ����������������� ����������������� � � ���� ���� �������� �������� �� �� ����������������� ����������������� � � �������� �������� �� �� ���� ���� ����������������� ����������������� � � �������� �������� ���� ���� ����������������� ����������������� �������� �������� ���� ���� � � ����������������� ����������������� �������� �������� ���� ���� � � ����� ����� ����������������� ����������������� �������� �������� � � ����������������� ����������������� ����� ����� �������� �������� � � � � ����������������� ����������������� ����� ����� �������� �������� � � � � �� �� ����������������� ����������������� ����� ����� �������� �������� � � ����������������� ����������������� ����� ����� �������� �������� � � ����������������� ����������������� ����� ����� �������� �������� � � ����������������� ����������������� ����� ����� �������� �������� ����������������� �������� �������� ����� ����� � � ����������������� �������� �������� � � �������� �������� ����� ����� �������� �������� � � �������� �������� ����� ����� �������� �������� � � �������� �������� ����� ����� �������� �������� �������� �������� ����� ����� �������� �������� �������� �������� � � ����� ����� �������� �������� �������� �������� � � ����� ����� �������� �������� �������� �������� ����� ����� �������� �������� �� �� �� �� �������� �������� ����� ����� �������� �������� �� �� �� �� • Presentation: Aut + ( M ) = � r, s | r 4 = s 4 = ( rs ) 2 = r 2 s 2 rs − 1 = 1 � • This map is chiral (no reflection). Jozef ˇ Sir´ aˇ n Open University and Slovak University of Technology Regular maps with a given automorphism group, and on a given surface AGT 2016 Pilsen 4 / 18

  4. Example of an orientably-regular map: K 5 on a torus �� �� � � ���� ���� �������� �������� �������� �������� �� �� � � ���� ���� �������� �������� �������� �������� � � �������� �������� �������� �������� ���� ���� � � �������� �������� �������� �������� ���� ���� � � �������� �������� �������� �������� ���� ���� �� �� �������� �������� �������� �������� �� �� ���� ���� � � � �� �� �� �� � �� �� �� �� ���� ���� �������� �������� �������� �������� � � ���� ���� �������� �������� �������� �������� ����������������� ����������������� ���� ���� �������� �������� ����������������� ����������������� � � ���� ���� �������� �������� ����������������� ����������������� � � ���� ���� �������� �������� ����������������� ����������������� � � ���� ���� �������� �������� �� �� ����������������� ����������������� � � �������� �������� �� �� ���� ���� ����������������� ����������������� � � �������� �������� ���� ���� ����������������� ����������������� �������� �������� ���� ���� � � ����������������� ����������������� �������� �������� ���� ���� � � ����� ����� ����������������� ����������������� �������� �������� � � ����������������� ����������������� ����� ����� �������� �������� � � � � ����������������� ����������������� ����� ����� �������� �������� � � � � �� �� ����������������� ����������������� ����� ����� �������� �������� � � ����������������� ����������������� ����� ����� �������� �������� � � ����������������� ����������������� ����� ����� �������� �������� � � ����������������� ����������������� ����� ����� �������� �������� ����������������� �������� �������� ����� ����� � � ����������������� �������� �������� � � �������� �������� ����� ����� �������� �������� � � �������� �������� ����� ����� �������� �������� � � �������� �������� ����� ����� �������� �������� �������� �������� ����� ����� �������� �������� �������� �������� � � ����� ����� �������� �������� �������� �������� � � ����� ����� �������� �������� �������� �������� ����� ����� �������� �������� �� �� �� �� �������� �������� ����� ����� �������� �������� �� �� �� �� • Presentation: Aut + ( M ) = � r, s | r 4 = s 4 = ( rs ) 2 = r 2 s 2 rs − 1 = 1 � • This map is chiral (no reflection). • Algebraic theory of reflexible and non-orientable regular maps - later. Jozef ˇ Sir´ aˇ n Open University and Slovak University of Technology Regular maps with a given automorphism group, and on a given surface AGT 2016 Pilsen 4 / 18

  5. Orientably-regular maps and exciting mathematics Jozef ˇ Sir´ aˇ n Open University and Slovak University of Technology Regular maps with a given automorphism group, and on a given surface AGT 2016 Pilsen 5 / 18

  6. Orientably-regular maps and exciting mathematics Up to isomorphism, 1-1 correspondence between: orientably-regular maps of type ( ℓ, m ) ; group presentations � r, s | r ℓ = s m = ( rs ) 2 = . . . = 1 � ; torsion-free normal subgroups of triangle groups T ℓ,m . Jozef ˇ Sir´ aˇ n Open University and Slovak University of Technology Regular maps with a given automorphism group, and on a given surface AGT 2016 Pilsen 5 / 18

  7. Orientably-regular maps and exciting mathematics Up to isomorphism, 1-1 correspondence between: orientably-regular maps of type ( ℓ, m ) ; group presentations � r, s | r ℓ = s m = ( rs ) 2 = . . . = 1 � ; torsion-free normal subgroups of triangle groups T ℓ,m . Maps, Riemann surfaces, and Galois theory: Jozef ˇ Sir´ aˇ n Open University and Slovak University of Technology Regular maps with a given automorphism group, and on a given surface AGT 2016 Pilsen 5 / 18

  8. Orientably-regular maps and exciting mathematics Up to isomorphism, 1-1 correspondence between: orientably-regular maps of type ( ℓ, m ) ; group presentations � r, s | r ℓ = s m = ( rs ) 2 = . . . = 1 � ; torsion-free normal subgroups of triangle groups T ℓ,m . Maps, Riemann surfaces, and Galois theory: A compact Riemann surface S can be uniformised by representing it in the form S ∼ = U /H for some Fuchsian group H < PSL(2 , R ) . Jozef ˇ Sir´ aˇ n Open University and Slovak University of Technology Regular maps with a given automorphism group, and on a given surface AGT 2016 Pilsen 5 / 18

  9. Orientably-regular maps and exciting mathematics Up to isomorphism, 1-1 correspondence between: orientably-regular maps of type ( ℓ, m ) ; group presentations � r, s | r ℓ = s m = ( rs ) 2 = . . . = 1 � ; torsion-free normal subgroups of triangle groups T ℓ,m . Maps, Riemann surfaces, and Galois theory: A compact Riemann surface S can be uniformised by representing it in the form S ∼ = U /H for some Fuchsian group H < PSL(2 , R ) . But S can also be defined by a complex polynomial eq’n P ( x, y ) = 0 as a many-valued function y = f ( x ) . Jozef ˇ Sir´ aˇ n Open University and Slovak University of Technology Regular maps with a given automorphism group, and on a given surface AGT 2016 Pilsen 5 / 18

