Regular maps with a given automorphism group, and on a given - - PowerPoint PPT Presentation

▶
regular maps with a given automorphism group and on a
SMART_READER_LITE
LIVE PREVIEW

Regular maps with a given automorphism group, and on a given - - PowerPoint PPT Presentation

Jan 03, 2023 102 likes •1.29k views

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

slide-1
SLIDE 1

Regular maps with a given automorphism group, and on a given surface

Jozef ˇ Sir´ aˇ n Open University and Slovak University of Technology AGT 2016 Pilsen

Jozef ˇ Sir´ aˇ n Open University and Slovak University of Technology AGT 2016 Pilsen Regular maps with a given automorphism group, and on a given surface 1 / 18

slide-2
SLIDE 2

Regular and orientably-regular maps – informally

Jozef ˇ Sir´ aˇ n Open University and Slovak University of Technology AGT 2016 Pilsen Regular maps with a given automorphism group, and on a given surface 2 / 18

slide-3
SLIDE 3

Regular and orientably-regular maps – informally

Jozef ˇ Sir´ aˇ n Open University and Slovak University of Technology AGT 2016 Pilsen Regular maps with a given automorphism group, and on a given surface 2 / 18

slide-4
SLIDE 4

Regular and orientably-regular maps – informally

Here, Aut+(M) and Aut(M) act regularly on arcs and flags, respectively.

Jozef ˇ Sir´ aˇ n Open University and Slovak University of Technology AGT 2016 Pilsen Regular maps with a given automorphism group, and on a given surface 2 / 18

slide-5
SLIDE 5

Regular and orientably-regular maps – informally

Here, Aut+(M) and Aut(M) act regularly on arcs and flags, respectively. Such maps (cellular embeddings of connected graphs) on arbitrary surfaces are called orientably-regular and regular (generalising the Platonic maps).

Jozef ˇ Sir´ aˇ n Open University and Slovak University of Technology AGT 2016 Pilsen Regular maps with a given automorphism group, and on a given surface 2 / 18

slide-6
SLIDE 6

Orientably-regular maps

Jozef ˇ Sir´ aˇ n Open University and Slovak University of Technology AGT 2016 Pilsen Regular maps with a given automorphism group, and on a given surface 3 / 18

slide-7
SLIDE 7

Orientably-regular maps

An orientable map M is orientably-regular if Aut+(M) is a transitive (and hence regular) permutation group on the arc set of M.

Jozef ˇ Sir´ aˇ n Open University and Slovak University of Technology AGT 2016 Pilsen Regular maps with a given automorphism group, and on a given surface 3 / 18

slide-8
SLIDE 8

Orientably-regular maps

An orientable map M is orientably-regular if Aut+(M) is a transitive (and hence regular) permutation group on the arc set of M. If r and s are rotations of M about the centre of a face and about an incident vertex, then G = Aut+(M) has a presentation of the form G = r, s | rℓ = sm = (rs)2 = . . . = 1 .

Jozef ˇ Sir´ aˇ n Open University and Slovak University of Technology AGT 2016 Pilsen Regular maps with a given automorphism group, and on a given surface 3 / 18

slide-9
SLIDE 9

Orientably-regular maps

An orientable map M is orientably-regular if Aut+(M) is a transitive (and hence regular) permutation group on the arc set of M. If r and s are rotations of M about the centre of a face and about an incident vertex, then G = Aut+(M) has a presentation of the form G = r, s | rℓ = sm = (rs)2 = . . . = 1 . The group G is then a quotient of the triangle group Tℓ,m = R, S | Rℓ = Sm = (RS)2 = 1 , i.e., G = Tℓ,m/K for a torsion-free K ⊳ Tℓ,m; equivalently, M = Uℓ,m/K, where Uℓ,m is an (ℓ, m)-tessellation of a simply connected surface.

Jozef ˇ Sir´ aˇ n Open University and Slovak University of Technology AGT 2016 Pilsen Regular maps with a given automorphism group, and on a given surface 3 / 18

slide-10
SLIDE 10

Orientably-regular maps

An orientable map M is orientably-regular if Aut+(M) is a transitive (and hence regular) permutation group on the arc set of M. If r and s are rotations of M about the centre of a face and about an incident vertex, then G = Aut+(M) has a presentation of the form G = r, s | rℓ = sm = (rs)2 = . . . = 1 . The group G is then a quotient of the triangle group Tℓ,m = R, S | Rℓ = Sm = (RS)2 = 1 , i.e., G = Tℓ,m/K for a torsion-free K ⊳ Tℓ,m; equivalently, M = Uℓ,m/K, where Uℓ,m is an (ℓ, m)-tessellation of a simply connected surface. Conversely, given any epimorphism from Tℓ,m onto a finite group G with torsion-free kernel, the corresponding orientably-regular map of type (ℓ, m) can be constructed using (right) cosets of the images of R, S and RS as faces, vertices and edges. (Works with cosets of r, s, rs.)