  10. Orientably-regular maps and exciting mathematics Up to isomorphism, 1-1 correspondence between: orientably-regular maps of type ( ℓ, m ) ; group presentations � r, s | r ℓ = s m = ( rs ) 2 = . . . = 1 � ; torsion-free normal subgroups of triangle groups T ℓ,m . Maps, Riemann surfaces, and Galois theory: A compact Riemann surface S can be uniformised by representing it in the form S ∼ = U /H for some Fuchsian group H < PSL(2 , R ) . But S can also be defined by a complex polynomial eq’n P ( x, y ) = 0 as a many-valued function y = f ( x ) . We have a tower of branched coverings: U → S → C . Jozef ˇ Sir´ aˇ n Open University and Slovak University of Technology Regular maps with a given automorphism group, and on a given surface AGT 2016 Pilsen 5 / 18

  11. Orientably-regular maps and exciting mathematics Up to isomorphism, 1-1 correspondence between: orientably-regular maps of type ( ℓ, m ) ; group presentations � r, s | r ℓ = s m = ( rs ) 2 = . . . = 1 � ; torsion-free normal subgroups of triangle groups T ℓ,m . Maps, Riemann surfaces, and Galois theory: A compact Riemann surface S can be uniformised by representing it in the form S ∼ = U /H for some Fuchsian group H < PSL(2 , R ) . But S can also be defined by a complex polynomial eq’n P ( x, y ) = 0 as a many-valued function y = f ( x ) . We have a tower of branched coverings: U → S → C . [Weil 1950, Belyj 1972]: S is definable by a P with algebraic coefficients if and only if S = U ℓ,m /K for some finite-index subgroup K of some T ℓ,m (loosely speaking, iff the complex structure on S “comes from a map”). Jozef ˇ Sir´ aˇ n Open University and Slovak University of Technology Regular maps with a given automorphism group, and on a given surface AGT 2016 Pilsen 5 / 18

  12. Orientably-regular maps and exciting mathematics Up to isomorphism, 1-1 correspondence between: orientably-regular maps of type ( ℓ, m ) ; group presentations � r, s | r ℓ = s m = ( rs ) 2 = . . . = 1 � ; torsion-free normal subgroups of triangle groups T ℓ,m . Maps, Riemann surfaces, and Galois theory: A compact Riemann surface S can be uniformised by representing it in the form S ∼ = U /H for some Fuchsian group H < PSL(2 , R ) . But S can also be defined by a complex polynomial eq’n P ( x, y ) = 0 as a many-valued function y = f ( x ) . We have a tower of branched coverings: U → S → C . [Weil 1950, Belyj 1972]: S is definable by a P with algebraic coefficients if and only if S = U ℓ,m /K for some finite-index subgroup K of some T ℓ,m (loosely speaking, iff the complex structure on S “comes from a map”). This way the absolute Galois group acts on maps! [Grothendieck 1981] Jozef ˇ Sir´ aˇ n Open University and Slovak University of Technology Regular maps with a given automorphism group, and on a given surface AGT 2016 Pilsen 5 / 18

  13. Orientably-regular maps and exciting mathematics Up to isomorphism, 1-1 correspondence between: orientably-regular maps of type ( ℓ, m ) ; group presentations � r, s | r ℓ = s m = ( rs ) 2 = . . . = 1 � ; torsion-free normal subgroups of triangle groups T ℓ,m . Maps, Riemann surfaces, and Galois theory: A compact Riemann surface S can be uniformised by representing it in the form S ∼ = U /H for some Fuchsian group H < PSL(2 , R ) . But S can also be defined by a complex polynomial eq’n P ( x, y ) = 0 as a many-valued function y = f ( x ) . We have a tower of branched coverings: U → S → C . [Weil 1950, Belyj 1972]: S is definable by a P with algebraic coefficients if and only if S = U ℓ,m /K for some finite-index subgroup K of some T ℓ,m (loosely speaking, iff the complex structure on S “comes from a map”). This way the absolute Galois group acts on maps! [Grothendieck 1981] Faithful on orientably-regular maps! [Gonz´ alez-Diez, Jaikin-Zapirain 2013] Jozef ˇ Sir´ aˇ n Open University and Slovak University of Technology Regular maps with a given automorphism group, and on a given surface AGT 2016 Pilsen 5 / 18

  14. Regular maps Jozef ˇ Sir´ aˇ n Open University and Slovak University of Technology Regular maps with a given automorphism group, and on a given surface AGT 2016 Pilsen 6 / 18

  15. Regular maps A map is regular if its automorphism group is regular on flags. Jozef ˇ Sir´ aˇ n Open University and Slovak University of Technology Regular maps with a given automorphism group, and on a given surface AGT 2016 Pilsen 6 / 18

  16. Regular maps A map is regular if its automorphism group is regular on flags. Jozef ˇ Sir´ aˇ n Open University and Slovak University of Technology Regular maps with a given automorphism group, and on a given surface AGT 2016 Pilsen 6 / 18

  17. 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 ) ℓ = ( zx ) m = ( xy ) 2 = . . . = 1 � Jozef ˇ Sir´ aˇ n Open University and Slovak University of Technology Regular maps with a given automorphism group, and on a given surface AGT 2016 Pilsen 6 / 18

  18. 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 ) ℓ = ( zx ) m = ( xy ) 2 = . . . = 1 � Conversely, every group with such a presentation determines a regular map. Jozef ˇ Sir´ aˇ n Open University and Slovak University of Technology Regular maps with a given automorphism group, and on a given surface AGT 2016 Pilsen 6 / 18

  19. Classification of (orientably-) regular maps Jozef ˇ Sir´ aˇ n Open University and Slovak University of Technology Regular maps with a given automorphism group, and on a given surface AGT 2016 Pilsen 7 / 18

  20. Classification of (orientably-) regular maps Classification of regular maps has been approached from three main directions: Jozef ˇ Sir´ aˇ n Open University and Slovak University of Technology Regular maps with a given automorphism group, and on a given surface AGT 2016 Pilsen 7 / 18

  21. Classification of (orientably-) regular maps Classification of regular maps has been approached from three main directions: by underlying graphs Jozef ˇ Sir´ aˇ n Open University and Slovak University of Technology Regular maps with a given automorphism group, and on a given surface AGT 2016 Pilsen 7 / 18

  22. Classification of (orientably-) regular maps Classification of regular maps has been approached from three main directions: by underlying graphs by automorphism groups Jozef ˇ Sir´ aˇ n Open University and Slovak University of Technology Regular maps with a given automorphism group, and on a given surface AGT 2016 Pilsen 7 / 18

  23. Classification of (orientably-) regular maps Classification of regular maps has been approached from three main directions: by underlying graphs by automorphism groups by supporting surfaces Jozef ˇ Sir´ aˇ n Open University and Slovak University of Technology Regular maps with a given automorphism group, and on a given surface AGT 2016 Pilsen 7 / 18

  24. Classification of (orientably-) regular maps Classification of regular maps has been approached from three main directions: by underlying graphs by automorphism groups by supporting surfaces Other approaches to the study of regular maps by a combination of graph-theoretic, algebraic, and topological means: Jozef ˇ Sir´ aˇ n Open University and Slovak University of Technology Regular maps with a given automorphism group, and on a given surface AGT 2016 Pilsen 7 / 18

  25. Classification of (orientably-) regular maps Classification of regular maps has been approached from three main directions: by underlying graphs by automorphism groups by supporting surfaces Other approaches to the study of regular maps by a combination of graph-theoretic, algebraic, and topological means: constructions using suitable graphs, groups, or tools (coverings) Jozef ˇ Sir´ aˇ n Open University and Slovak University of Technology Regular maps with a given automorphism group, and on a given surface AGT 2016 Pilsen 7 / 18

  26. Classification of (orientably-) regular maps Classification of regular maps has been approached from three main directions: by underlying graphs by automorphism groups by supporting surfaces Other approaches to the study of regular maps by a combination of graph-theoretic, algebraic, and topological means: constructions using suitable graphs, groups, or tools (coverings) structural investigation (short cycles, representativity – planar width) Jozef ˇ Sir´ aˇ n Open University and Slovak University of Technology Regular maps with a given automorphism group, and on a given surface AGT 2016 Pilsen 7 / 18

  27. Classification of (orientably-) regular maps Classification of regular maps has been approached from three main directions: by underlying graphs by automorphism groups by supporting surfaces Other approaches to the study of regular maps by a combination of graph-theoretic, algebraic, and topological means: constructions using suitable graphs, groups, or tools (coverings) structural investigation (short cycles, representativity – planar width) imposing additional algebraic structure – regular Cayley maps Jozef ˇ Sir´ aˇ n Open University and Slovak University of Technology Regular maps with a given automorphism group, and on a given surface AGT 2016 Pilsen 7 / 18

  28. Classification of (orientably-) regular maps Classification of regular maps has been approached from three main directions: by underlying graphs by automorphism groups by supporting surfaces Other approaches to the study of regular maps by a combination of graph-theoretic, algebraic, and topological means: constructions using suitable graphs, groups, or tools (coverings) structural investigation (short cycles, representativity – planar width) imposing additional algebraic structure – regular Cayley maps research motivated by computer-aided results Jozef ˇ Sir´ aˇ n Open University and Slovak University of Technology Regular maps with a given automorphism group, and on a given surface AGT 2016 Pilsen 7 / 18

  29. Classification by underlying graphs: Jozef ˇ Sir´ aˇ n Open University and Slovak University of Technology Regular maps with a given automorphism group, and on a given surface AGT 2016 Pilsen 8 / 18

  30. Classification by underlying graphs: complete characterization of graphs underlying (orientably) regular maps in terms of existence of ‘suitable’ subgroups of the graph automorphism groups [Gardiner, Nedela, ˇ S and ˇ Skoviera 1999] Jozef ˇ Sir´ aˇ n Open University and Slovak University of Technology Regular maps with a given automorphism group, and on a given surface AGT 2016 Pilsen 8 / 18

  31. Classification by underlying graphs: complete characterization of graphs underlying (orientably) regular maps in terms of existence of ‘suitable’ subgroups of the graph automorphism groups [Gardiner, Nedela, ˇ S and ˇ Skoviera 1999] complete graphs [Biggs 1974] and [James and Jones 1984] in the orientable case; [Wilson 1989] in the non-orientable case Jozef ˇ Sir´ aˇ n Open University and Slovak University of Technology Regular maps with a given automorphism group, and on a given surface AGT 2016 Pilsen 8 / 18

  32. Classification by underlying graphs: complete characterization of graphs underlying (orientably) regular maps in terms of existence of ‘suitable’ subgroups of the graph automorphism groups [Gardiner, Nedela, ˇ S and ˇ Skoviera 1999] complete graphs [Biggs 1974] and [James and Jones 1984] in the orientable case; [Wilson 1989] in the non-orientable case complete bipartite graphs – recent major work of [Jones 2012] in the orientable case, with special cases settled earlier by multiple authors; the non-orientable case [Kwak and Kwon 2011] Jozef ˇ Sir´ aˇ n Open University and Slovak University of Technology Regular maps with a given automorphism group, and on a given surface AGT 2016 Pilsen 8 / 18

  33. Classification by underlying graphs: complete characterization of graphs underlying (orientably) regular maps in terms of existence of ‘suitable’ subgroups of the graph automorphism groups [Gardiner, Nedela, ˇ S and ˇ Skoviera 1999] complete graphs [Biggs 1974] and [James and Jones 1984] in the orientable case; [Wilson 1989] in the non-orientable case complete bipartite graphs – recent major work of [Jones 2012] in the orientable case, with special cases settled earlier by multiple authors; the non-orientable case [Kwak and Kwon 2011] K p,p,...,p in the orientable case [Du, Kwak and Nedela 2005] Jozef ˇ Sir´ aˇ n Open University and Slovak University of Technology Regular maps with a given automorphism group, and on a given surface AGT 2016 Pilsen 8 / 18

  34. Classification by underlying graphs: complete characterization of graphs underlying (orientably) regular maps in terms of existence of ‘suitable’ subgroups of the graph automorphism groups [Gardiner, Nedela, ˇ S and ˇ Skoviera 1999] complete graphs [Biggs 1974] and [James and Jones 1984] in the orientable case; [Wilson 1989] in the non-orientable case complete bipartite graphs – recent major work of [Jones 2012] in the orientable case, with special cases settled earlier by multiple authors; the non-orientable case [Kwak and Kwon 2011] K p,p,...,p in the orientable case [Du, Kwak and Nedela 2005] Q n [Breda, Catalano, Conder, Kwak, Kwon, Nedela, Wilson 2012] Jozef ˇ Sir´ aˇ n Open University and Slovak University of Technology Regular maps with a given automorphism group, and on a given surface AGT 2016 Pilsen 8 / 18

  35. Classification by underlying graphs: complete characterization of graphs underlying (orientably) regular maps in terms of existence of ‘suitable’ subgroups of the graph automorphism groups [Gardiner, Nedela, ˇ S and ˇ Skoviera 1999] complete graphs [Biggs 1974] and [James and Jones 1984] in the orientable case; [Wilson 1989] in the non-orientable case complete bipartite graphs – recent major work of [Jones 2012] in the orientable case, with special cases settled earlier by multiple authors; the non-orientable case [Kwak and Kwon 2011] K p,p,...,p in the orientable case [Du, Kwak and Nedela 2005] Q n [Breda, Catalano, Conder, Kwak, Kwon, Nedela, Wilson 2012] merged Johnson graphs in the orientable case [Jones 2005] Jozef ˇ Sir´ aˇ n Open University and Slovak University of Technology Regular maps with a given automorphism group, and on a given surface AGT 2016 Pilsen 8 / 18