Jozef ˇ Sir´ aˇ n Open University and Slovak University of Technology AGT 2016 Pilsen Regular maps with a given automorphism group, and on a given surface 3 / 18

slide-11
SLIDE 11

Example of an orientably-regular map: K5 on a torus

  • Jozef ˇ

Sir´ aˇ n Open University and Slovak University of Technology AGT 2016 Pilsen Regular maps with a given automorphism group, and on a given surface 4 / 18

slide-12
SLIDE 12

Example of an orientably-regular map: K5 on a torus

  • Presentation: Aut+(M) = r, s | r4 = s4 = (rs)2 = r2s2rs−1 = 1

Jozef ˇ Sir´ aˇ n Open University and Slovak University of Technology AGT 2016 Pilsen Regular maps with a given automorphism group, and on a given surface 4 / 18

slide-13
SLIDE 13

Example of an orientably-regular map: K5 on a torus

  • Presentation: Aut+(M) = r, s | r4 = s4 = (rs)2 = r2s2rs−1 = 1
  • This map is chiral (no reflection).

Jozef ˇ Sir´ aˇ n Open University and Slovak University of Technology AGT 2016 Pilsen Regular maps with a given automorphism group, and on a given surface 4 / 18

slide-14
SLIDE 14

Example of an orientably-regular map: K5 on a torus

  • Presentation: Aut+(M) = r, s | r4 = s4 = (rs)2 = r2s2rs−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 AGT 2016 Pilsen Regular maps with a given automorphism group, and on a given surface 4 / 18

slide-15
SLIDE 15

Orientably-regular maps and exciting mathematics

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

slide-16
SLIDE 16

Orientably-regular maps and exciting mathematics

Up to isomorphism, 1-1 correspondence between:

  • rientably-regular maps of type (ℓ, m);

group presentations r, s | rℓ = sm = (rs)2 = . . . = 1; torsion-free normal subgroups of triangle groups Tℓ,m.

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

slide-17
SLIDE 17

Orientably-regular maps and exciting mathematics

Up to isomorphism, 1-1 correspondence between:

  • rientably-regular maps of type (ℓ, m);

group presentations r, s | rℓ = sm = (rs)2 = . . . = 1; torsion-free normal subgroups of triangle groups Tℓ,m. Maps, Riemann surfaces, and Galois theory:

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

slide-18
SLIDE 18

Orientably-regular maps and exciting mathematics

Up to isomorphism, 1-1 correspondence between:

  • rientably-regular maps of type (ℓ, m);

group presentations r, s | rℓ = sm = (rs)2 = . . . = 1; torsion-free normal subgroups of triangle groups Tℓ,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 AGT 2016 Pilsen Regular maps with a given automorphism group, and on a given surface 5 / 18

slide-19
SLIDE 19

Orientably-regular maps and exciting mathematics

Up to isomorphism, 1-1 correspondence between:

  • rientably-regular maps of type (ℓ, m);

group presentations r, s | rℓ = sm = (rs)2 = . . . = 1; torsion-free normal subgroups of triangle groups Tℓ,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 AGT 2016 Pilsen Regular maps with a given automorphism group, and on a given surface 5 / 18

slide-20
SLIDE 20

Orientably-regular maps and exciting mathematics

Up to isomorphism, 1-1 correspondence between:

  • rientably-regular maps of type (ℓ, m);

group presentations r, s | rℓ = sm = (rs)2 = . . . = 1; torsion-free normal subgroups of triangle groups Tℓ,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 AGT 2016 Pilsen Regular maps with a given automorphism group, and on a given surface 5 / 18

slide-21
SLIDE 21

Orientably-regular maps and exciting mathematics

Up to isomorphism, 1-1 correspondence between:

  • rientably-regular maps of type (ℓ, m);

group presentations r, s | rℓ = sm = (rs)2 = . . . = 1; torsion-free normal subgroups of triangle groups Tℓ,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 = Uℓ,m/K for some finite-index subgroup K of some Tℓ,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 AGT 2016 Pilsen Regular maps with a given automorphism group, and on a given surface 5 / 18

slide-22
SLIDE 22

Orientably-regular maps and exciting mathematics

Up to isomorphism, 1-1 correspondence between:

  • rientably-regular maps of type (ℓ, m);

group presentations r, s | rℓ = sm = (rs)2 = . . . = 1; torsion-free normal subgroups of triangle groups Tℓ,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 = Uℓ,m/K for some finite-index subgroup K of some Tℓ,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 AGT 2016 Pilsen Regular maps with a given automorphism group, and on a given surface 5 / 18

slide-23
SLIDE 23

Orientably-regular maps and exciting mathematics

Up to isomorphism, 1-1 correspondence between:

  • rientably-regular maps of type (ℓ, m);

group presentations r, s | rℓ = sm = (rs)2 = . . . = 1; torsion-free normal subgroups of triangle groups Tℓ,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 = Uℓ,m/K for some finite-index subgroup K of some Tℓ,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 AGT 2016 Pilsen Regular maps with a given automorphism group, and on a given surface 5 / 18

slide-24
SLIDE 24

Regular maps

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

slide-25
SLIDE 25

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 AGT 2016 Pilsen Regular maps with a given automorphism group, and on a given surface 6 / 18

slide-26
SLIDE 26

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 AGT 2016 Pilsen Regular maps with a given automorphism group, and on a given surface 6 / 18

slide-27
SLIDE 27

Regular maps

A map is regular if its automorphism group is regular on flags. Aut(M) = x, y, z| x2 = y2 = z2 = (yz)ℓ = (zx)m = (xy)2 = . . . = 1