  36. Classification of regular maps by automorphism groups Jozef ˇ Sir´ aˇ n Open University and Slovak University of Technology Regular maps with a given automorphism group, and on a given surface AGT 2016 Pilsen 9 / 18

  37. Classification of regular maps by automorphism groups regular maps with nilpotent groups of class ≤ 3 ; orientably-regular maps with simple graphs on nilpotent groups of class c are quotients c, Nedela, ˇ of a single such map [Du, Conder, Malniˇ Skoviera, ...] Jozef ˇ Sir´ aˇ n Open University and Slovak University of Technology Regular maps with a given automorphism group, and on a given surface AGT 2016 Pilsen 9 / 18

  38. Classification of regular maps by automorphism groups regular maps with nilpotent groups of class ≤ 3 ; orientably-regular maps with simple graphs on nilpotent groups of class c are quotients c, Nedela, ˇ of a single such map [Du, Conder, Malniˇ Skoviera, ...] regular maps with almost-Sylow-cyclic automorphism groups (every odd-order Sylow subgroup is cyclic and the even-order one cnik and ˇ has a cyclic subgroup of index 2 ) [Conder, Potoˇ S 2010] – in the solvable case independent of [Zassenhaus 1936] Jozef ˇ Sir´ aˇ n Open University and Slovak University of Technology Regular maps with a given automorphism group, and on a given surface AGT 2016 Pilsen 9 / 18

  39. Classification of regular maps by automorphism groups regular maps with nilpotent groups of class ≤ 3 ; orientably-regular maps with simple graphs on nilpotent groups of class c are quotients c, Nedela, ˇ of a single such map [Du, Conder, Malniˇ Skoviera, ...] regular maps with almost-Sylow-cyclic automorphism groups (every odd-order Sylow subgroup is cyclic and the even-order one cnik and ˇ has a cyclic subgroup of index 2 ) [Conder, Potoˇ S 2010] – in the solvable case independent of [Zassenhaus 1936] orientably regular maps with automorphism groups isomorphic to PSL(2 , q ) and PGL(2 , q ) [McBeath 1967, Sah 1969] Jozef ˇ Sir´ aˇ n Open University and Slovak University of Technology Regular maps with a given automorphism group, and on a given surface AGT 2016 Pilsen 9 / 18

  40. Classification of regular maps by automorphism groups regular maps with nilpotent groups of class ≤ 3 ; orientably-regular maps with simple graphs on nilpotent groups of class c are quotients c, Nedela, ˇ of a single such map [Du, Conder, Malniˇ Skoviera, ...] regular maps with almost-Sylow-cyclic automorphism groups (every odd-order Sylow subgroup is cyclic and the even-order one cnik and ˇ has a cyclic subgroup of index 2 ) [Conder, Potoˇ S 2010] – in the solvable case independent of [Zassenhaus 1936] orientably regular maps with automorphism groups isomorphic to PSL(2 , q ) and PGL(2 , q ) [McBeath 1967, Sah 1969] non-orientable regular maps with automorphism groups isomorphic to cnik and ˇ PSL(2 , q ) and PGL(2 , q ) [Conder, Potoˇ S 2008] Jozef ˇ Sir´ aˇ n Open University and Slovak University of Technology Regular maps with a given automorphism group, and on a given surface AGT 2016 Pilsen 9 / 18

  41. Classification of regular maps by automorphism groups regular maps with nilpotent groups of class ≤ 3 ; orientably-regular maps with simple graphs on nilpotent groups of class c are quotients c, Nedela, ˇ of a single such map [Du, Conder, Malniˇ Skoviera, ...] regular maps with almost-Sylow-cyclic automorphism groups (every odd-order Sylow subgroup is cyclic and the even-order one cnik and ˇ has a cyclic subgroup of index 2 ) [Conder, Potoˇ S 2010] – in the solvable case independent of [Zassenhaus 1936] orientably regular maps with automorphism groups isomorphic to PSL(2 , q ) and PGL(2 , q ) [McBeath 1967, Sah 1969] non-orientable regular maps with automorphism groups isomorphic to cnik and ˇ PSL(2 , q ) and PGL(2 , q ) [Conder, Potoˇ S 2008] Suzuki simple groups for maps of type (4 , 5) [Jones 1993] Jozef ˇ Sir´ aˇ n Open University and Slovak University of Technology Regular maps with a given automorphism group, and on a given surface AGT 2016 Pilsen 9 / 18

  42. Classification of regular maps by automorphism groups regular maps with nilpotent groups of class ≤ 3 ; orientably-regular maps with simple graphs on nilpotent groups of class c are quotients c, Nedela, ˇ of a single such map [Du, Conder, Malniˇ Skoviera, ...] regular maps with almost-Sylow-cyclic automorphism groups (every odd-order Sylow subgroup is cyclic and the even-order one cnik and ˇ has a cyclic subgroup of index 2 ) [Conder, Potoˇ S 2010] – in the solvable case independent of [Zassenhaus 1936] orientably regular maps with automorphism groups isomorphic to PSL(2 , q ) and PGL(2 , q ) [McBeath 1967, Sah 1969] non-orientable regular maps with automorphism groups isomorphic to cnik and ˇ PSL(2 , q ) and PGL(2 , q ) [Conder, Potoˇ S 2008] Suzuki simple groups for maps of type (4 , 5) [Jones 1993] Ree simple groups for maps of type (3 , 7) , (3 , 9) and (3 , p ) for primes p ≡ − 1 mod 12 [Jones 1994] Jozef ˇ Sir´ aˇ n Open University and Slovak University of Technology Regular maps with a given automorphism group, and on a given surface AGT 2016 Pilsen 9 / 18

  43. Twisted linear fractional groups Jozef ˇ Sir´ aˇ n Open University and Slovak University of Technology Regular maps with a given automorphism group, and on a given surface AGT 2016 Pilsen 10 / 18

  44. Twisted linear fractional groups F – a field, S F and N F – non-zero squares and non-squares. The groups PSL(2 , F ) and PGL(2 , F ) consist of permutations of F ∪ {∞} given by z �→ az + b if ad − bc ∈ S F , resp . ad − bc ∈ S F ∪ N F . cz + d Jozef ˇ Sir´ aˇ n Open University and Slovak University of Technology Regular maps with a given automorphism group, and on a given surface AGT 2016 Pilsen 10 / 18

  45. Twisted linear fractional groups F – a field, S F and N F – non-zero squares and non-squares. The groups PSL(2 , F ) and PGL(2 , F ) consist of permutations of F ∪ {∞} given by z �→ az + b if ad − bc ∈ S F , resp . ad − bc ∈ S F ∪ N F . cz + d If F = GF( q 2 ) , q = p f , p odd, and if σ : x �→ x q is the automorphism of F of order 2 , the twisted linear fractional group M ( q 2 ) consists of (untwisted and twisted) permutations of F ∪ {∞} defined by z �→ az σ + b z �→ az + b cz + d if ad − bc ∈ S F and cz σ + d if ad − bc ∈ N F . Jozef ˇ Sir´ aˇ n Open University and Slovak University of Technology Regular maps with a given automorphism group, and on a given surface AGT 2016 Pilsen 10 / 18