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

slide-28
SLIDE 28

Regular maps

A map is regular if its automorphism group is regular on flags. Aut(M) = x, y, z| x2 = y2 = z2 = (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 AGT 2016 Pilsen Regular maps with a given automorphism group, and on a given surface 6 / 18

slide-29
SLIDE 29

Classification of (orientably-) regular maps

Jozef ˇ Sir´ aˇ n Open University and Slovak University of Technology AGT 2016 Pilsen Regular maps with a given automorphism group, and on a given surface 7 / 18

slide-30
SLIDE 30

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 AGT 2016 Pilsen Regular maps with a given automorphism group, and on a given surface 7 / 18

slide-31
SLIDE 31

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 AGT 2016 Pilsen Regular maps with a given automorphism group, and on a given surface 7 / 18

slide-32
SLIDE 32

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 AGT 2016 Pilsen Regular maps with a given automorphism group, and on a given surface 7 / 18

slide-33
SLIDE 33

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 AGT 2016 Pilsen Regular maps with a given automorphism group, and on a given surface 7 / 18

slide-34
SLIDE 34

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 AGT 2016 Pilsen Regular maps with a given automorphism group, and on a given surface 7 / 18

slide-35
SLIDE 35

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 AGT 2016 Pilsen Regular maps with a given automorphism group, and on a given surface 7 / 18

slide-36
SLIDE 36

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 AGT 2016 Pilsen Regular maps with a given automorphism group, and on a given surface 7 / 18

slide-37
SLIDE 37

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 AGT 2016 Pilsen Regular maps with a given automorphism group, and on a given surface 7 / 18

slide-38
SLIDE 38

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 AGT 2016 Pilsen Regular maps with a given automorphism group, and on a given surface 7 / 18

slide-39
SLIDE 39

Classification by underlying graphs:

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

slide-40
SLIDE 40

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 AGT 2016 Pilsen Regular maps with a given automorphism group, and on a given surface 8 / 18

slide-41
SLIDE 41

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

  • rientable case; [Wilson 1989] in the non-orientable case

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

slide-42
SLIDE 42

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

  • rientable case; [Wilson 1989] in the non-orientable case

complete bipartite graphs – recent major work of [Jones 2012] in the

  • rientable 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 AGT 2016 Pilsen Regular maps with a given automorphism group, and on a given surface 8 / 18

slide-43
SLIDE 43

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

  • rientable case; [Wilson 1989] in the non-orientable case

complete bipartite graphs – recent major work of [Jones 2012] in the

  • rientable case, with special cases settled earlier by multiple authors;

the non-orientable case [Kwak and Kwon 2011] Kp,p,...,p in the orientable case [Du, Kwak and Nedela 2005]

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

slide-44
SLIDE 44

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

  • rientable case; [Wilson 1989] in the non-orientable case

complete bipartite graphs – recent major work of [Jones 2012] in the

  • rientable case, with special cases settled earlier by multiple authors;

the non-orientable case [Kwak and Kwon 2011] Kp,p,...,p in the orientable case [Du, Kwak and Nedela 2005] Qn [Breda, Catalano, Conder, Kwak, Kwon, Nedela, Wilson 2012]

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

slide-45
SLIDE 45

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

  • rientable case; [Wilson 1989] in the non-orientable case

complete bipartite graphs – recent major work of [Jones 2012] in the

  • rientable case, with special cases settled earlier by multiple authors;

the non-orientable case [Kwak and Kwon 2011] Kp,p,...,p in the orientable case [Du, Kwak and Nedela 2005] Qn [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 AGT 2016 Pilsen Regular maps with a given automorphism group, and on a given surface 8 / 18

slide-46
SLIDE 46

Classification of regular maps by automorphism groups

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

slide-47
SLIDE 47

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

  • f a single such map [Du, Conder, Malniˇ

c, Nedela, ˇ Skoviera, ...]

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

slide-48
SLIDE 48

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

  • f a single such map [Du, Conder, Malniˇ

c, Nedela, ˇ Skoviera, ...] regular maps with almost-Sylow-cyclic automorphism groups (every odd-order Sylow subgroup is cyclic and the even-order one has a cyclic subgroup of index 2) [Conder, Potoˇ cnik and ˇ S 2010] – in the solvable case independent of [Zassenhaus 1936]

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

slide-49
SLIDE 49

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

  • f a single such map [Du, Conder, Malniˇ

c, Nedela, ˇ Skoviera, ...] regular maps with almost-Sylow-cyclic automorphism groups (every odd-order Sylow subgroup is cyclic and the even-order one has a cyclic subgroup of index 2) [Conder, Potoˇ cnik and ˇ S 2010] – in the solvable case independent of [Zassenhaus 1936]