  46. Twisted linear fractional groups F – a field, S F and N F – non-zero squares and non-squares. The groups PSL(2 , F ) and PGL(2 , F ) consist of permutations of F ∪ {∞} given by z �→ az + b if ad − bc ∈ S F , resp . ad − bc ∈ S F ∪ N F . cz + d If F = GF( q 2 ) , q = p f , p odd, and if σ : x �→ x q is the automorphism of F of order 2 , the twisted linear fractional group M ( q 2 ) consists of (untwisted and twisted) permutations of F ∪ {∞} defined by z �→ az σ + b z �→ az + b cz + d if ad − bc ∈ S F and cz σ + d if ad − bc ∈ N F . Zassenhaus (1936). Jozef ˇ Sir´ aˇ n Open University and Slovak University of Technology Regular maps with a given automorphism group, and on a given surface AGT 2016 Pilsen 10 / 18

  47. Twisted linear fractional groups F – a field, S F and N F – non-zero squares and non-squares. The groups PSL(2 , F ) and PGL(2 , F ) consist of permutations of F ∪ {∞} given by z �→ az + b if ad − bc ∈ S F , resp . ad − bc ∈ S F ∪ N F . cz + d If F = GF( q 2 ) , q = p f , p odd, and if σ : x �→ x q is the automorphism of F of order 2 , the twisted linear fractional group M ( q 2 ) consists of (untwisted and twisted) permutations of F ∪ {∞} defined by z �→ az σ + b z �→ az + b cz + d if ad − bc ∈ S F and cz σ + d if ad − bc ∈ N F . Zassenhaus (1936). We worked out a lot of facts about conjugacy classes and canonical representatives of elements of M ( q 2 ) > 2 PSL(2 , q 2 ) – e.g. all twisted elements have order divisible by 4 and dividing 2( q ± 1) , etc. Jozef ˇ Sir´ aˇ n Open University and Slovak University of Technology Regular maps with a given automorphism group, and on a given surface AGT 2016 Pilsen 10 / 18

  48. Enumeration results [with Erskine and Hriˇ n´ akov´ a] Jozef ˇ Sir´ aˇ n Open University and Slovak University of Technology Regular maps with a given automorphism group, and on a given surface AGT 2016 Pilsen 11 / 18

  49. Enumeration results [with Erskine and Hriˇ n´ akov´ a] Fact: Enumeration of orientably-regular maps M , Aut + ( M ) ∼ = G , �→ enumeration of triples ( G, r, s ) , G = � r, s ; r ℓ = s m = ( rs ) 2 = . . . = 1 � , up to conjugation in Aut( G ) , that is, by considering triples ( G, r, s ) and ( G, r ′ , s ′ ) equivalent if there is an automorphism of G : ( r, s ) �→ ( r ′ , s ′ ) . Jozef ˇ Sir´ aˇ n Open University and Slovak University of Technology Regular maps with a given automorphism group, and on a given surface AGT 2016 Pilsen 11 / 18

  50. Enumeration results [with Erskine and Hriˇ n´ akov´ a] Fact: Enumeration of orientably-regular maps M , Aut + ( M ) ∼ = G , �→ enumeration of triples ( G, r, s ) , G = � r, s ; r ℓ = s m = ( rs ) 2 = . . . = 1 � , up to conjugation in Aut( G ) , that is, by considering triples ( G, r, s ) and ( G, r ′ , s ′ ) equivalent if there is an automorphism of G : ( r, s ) �→ ( r ′ , s ′ ) . Theorem. Let q = p f , f = 2 n o ; p, o odd. The number of orientably-regular maps M with Aut + ( M ) ∼ = M ( q 2 ) is, up to isomorphism, equal to 1 � µ ( o/d ) h (2 n d ) , f d | o where h ( x ) = ( p 2 x − 1)( p 2 x − 2) / 8 and µ is the M¨ obius function. Jozef ˇ Sir´ aˇ n Open University and Slovak University of Technology Regular maps with a given automorphism group, and on a given surface AGT 2016 Pilsen 11 / 18

  51. Enumeration results [with Erskine and Hriˇ n´ akov´ a] Fact: Enumeration of orientably-regular maps M , Aut + ( M ) ∼ = G , �→ enumeration of triples ( G, r, s ) , G = � r, s ; r ℓ = s m = ( rs ) 2 = . . . = 1 � , up to conjugation in Aut( G ) , that is, by considering triples ( G, r, s ) and ( G, r ′ , s ′ ) equivalent if there is an automorphism of G : ( r, s ) �→ ( r ′ , s ′ ) . Theorem. Let q = p f , f = 2 n o ; p, o odd. The number of orientably-regular maps M with Aut + ( M ) ∼ = M ( q 2 ) is, up to isomorphism, equal to 1 � µ ( o/d ) h (2 n d ) , f d | o where h ( x ) = ( p 2 x − 1)( p 2 x − 2) / 8 and µ is the M¨ obius function. Theorem. The number of reflexible maps M with Aut + ( M ) ∼ = M ( q 2 ) is 1 � µ ( o/d ) k (2 n d ) , f d | o where k ( x ) = ( p 2 x − 1)(3 p x − 2) / 8 and µ is the M¨ obius function. Jozef ˇ Sir´ aˇ n Open University and Slovak University of Technology Regular maps with a given automorphism group, and on a given surface AGT 2016 Pilsen 11 / 18

  52. Remarks Jozef ˇ Sir´ aˇ n Open University and Slovak University of Technology Regular maps with a given automorphism group, and on a given surface AGT 2016 Pilsen 12 / 18

  53. Remarks The results are strikingly different from those for the groups PGL(2 , q ) : Jozef ˇ Sir´ aˇ n Open University and Slovak University of Technology Regular maps with a given automorphism group, and on a given surface AGT 2016 Pilsen 12 / 18

  54. Remarks The results are strikingly different from those for the groups PGL(2 , q ) : all the orientably-regular maps for PGL(2 , q ) are reflexible, while this is not the case for M ( q 2 ) ; Jozef ˇ Sir´ aˇ n Open University and Slovak University of Technology Regular maps with a given automorphism group, and on a given surface AGT 2016 Pilsen 12 / 18

  55. Remarks The results are strikingly different from those for the groups PGL(2 , q ) : all the orientably-regular maps for PGL(2 , q ) are reflexible, while this is not the case for M ( q 2 ) ; groups PGL(2 , q ) are also automorphism groups of non-orientable regular maps, while the groups M ( q 2 ) are not; Jozef ˇ Sir´ aˇ n Open University and Slovak University of Technology Regular maps with a given automorphism group, and on a given surface AGT 2016 Pilsen 12 / 18

  56. Remarks The results are strikingly different from those for the groups PGL(2 , q ) : all the orientably-regular maps for PGL(2 , q ) are reflexible, while this is not the case for M ( q 2 ) ; groups PGL(2 , q ) are also automorphism groups of non-orientable regular maps, while the groups M ( q 2 ) are not; for any even ℓ, m ≥ 4 not both equal to 4 there are orientably-regular maps of type ( ℓ, m ) with automorphism group PGL(2 , q ) for infinitely many values of q , while for ℓ, m ≡ 0 (mod 8) and ℓ �≡ m (mod 16) there is no orientably-regular map of type ( ℓ, m ) on M ( q 2 ) for any q . Jozef ˇ Sir´ aˇ n Open University and Slovak University of Technology Regular maps with a given automorphism group, and on a given surface AGT 2016 Pilsen 12 / 18

  57. Remarks The results are strikingly different from those for the groups PGL(2 , q ) : all the orientably-regular maps for PGL(2 , q ) are reflexible, while this is not the case for M ( q 2 ) ; groups PGL(2 , q ) are also automorphism groups of non-orientable regular maps, while the groups M ( q 2 ) are not; for any even ℓ, m ≥ 4 not both equal to 4 there are orientably-regular maps of type ( ℓ, m ) with automorphism group PGL(2 , q ) for infinitely many values of q , while for ℓ, m ≡ 0 (mod 8) and ℓ �≡ m (mod 16) there is no orientably-regular map of type ( ℓ, m ) on M ( q 2 ) for any q . Frobenius 1896: The number of solutions ( x 1 , x 2 , . . . , x k ) of the equation x 1 x 2 · · · x k = 1 with x i in a conjugacy class C i of a finite group G is |C 1 | · · · |C k | χ ( x 1 ) · · · χ ( x k ) � χ (1) k − 2 | G | χ χ ... irreducible complex characters of G . ( x 1 = r , x 2 = s , x 3 = ( rs ) − 1 ) Jozef ˇ Sir´ aˇ n Open University and Slovak University of Technology Regular maps with a given automorphism group, and on a given surface AGT 2016 Pilsen 12 / 18

  58. Regular maps on a compact surface by 2001 Jozef ˇ Sir´ aˇ n Open University and Slovak University of Technology Regular maps with a given automorphism group, and on a given surface AGT 2016 Pilsen 13 / 18

  59. Regular maps on a compact surface by 2001 Sphere: Jozef ˇ Sir´ aˇ n Open University and Slovak University of Technology Regular maps with a given automorphism group, and on a given surface AGT 2016 Pilsen 13 / 18

  60. Regular maps on a compact surface by 2001 Sphere: Platonic maps (and ∞ of trivial maps) Jozef ˇ Sir´ aˇ n Open University and Slovak University of Technology Regular maps with a given automorphism group, and on a given surface AGT 2016 Pilsen 13 / 18

  61. Regular maps on a compact surface by 2001 Sphere: Platonic maps (and ∞ of trivial maps) Projective plane: Jozef ˇ Sir´ aˇ n Open University and Slovak University of Technology Regular maps with a given automorphism group, and on a given surface AGT 2016 Pilsen 13 / 18

  62. Regular maps on a compact surface by 2001 Sphere: Platonic maps (and ∞ of trivial maps) Projective plane: Petersen, K 4 , duals (and ∞ of trivial maps) Jozef ˇ Sir´ aˇ n Open University and Slovak University of Technology Regular maps with a given automorphism group, and on a given surface AGT 2016 Pilsen 13 / 18

  63. Regular maps on a compact surface by 2001 Sphere: Platonic maps (and ∞ of trivial maps) Projective plane: Petersen, K 4 , duals (and ∞ of trivial maps) Torus: Jozef ˇ Sir´ aˇ n Open University and Slovak University of Technology Regular maps with a given automorphism group, and on a given surface AGT 2016 Pilsen 13 / 18

  64. Regular maps on a compact surface by 2001 Sphere: Platonic maps (and ∞ of trivial maps) Projective plane: Petersen, K 4 , duals (and ∞ of trivial maps) Torus: Infinitely many non trivial regular maps Jozef ˇ Sir´ aˇ n Open University and Slovak University of Technology Regular maps with a given automorphism group, and on a given surface AGT 2016 Pilsen 13 / 18

  65. Regular maps on a compact surface by 2001 Sphere: Platonic maps (and ∞ of trivial maps) Projective plane: Petersen, K 4 , duals (and ∞ of trivial maps) Torus: Infinitely many non trivial regular maps Klein bottle: Jozef ˇ Sir´ aˇ n Open University and Slovak University of Technology Regular maps with a given automorphism group, and on a given surface AGT 2016 Pilsen 13 / 18

  66. Regular maps on a compact surface by 2001 Sphere: Platonic maps (and ∞ of trivial maps) Projective plane: Petersen, K 4 , duals (and ∞ of trivial maps) Torus: Infinitely many non trivial regular maps Klein bottle: No regular map at all! Jozef ˇ Sir´ aˇ n Open University and Slovak University of Technology Regular maps with a given automorphism group, and on a given surface AGT 2016 Pilsen 13 / 18

  67. Regular maps on a compact surface by 2001 Sphere: Platonic maps (and ∞ of trivial maps) Projective plane: Petersen, K 4 , duals (and ∞ of trivial maps) Torus: Infinitely many non trivial regular maps Klein bottle: No regular map at all! If G = � x, y, z | x 2 = y 2 = z 2 = ( yz ) ℓ = ( zx ) m = ( xy ) 2 = . . . = 1 � gives a regular map of type ( ℓ, m ) on a compact surface with Euler char. χ , then 4 ℓm | G | = ℓm − 2 ℓ − 2 m ( − χ ) Jozef ˇ Sir´ aˇ n Open University and Slovak University of Technology Regular maps with a given automorphism group, and on a given surface AGT 2016 Pilsen 13 / 18

  68. Regular maps on a compact surface by 2001 Sphere: Platonic maps (and ∞ of trivial maps) Projective plane: Petersen, K 4 , duals (and ∞ of trivial maps) Torus: Infinitely many non trivial regular maps Klein bottle: No regular map at all! If G = � x, y, z | x 2 = y 2 = z 2 = ( yz ) ℓ = ( zx ) m = ( xy ) 2 = . . . = 1 � gives a regular map of type ( ℓ, m ) on a compact surface with Euler char. χ , then 4 ℓm | G | = ℓm − 2 ℓ − 2 m ( − χ ) Every surface with χ < 0 supports just a finite number of regular maps. Jozef ˇ Sir´ aˇ n Open University and Slovak University of Technology Regular maps with a given automorphism group, and on a given surface AGT 2016 Pilsen 13 / 18