  • rientably 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 AGT 2016 Pilsen Regular maps with a given automorphism group, and on a given surface 9 / 18

slide-50
SLIDE 50

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

  • f a single such map [Du, Conder, Malniˇ

c, Nedela, ˇ Skoviera, ...] regular maps with almost-Sylow-cyclic automorphism groups (every odd-order Sylow subgroup is cyclic and the even-order one has a cyclic subgroup of index 2) [Conder, Potoˇ cnik and ˇ S 2010] – in the solvable case independent of [Zassenhaus 1936]

  • rientably 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 PSL(2, q) and PGL(2, q) [Conder, Potoˇ cnik and ˇ S 2008]

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

slide-51
SLIDE 51

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

  • f a single such map [Du, Conder, Malniˇ

c, Nedela, ˇ Skoviera, ...] regular maps with almost-Sylow-cyclic automorphism groups (every odd-order Sylow subgroup is cyclic and the even-order one has a cyclic subgroup of index 2) [Conder, Potoˇ cnik and ˇ S 2010] – in the solvable case independent of [Zassenhaus 1936]

  • rientably 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 PSL(2, q) and PGL(2, q) [Conder, Potoˇ cnik and ˇ 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 AGT 2016 Pilsen Regular maps with a given automorphism group, and on a given surface 9 / 18

slide-52
SLIDE 52

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

  • f a single such map [Du, Conder, Malniˇ

c, Nedela, ˇ Skoviera, ...] regular maps with almost-Sylow-cyclic automorphism groups (every odd-order Sylow subgroup is cyclic and the even-order one has a cyclic subgroup of index 2) [Conder, Potoˇ cnik and ˇ S 2010] – in the solvable case independent of [Zassenhaus 1936]

  • rientably 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 PSL(2, q) and PGL(2, q) [Conder, Potoˇ cnik and ˇ 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 AGT 2016 Pilsen Regular maps with a given automorphism group, and on a given surface 9 / 18

slide-53
SLIDE 53

Twisted linear fractional groups

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

slide-54
SLIDE 54

Twisted linear fractional groups

F – a field, SF and NF – 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 cz + d if ad − bc ∈ SF ,

  • resp. ad − bc ∈ SF ∪ NF .

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

slide-55
SLIDE 55

Twisted linear fractional groups

F – a field, SF and NF – 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 cz + d if ad − bc ∈ SF ,

  • resp. ad − bc ∈ SF ∪ NF .

If F = GF(q2), q = pf, p odd, and if σ : x → xq is the automorphism of F of order 2, the twisted linear fractional group M(q2) consists of (untwisted and twisted) permutations of F ∪ {∞} defined by z → az + b cz + d if ad−bc ∈ SF and z → azσ + b czσ + d if ad−bc ∈ NF .

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

slide-56
SLIDE 56

Twisted linear fractional groups

F – a field, SF and NF – 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 cz + d if ad − bc ∈ SF ,

  • resp. ad − bc ∈ SF ∪ NF .

If F = GF(q2), q = pf, p odd, and if σ : x → xq is the automorphism of F of order 2, the twisted linear fractional group M(q2) consists of (untwisted and twisted) permutations of F ∪ {∞} defined by z → az + b cz + d if ad−bc ∈ SF and z → azσ + b czσ + d if ad−bc ∈ NF . Zassenhaus (1936).

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

slide-57
SLIDE 57

Twisted linear fractional groups

F – a field, SF and NF – 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 cz + d if ad − bc ∈ SF ,

  • resp. ad − bc ∈ SF ∪ NF .

If F = GF(q2), q = pf, p odd, and if σ : x → xq is the automorphism of F of order 2, the twisted linear fractional group M(q2) consists of (untwisted and twisted) permutations of F ∪ {∞} defined by z → az + b cz + d if ad−bc ∈ SF and z → azσ + b czσ + d if ad−bc ∈ NF . Zassenhaus (1936). We worked out a lot of facts about conjugacy classes and canonical representatives of elements of M(q2) >2 PSL(2, q2) – 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 AGT 2016 Pilsen Regular maps with a given automorphism group, and on a given surface 10 / 18

slide-58
SLIDE 58

Enumeration results [with Erskine and Hriˇ n´ akov´ a]

Jozef ˇ Sir´ aˇ n Open University and Slovak University of Technology AGT 2016 Pilsen Regular maps with a given automorphism group, and on a given surface 11 / 18

slide-59
SLIDE 59

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; rℓ = sm = (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 AGT 2016 Pilsen Regular maps with a given automorphism group, and on a given surface 11 / 18

slide-60
SLIDE 60

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; rℓ = sm = (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 = pf, f = 2no; p, o odd. The number of orientably-regular

maps M with Aut+(M) ∼ = M(q2) is, up to isomorphism, equal to 1 f

  • d|o

µ(o/d)h(2nd) , where h(x) = (p2x − 1)(p2x − 2)/8 and µ is the M¨

  • bius function.

Jozef ˇ Sir´ aˇ n Open University and Slovak University of Technology AGT 2016 Pilsen Regular maps with a given automorphism group, and on a given surface 11 / 18

slide-61
SLIDE 61

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; rℓ = sm = (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 = pf, f = 2no; p, o odd. The number of orientably-regular

maps M with Aut+(M) ∼ = M(q2) is, up to isomorphism, equal to 1 f

  • d|o

µ(o/d)h(2nd) , where h(x) = (p2x − 1)(p2x − 2)/8 and µ is the M¨

  • bius function.
  • Theorem. The number of reflexible maps M with Aut+(M) ∼

= M(q2) is 1 f

  • d|o

µ(o/d)k(2nd) , where k(x) = (p2x − 1)(3px − 2)/8 and µ is the M¨

  • bius function.

Jozef ˇ Sir´ aˇ n Open University and Slovak University of Technology AGT 2016 Pilsen Regular maps with a given automorphism group, and on a given surface 11 / 18

slide-62
SLIDE 62

Remarks

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

slide-63
SLIDE 63

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 AGT 2016 Pilsen Regular maps with a given automorphism group, and on a given surface 12 / 18

slide-64
SLIDE 64

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(q2);

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

slide-65
SLIDE 65

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(q2); groups PGL(2, q) are also automorphism groups of non-orientable regular maps, while the groups M(q2) are not;

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

slide-66
SLIDE 66

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(q2); groups PGL(2, q) are also automorphism groups of non-orientable regular maps, while the groups M(q2) 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(q2) for any q.

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

slide-67
SLIDE 67

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(q2); groups PGL(2, q) are also automorphism groups of non-orientable regular maps, while the groups M(q2) 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(q2) for any q. Frobenius 1896: The number of solutions (x1, x2, . . . , xk) of the equation x1x2 · · · xk = 1 with xi in a conjugacy class Ci of a finite group G is |C1| · · · |Ck| |G|

  • χ

χ(x1) · · · χ(xk) χ(1)k−2 χ ... irreducible complex characters of G. (x1 = r, x2 = s, x3 = (rs)−1)

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

slide-68
SLIDE 68

Regular maps on a compact surface by 2001

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

slide-69
SLIDE 69

Regular maps on a compact surface by 2001

Sphere:

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

slide-70
SLIDE 70

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 AGT 2016 Pilsen Regular maps with a given automorphism group, and on a given surface 13 / 18

slide-71
SLIDE 71

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 AGT 2016 Pilsen Regular maps with a given automorphism group, and on a given surface 13 / 18

slide-72
SLIDE 72

Regular maps on a compact surface by 2001

Sphere: Platonic maps (and ∞ of trivial maps) Projective plane: Petersen, K4, duals (and ∞ of trivial maps)

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

slide-73
SLIDE 73

Regular maps on a compact surface by 2001

Sphere: Platonic maps (and ∞ of trivial maps) Projective plane: Petersen, K4, duals (and ∞ of trivial maps) Torus:

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

slide-74
SLIDE 74

Regular maps on a compact surface by 2001

Sphere: Platonic maps (and ∞ of trivial maps) Projective plane: Petersen, K4, duals (and ∞ of trivial maps) Torus: Infinitely many nontrivial regular maps

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

slide-75
SLIDE 75

Regular maps on a compact surface by 2001

Sphere: Platonic maps (and ∞ of trivial maps) Projective plane: Petersen, K4, duals (and ∞ of trivial maps) Torus: Infinitely many nontrivial regular maps Klein bottle:

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

slide-76
SLIDE 76

Regular maps on a compact surface by 2001

Sphere: Platonic maps (and ∞ of trivial maps) Projective plane: Petersen, K4, duals (and ∞ of trivial maps) Torus: Infinitely many nontrivial regular maps Klein bottle: No regular map at all!

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

slide-77
SLIDE 77

Regular maps on a compact surface by 2001

Sphere: Platonic maps (and ∞ of trivial maps) Projective plane: Petersen, K4, duals (and ∞ of trivial maps) Torus: Infinitely many nontrivial regular maps Klein bottle: No regular map at all! If G = x, y, z| x2 = y2 = z2 = (yz)ℓ = (zx)m = (xy)2 = . . . = 1 gives a regular map of type (ℓ, m) on a compact surface with Euler char. χ, then |G| = 4ℓm ℓm − 2ℓ − 2m(−χ)

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

slide-78
SLIDE 78

Regular maps on a compact surface by 2001

Sphere: Platonic maps (and ∞ of trivial maps) Projective plane: Petersen, K4, duals (and ∞ of trivial maps) Torus: Infinitely many nontrivial regular maps Klein bottle: No regular map at all! If G = x, y, z| x2 = y2 = z2 = (yz)ℓ = (zx)m = (xy)2 = . . . = 1 gives a regular map of type (ℓ, m) on a compact surface with Euler char. χ, then |G| = 4ℓm ℓm − 2ℓ − 2m(−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 AGT 2016 Pilsen Regular maps with a given automorphism group, and on a given surface 13 / 18

slide-79
SLIDE 79

Regular maps on a compact surface by 2001

Sphere: Platonic maps (and ∞ of trivial maps) Projective plane: Petersen, K4, duals (and ∞ of trivial maps) Torus: Infinitely many nontrivial regular maps Klein bottle: No regular map at all! If G = x, y, z| x2 = y2 = z2 = (yz)ℓ = (zx)m = (xy)2 = . . . = 1 gives a regular map of type (ℓ, m) on a compact surface with Euler char. χ, then |G| = 4ℓm ℓm − 2ℓ − 2m(−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 AGT 2016 Pilsen Regular maps with a given automorphism group, and on a given surface 13 / 18