  69. Regular maps on a compact surface by 2001 Sphere: Platonic maps (and ∞ of trivial maps) Projective plane: Petersen, K 4 , duals (and ∞ of trivial maps) Torus: Infinitely many non trivial regular maps Klein bottle: No regular map at all! If G = � x, y, z | x 2 = y 2 = z 2 = ( yz ) ℓ = ( zx ) m = ( xy ) 2 = . . . = 1 � gives a regular map of type ( ℓ, m ) on a compact surface with Euler char. χ , then 4 ℓm | G | = ℓm − 2 ℓ − 2 m ( − χ ) Every surface with χ < 0 supports just a finite number of regular maps. State-of-the-art in the classification of regular maps by 2001: Jozef ˇ Sir´ aˇ n Open University and Slovak University of Technology Regular maps with a given automorphism group, and on a given surface AGT 2016 Pilsen 13 / 18

  70. Regular maps on a compact surface by 2001 Sphere: Platonic maps (and ∞ of trivial maps) Projective plane: Petersen, K 4 , duals (and ∞ of trivial maps) Torus: Infinitely many non trivial regular maps Klein bottle: No regular map at all! If G = � x, y, z | x 2 = y 2 = z 2 = ( yz ) ℓ = ( zx ) m = ( xy ) 2 = . . . = 1 � gives a regular map of type ( ℓ, m ) on a compact surface with Euler char. χ , then 4 ℓm | G | = ℓm − 2 ℓ − 2 m ( − χ ) Every surface with χ < 0 supports just a finite number of regular maps. State-of-the-art in the classification of regular maps by 2001: By hand for χ ≥ − 8 [numerous authors] Jozef ˇ Sir´ aˇ n Open University and Slovak University of Technology Regular maps with a given automorphism group, and on a given surface AGT 2016 Pilsen 13 / 18

  71. Regular maps on a compact surface by 2001 Sphere: Platonic maps (and ∞ of trivial maps) Projective plane: Petersen, K 4 , duals (and ∞ of trivial maps) Torus: Infinitely many non trivial regular maps Klein bottle: No regular map at all! If G = � x, y, z | x 2 = y 2 = z 2 = ( yz ) ℓ = ( zx ) m = ( xy ) 2 = . . . = 1 � gives a regular map of type ( ℓ, m ) on a compact surface with Euler char. χ , then 4 ℓm | G | = ℓm − 2 ℓ − 2 m ( − χ ) Every surface with χ < 0 supports just a finite number of regular maps. State-of-the-art in the classification of regular maps by 2001: By hand for χ ≥ − 8 [numerous authors] A computer-assisted classification for χ ≥ − 28 [Conder, Dobcs´ anyi 2001] Jozef ˇ Sir´ aˇ n Open University and Slovak University of Technology Regular maps with a given automorphism group, and on a given surface AGT 2016 Pilsen 13 / 18

  72. Classification of regular maps for infinitely many surfaces Jozef ˇ Sir´ aˇ n Open University and Slovak University of Technology Regular maps with a given automorphism group, and on a given surface AGT 2016 Pilsen 14 / 18

  73. Classification of regular maps for infinitely many surfaces • χ = − p for every prime p [Breda, Nedela,ˇ S 2005] Jozef ˇ Sir´ aˇ n Open University and Slovak University of Technology Regular maps with a given automorphism group, and on a given surface AGT 2016 Pilsen 14 / 18

  74. Classification of regular maps for infinitely many surfaces • χ = − p for every prime p [Breda, Nedela,ˇ S 2005] • χ = − 2 p and orientable + ‘large’ [Belolipetsky, Jones 2005] Jozef ˇ Sir´ aˇ n Open University and Slovak University of Technology Regular maps with a given automorphism group, and on a given surface AGT 2016 Pilsen 14 / 18

  75. Classification of regular maps for infinitely many surfaces • χ = − p for every prime p [Breda, Nedela,ˇ S 2005] • χ = − 2 p and orientable + ‘large’ [Belolipetsky, Jones 2005] • χ = − 2 p and orientable, all [Conder, Tucker, ˇ S 2010] Jozef ˇ Sir´ aˇ n Open University and Slovak University of Technology Regular maps with a given automorphism group, and on a given surface AGT 2016 Pilsen 14 / 18

  76. Classification of regular maps for infinitely many surfaces • χ = − p for every prime p [Breda, Nedela,ˇ S 2005] • χ = − 2 p and orientable + ‘large’ [Belolipetsky, Jones 2005] • χ = − 2 p and orientable, all [Conder, Tucker, ˇ S 2010] • χ = − p 2 [Conder, Potoˇ cnik, ˇ S 2010] Jozef ˇ Sir´ aˇ n Open University and Slovak University of Technology Regular maps with a given automorphism group, and on a given surface AGT 2016 Pilsen 14 / 18

  77. Classification of regular maps for infinitely many surfaces • χ = − p for every prime p [Breda, Nedela,ˇ S 2005] • χ = − 2 p and orientable + ‘large’ [Belolipetsky, Jones 2005] • χ = − 2 p and orientable, all [Conder, Tucker, ˇ S 2010] • χ = − p 2 [Conder, Potoˇ cnik, ˇ S 2010] • χ = − 3 p [Conder, Nedela, ˇ S 2012] Jozef ˇ Sir´ aˇ n Open University and Slovak University of Technology Regular maps with a given automorphism group, and on a given surface AGT 2016 Pilsen 14 / 18

  78. Classification of regular maps for infinitely many surfaces • χ = − p for every prime p [Breda, Nedela,ˇ S 2005] • χ = − 2 p and orientable + ‘large’ [Belolipetsky, Jones 2005] • χ = − 2 p and orientable, all [Conder, Tucker, ˇ S 2010] • χ = − p 2 [Conder, Potoˇ cnik, ˇ S 2010] • χ = − 3 p [Conder, Nedela, ˇ S 2012] Classification for some families of orientably regular maps with χ = 2 − 2 g : Jozef ˇ Sir´ aˇ n Open University and Slovak University of Technology Regular maps with a given automorphism group, and on a given surface AGT 2016 Pilsen 14 / 18

  79. Classification of regular maps for infinitely many surfaces • χ = − p for every prime p [Breda, Nedela,ˇ S 2005] • χ = − 2 p and orientable + ‘large’ [Belolipetsky, Jones 2005] • χ = − 2 p and orientable, all [Conder, Tucker, ˇ S 2010] • χ = − p 2 [Conder, Potoˇ cnik, ˇ S 2010] • χ = − 3 p [Conder, Nedela, ˇ S 2012] Classification for some families of orientably regular maps with χ = 2 − 2 g : • Or.-reg. M with ( g − 1 , | Aut + ( M ) | ) = 1 [Conder, Tucker, ˇ S 2010] Jozef ˇ Sir´ aˇ n Open University and Slovak University of Technology Regular maps with a given automorphism group, and on a given surface AGT 2016 Pilsen 14 / 18

  80. Classification of regular maps for infinitely many surfaces • χ = − p for every prime p [Breda, Nedela,ˇ S 2005] • χ = − 2 p and orientable + ‘large’ [Belolipetsky, Jones 2005] • χ = − 2 p and orientable, all [Conder, Tucker, ˇ S 2010] • χ = − p 2 [Conder, Potoˇ cnik, ˇ S 2010] • χ = − 3 p [Conder, Nedela, ˇ S 2012] Classification for some families of orientably regular maps with χ = 2 − 2 g : • Or.-reg. M with ( g − 1 , | Aut + ( M ) | ) = 1 [Conder, Tucker, ˇ S 2010] • Or.-reg. M with g − 1 = p | | Aut + ( M ) | [Conder, Tucker, ˇ S 2010] Jozef ˇ Sir´ aˇ n Open University and Slovak University of Technology Regular maps with a given automorphism group, and on a given surface AGT 2016 Pilsen 14 / 18

  81. Classification of regular maps for infinitely many surfaces • χ = − p for every prime p [Breda, Nedela,ˇ S 2005] • χ = − 2 p and orientable + ‘large’ [Belolipetsky, Jones 2005] • χ = − 2 p and orientable, all [Conder, Tucker, ˇ S 2010] • χ = − p 2 [Conder, Potoˇ cnik, ˇ S 2010] • χ = − 3 p [Conder, Nedela, ˇ S 2012] Classification for some families of orientably regular maps with χ = 2 − 2 g : • Or.-reg. M with ( g − 1 , | Aut + ( M ) | ) = 1 [Conder, Tucker, ˇ S 2010] • Or.-reg. M with g − 1 = p | | Aut + ( M ) | [Conder, Tucker, ˇ S 2010] Classification for ‘small’ genera carried over to χ ≥ − 600 with the help of more powerful computational methods [Conder 2013]; orientably-regular maps with ≤ 3 , 000 edges and non-orientable regular maps with at most 1 , 500 edges done by Potoˇ cnik, Spiga and Verret 2015]. Jozef ˇ Sir´ aˇ n Open University and Slovak University of Technology Regular maps with a given automorphism group, and on a given surface AGT 2016 Pilsen 14 / 18

  82. Gaps in the nonorientable genus spectrum Jozef ˇ Sir´ aˇ n Open University and Slovak University of Technology Regular maps with a given automorphism group, and on a given surface AGT 2016 Pilsen 15 / 18

  83. Gaps in the nonorientable genus spectrum Well known: For every g > 0 there exists a regular map on an orientable surface of genus g ; for instance, of type (4 g, 4 g ) . Jozef ˇ Sir´ aˇ n Open University and Slovak University of Technology Regular maps with a given automorphism group, and on a given surface AGT 2016 Pilsen 15 / 18

  84. Gaps in the nonorientable genus spectrum Well known: For every g > 0 there exists a regular map on an orientable surface of genus g ; for instance, of type (4 g, 4 g ) . A gap is a value of χ for which a nonorientable surface of Euler characteristic χ carries no regular map at all. Jozef ˇ Sir´ aˇ n Open University and Slovak University of Technology Regular maps with a given automorphism group, and on a given surface AGT 2016 Pilsen 15 / 18

  85. Gaps in the nonorientable genus spectrum Well known: For every g > 0 there exists a regular map on an orientable surface of genus g ; for instance, of type (4 g, 4 g ) . A gap is a value of χ for which a nonorientable surface of Euler characteristic χ carries no regular map at all. Known infinite families of gaps: Jozef ˇ Sir´ aˇ n Open University and Slovak University of Technology Regular maps with a given automorphism group, and on a given surface AGT 2016 Pilsen 15 / 18

  86. Gaps in the nonorientable genus spectrum Well known: For every g > 0 there exists a regular map on an orientable surface of genus g ; for instance, of type (4 g, 4 g ) . A gap is a value of χ for which a nonorientable surface of Euler characteristic χ carries no regular map at all. Known infinite families of gaps: • χ = − p for primes p ≡ 1 mod 12 , p � = 13 [Breda, Nedela, ˇ S 2005] Jozef ˇ Sir´ aˇ n Open University and Slovak University of Technology Regular maps with a given automorphism group, and on a given surface AGT 2016 Pilsen 15 / 18

  87. Gaps in the nonorientable genus spectrum Well known: For every g > 0 there exists a regular map on an orientable surface of genus g ; for instance, of type (4 g, 4 g ) . A gap is a value of χ for which a nonorientable surface of Euler characteristic χ carries no regular map at all. Known infinite families of gaps: • χ = − p for primes p ≡ 1 mod 12 , p � = 13 [Breda, Nedela, ˇ S 2005] • χ = − p 2 for all primes p > 7 [Conder, Potoˇ cnik, ˇ S 2010] Jozef ˇ Sir´ aˇ n Open University and Slovak University of Technology Regular maps with a given automorphism group, and on a given surface AGT 2016 Pilsen 15 / 18

  88. Gaps in the nonorientable genus spectrum Well known: For every g > 0 there exists a regular map on an orientable surface of genus g ; for instance, of type (4 g, 4 g ) . A gap is a value of χ for which a nonorientable surface of Euler characteristic χ carries no regular map at all. Known infinite families of gaps: • χ = − p for primes p ≡ 1 mod 12 , p � = 13 [Breda, Nedela, ˇ S 2005] • χ = − p 2 for all primes p > 7 [Conder, Potoˇ cnik, ˇ S 2010] • χ = − 3 p for all p > 11 such that p ≡ 3 mod 4 and p �≡ 55 mod 84 [Conder, Nedela, ˇ S 2012] Jozef ˇ Sir´ aˇ n Open University and Slovak University of Technology Regular maps with a given automorphism group, and on a given surface AGT 2016 Pilsen 15 / 18

  89. Gaps in the nonorientable genus spectrum Well known: For every g > 0 there exists a regular map on an orientable surface of genus g ; for instance, of type (4 g, 4 g ) . A gap is a value of χ for which a nonorientable surface of Euler characteristic χ carries no regular map at all. Known infinite families of gaps: • χ = − p for primes p ≡ 1 mod 12 , p � = 13 [Breda, Nedela, ˇ S 2005] • χ = − p 2 for all primes p > 7 [Conder, Potoˇ cnik, ˇ S 2010] • χ = − 3 p for all p > 11 such that p ≡ 3 mod 4 and p �≡ 55 mod 84 [Conder, Nedela, ˇ S 2012] More than 3 / 4 of values of χ are non-gaps [Conder, Everitt 1995]. Jozef ˇ Sir´ aˇ n Open University and Slovak University of Technology Regular maps with a given automorphism group, and on a given surface AGT 2016 Pilsen 15 / 18

  90. Regular maps with odd χ [with Conder, Gill and Short] Jozef ˇ Sir´ aˇ n Open University and Slovak University of Technology Regular maps with a given automorphism group, and on a given surface AGT 2016 Pilsen 16 / 18

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