slide-80
SLIDE 80

Regular maps on a compact surface by 2001

Sphere: Platonic maps (and ∞ of trivial maps) Projective plane: Petersen, K4, duals (and ∞ of trivial maps) Torus: Infinitely many nontrivial regular maps Klein bottle: No regular map at all! If G = x, y, z| x2 = y2 = z2 = (yz)ℓ = (zx)m = (xy)2 = . . . = 1 gives a regular map of type (ℓ, m) on a compact surface with Euler char. χ, then |G| = 4ℓm ℓm − 2ℓ − 2m(−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 AGT 2016 Pilsen Regular maps with a given automorphism group, and on a given surface 13 / 18

slide-81
SLIDE 81

Regular maps on a compact surface by 2001

Sphere: Platonic maps (and ∞ of trivial maps) Projective plane: Petersen, K4, duals (and ∞ of trivial maps) Torus: Infinitely many nontrivial regular maps Klein bottle: No regular map at all! If G = x, y, z| x2 = y2 = z2 = (yz)ℓ = (zx)m = (xy)2 = . . . = 1 gives a regular map of type (ℓ, m) on a compact surface with Euler char. χ, then |G| = 4ℓm ℓm − 2ℓ − 2m(−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 AGT 2016 Pilsen Regular maps with a given automorphism group, and on a given surface 13 / 18

slide-82
SLIDE 82

Classification of regular maps for infinitely many surfaces

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

slide-83
SLIDE 83

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 AGT 2016 Pilsen Regular maps with a given automorphism group, and on a given surface 14 / 18

slide-84
SLIDE 84

Classification of regular maps for infinitely many surfaces

  • χ = −p for every prime p [Breda, Nedela,ˇ

S 2005]

  • χ = −2p and orientable + ‘large’ [Belolipetsky, Jones 2005]

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

slide-85
SLIDE 85

Classification of regular maps for infinitely many surfaces

  • χ = −p for every prime p [Breda, Nedela,ˇ

S 2005]

  • χ = −2p and orientable + ‘large’ [Belolipetsky, Jones 2005]
  • χ = −2p and orientable, all [Conder, Tucker, ˇ

S 2010]

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

slide-86
SLIDE 86

Classification of regular maps for infinitely many surfaces

  • χ = −p for every prime p [Breda, Nedela,ˇ

S 2005]

  • χ = −2p and orientable + ‘large’ [Belolipetsky, Jones 2005]
  • χ = −2p and orientable, all [Conder, Tucker, ˇ

S 2010]

  • χ = −p2 [Conder, Potoˇ

cnik, ˇ S 2010]

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

slide-87
SLIDE 87

Classification of regular maps for infinitely many surfaces

  • χ = −p for every prime p [Breda, Nedela,ˇ

S 2005]

  • χ = −2p and orientable + ‘large’ [Belolipetsky, Jones 2005]
  • χ = −2p and orientable, all [Conder, Tucker, ˇ

S 2010]

  • χ = −p2 [Conder, Potoˇ

cnik, ˇ S 2010]

  • χ = −3p [Conder, Nedela, ˇ

S 2012]

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

slide-88
SLIDE 88

Classification of regular maps for infinitely many surfaces

  • χ = −p for every prime p [Breda, Nedela,ˇ

S 2005]

  • χ = −2p and orientable + ‘large’ [Belolipetsky, Jones 2005]
  • χ = −2p and orientable, all [Conder, Tucker, ˇ

S 2010]

  • χ = −p2 [Conder, Potoˇ

cnik, ˇ S 2010]

  • χ = −3p [Conder, Nedela, ˇ

S 2012] Classification for some families of orientably regular maps with χ = 2 − 2g:

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

slide-89
SLIDE 89

Classification of regular maps for infinitely many surfaces

  • χ = −p for every prime p [Breda, Nedela,ˇ

S 2005]

  • χ = −2p and orientable + ‘large’ [Belolipetsky, Jones 2005]
  • χ = −2p and orientable, all [Conder, Tucker, ˇ

S 2010]

  • χ = −p2 [Conder, Potoˇ

cnik, ˇ S 2010]

  • χ = −3p [Conder, Nedela, ˇ

S 2012] Classification for some families of orientably regular maps with χ = 2 − 2g:

  • 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 AGT 2016 Pilsen Regular maps with a given automorphism group, and on a given surface 14 / 18

slide-90
SLIDE 90

Classification of regular maps for infinitely many surfaces

  • χ = −p for every prime p [Breda, Nedela,ˇ

S 2005]

  • χ = −2p and orientable + ‘large’ [Belolipetsky, Jones 2005]
  • χ = −2p and orientable, all [Conder, Tucker, ˇ

S 2010]

  • χ = −p2 [Conder, Potoˇ

cnik, ˇ S 2010]

  • χ = −3p [Conder, Nedela, ˇ

S 2012] Classification for some families of orientably regular maps with χ = 2 − 2g:

  • 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 AGT 2016 Pilsen Regular maps with a given automorphism group, and on a given surface 14 / 18

slide-91
SLIDE 91

Classification of regular maps for infinitely many surfaces

  • χ = −p for every prime p [Breda, Nedela,ˇ

S 2005]

  • χ = −2p and orientable + ‘large’ [Belolipetsky, Jones 2005]
  • χ = −2p and orientable, all [Conder, Tucker, ˇ

S 2010]

  • χ = −p2 [Conder, Potoˇ

cnik, ˇ S 2010]

  • χ = −3p [Conder, Nedela, ˇ

S 2012] Classification for some families of orientably regular maps with χ = 2 − 2g:

  • 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 AGT 2016 Pilsen Regular maps with a given automorphism group, and on a given surface 14 / 18

slide-92
SLIDE 92

Gaps in the nonorientable genus spectrum

Jozef ˇ Sir´ aˇ n Open University and Slovak University of Technology AGT 2016 Pilsen Regular maps with a given automorphism group, and on a given surface 15 / 18

slide-93
SLIDE 93

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 (4g, 4g).

Jozef ˇ Sir´ aˇ n Open University and Slovak University of Technology AGT 2016 Pilsen Regular maps with a given automorphism group, and on a given surface 15 / 18

slide-94
SLIDE 94

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 (4g, 4g). 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 AGT 2016 Pilsen Regular maps with a given automorphism group, and on a given surface 15 / 18

slide-95
SLIDE 95

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 (4g, 4g). 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 AGT 2016 Pilsen Regular maps with a given automorphism group, and on a given surface 15 / 18

slide-96
SLIDE 96

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 (4g, 4g). 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 AGT 2016 Pilsen Regular maps with a given automorphism group, and on a given surface 15 / 18

slide-97
SLIDE 97

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 (4g, 4g). 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 = −p2 for all primes p > 7 [Conder, Potoˇ

cnik, ˇ S 2010]

Jozef ˇ Sir´ aˇ n Open University and Slovak University of Technology AGT 2016 Pilsen Regular maps with a given automorphism group, and on a given surface 15 / 18

slide-98
SLIDE 98

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 (4g, 4g). 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 = −p2 for all primes p > 7 [Conder, Potoˇ

cnik, ˇ S 2010]

  • χ = −3p 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 AGT 2016 Pilsen Regular maps with a given automorphism group, and on a given surface 15 / 18

slide-99
SLIDE 99

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 (4g, 4g). 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 = −p2 for all primes p > 7 [Conder, Potoˇ

cnik, ˇ S 2010]

  • χ = −3p 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 AGT 2016 Pilsen Regular maps with a given automorphism group, and on a given surface 15 / 18

slide-100
SLIDE 100

Regular maps with odd χ [with Conder, Gill and Short]

Jozef ˇ Sir´ aˇ n Open University and Slovak University of Technology AGT 2016 Pilsen Regular maps with a given automorphism group, and on a given surface 16 / 18

slide-101
SLIDE 101

Regular maps with odd χ [with Conder, Gill and Short]

If M is regular with odd χ, then Aut(M) has dihedral Sylow 2-subgroups.

Jozef ˇ Sir´ aˇ n Open University and Slovak University of Technology AGT 2016 Pilsen Regular maps with a given automorphism group, and on a given surface 16 / 18

slide-102
SLIDE 102

Regular maps with odd χ [with Conder, Gill and Short]

If M is regular with odd χ, then Aut(M) has dihedral Sylow 2-subgroups. [Gorenstein, Walter 1965]: If G has dihedral Sylow 2-subgroups and O is the odd part of G, then G/O is isomorphic to a Sylow 2-subgroup of G, or A7, or else a to subgroup of Aut(PSL(2, q)) containing PSL(2, q), q odd.

Jozef ˇ Sir´ aˇ n Open University and Slovak University of Technology AGT 2016 Pilsen Regular maps with a given automorphism group, and on a given surface 16 / 18

slide-103
SLIDE 103

Regular maps with odd χ [with Conder, Gill and Short]

If M is regular with odd χ, then Aut(M) has dihedral Sylow 2-subgroups. [Gorenstein, Walter 1965]: If G has dihedral Sylow 2-subgroups and O is the odd part of G, then G/O is isomorphic to a Sylow 2-subgroup of G, or A7, or else a to subgroup of Aut(PSL(2, q)) containing PSL(2, q), q odd.

  • Theorem. Let G be the automorphism group of a regular map with

χ odd. Then, one of the following cases occurs:

Jozef ˇ Sir´ aˇ n Open University and Slovak University of Technology AGT 2016 Pilsen Regular maps with a given automorphism group, and on a given surface 16 / 18

slide-104
SLIDE 104

Regular maps with odd χ [with Conder, Gill and Short]

If M is regular with odd χ, then Aut(M) has dihedral Sylow 2-subgroups. [Gorenstein, Walter 1965]: If G has dihedral Sylow 2-subgroups and O is the odd part of G, then G/O is isomorphic to a Sylow 2-subgroup of G, or A7, or else a to subgroup of Aut(PSL(2, q)) containing PSL(2, q), q odd.

  • Theorem. Let G be the automorphism group of a regular map with

χ odd. Then, one of the following cases occurs: (a) G/O is isomorphic to A7 or PSL(2, q) for some odd prime power q, and if χ = −rn for an odd prime r, then G/O ≃ PSL(2, q) or PGL(2, q).

Jozef ˇ Sir´ aˇ n Open University and Slovak University of Technology AGT 2016 Pilsen Regular maps with a given automorphism group, and on a given surface 16 / 18

slide-105
SLIDE 105

Regular maps with odd χ [with Conder, Gill and Short]

If M is regular with odd χ, then Aut(M) has dihedral Sylow 2-subgroups. [Gorenstein, Walter 1965]: If G has dihedral Sylow 2-subgroups and O is the odd part of G, then G/O is isomorphic to a Sylow 2-subgroup of G, or A7, or else a to subgroup of Aut(PSL(2, q)) containing PSL(2, q), q odd.

  • Theorem. Let G be the automorphism group of a regular map with

χ odd. Then, one of the following cases occurs: (a) G/O is isomorphic to A7 or PSL(2, q) for some odd prime power q, and if χ = −rn for an odd prime r, then G/O ≃ PSL(2, q) or PGL(2, q). (b) G has an odd-order normal subgroup N such that G/N ≃ 3 · A7.

Jozef ˇ Sir´ aˇ n Open University and Slovak University of Technology AGT 2016 Pilsen Regular maps with a given automorphism group, and on a given surface 16 / 18

slide-106
SLIDE 106

Regular maps with odd χ [with Conder, Gill and Short]

If M is regular with odd χ, then Aut(M) has dihedral Sylow 2-subgroups. [Gorenstein, Walter 1965]: If G has dihedral Sylow 2-subgroups and O is the odd part of G, then G/O is isomorphic to a Sylow 2-subgroup of G, or A7, or else a to subgroup of Aut(PSL(2, q)) containing PSL(2, q), q odd.

  • Theorem. Let G be the automorphism group of a regular map with

χ odd. Then, one of the following cases occurs: (a) G/O is isomorphic to A7 or PSL(2, q) for some odd prime power q, and if χ = −rn for an odd prime r, then G/O ≃ PSL(2, q) or PGL(2, q). (b) G has an odd-order normal subgroup N such that G/N ≃ 3 · A7. (c) G has an odd-order normal N such that G/N ≃ (PSL(2, q) × C) · 2.

Jozef ˇ Sir´ aˇ n Open University and Slovak University of Technology AGT 2016 Pilsen Regular maps with a given automorphism group, and on a given surface 16 / 18

slide-107
SLIDE 107

Regular maps with odd χ [with Conder, Gill and Short]

If M is regular with odd χ, then Aut(M) has dihedral Sylow 2-subgroups. [Gorenstein, Walter 1965]: If G has dihedral Sylow 2-subgroups and O is the odd part of G, then G/O is isomorphic to a Sylow 2-subgroup of G, or A7, or else a to subgroup of Aut(PSL(2, q)) containing PSL(2, q), q odd.

  • Theorem. Let G be the automorphism group of a regular map with

χ odd. Then, one of the following cases occurs: (a) G/O is isomorphic to A7 or PSL(2, q) for some odd prime power q, and if χ = −rn for an odd prime r, then G/O ≃ PSL(2, q) or PGL(2, q). (b) G has an odd-order normal subgroup N such that G/N ≃ 3 · A7. (c) G has an odd-order normal N such that G/N ≃ (PSL(2, q) × C) · 2. Moreover, if G is not solvable, then r ∈ {3, 5, 7, 13} or n ≥ log2 r + 2, and if G is solvable, then G is almost Sylow-cyclic or n ≥ min{log2 r − 2, 7}.

Jozef ˇ Sir´ aˇ n Open University and Slovak University of Technology AGT 2016 Pilsen Regular maps with a given automorphism group, and on a given surface 16 / 18

slide-108
SLIDE 108

Regular maps, χ = −rn: The case G/O ≃ PSL(2, q)

Jozef ˇ Sir´ aˇ n Open University and Slovak University of Technology AGT 2016 Pilsen Regular maps with a given automorphism group, and on a given surface 17 / 18

slide-109
SLIDE 109

Regular maps, χ = −rn: The case G/O ≃ PSL(2, q)

  • Theorem. For any regular map M of Euler characteristic χ < 0 and

for any odd integer s there is a smooth cover ˜ M of M that has Euler characteristic s1−χχ.

Jozef ˇ Sir´ aˇ n Open University and Slovak University of Technology AGT 2016 Pilsen Regular maps with a given automorphism group, and on a given surface 17 / 18

slide-110
SLIDE 110

Regular maps, χ = −rn: The case G/O ≃ PSL(2, q)

  • Theorem. For any regular map M of Euler characteristic χ < 0 and

for any odd integer s there is a smooth cover ˜ M of M that has Euler characteristic s1−χχ.

  • Theorem. Let G be the automorphism group of a regular map of type

(k, m) with χ = −rn for an odd prime r. If G/O ≃ PSL(2, q) for some prime power q, then one of the following cases occur: r q (k, m) 3 5 (5, 5) 3 5 (3, 15) 7 13 (3,13) 13 13 (3, 7)

Jozef ˇ Sir´ aˇ n Open University and Slovak University of Technology AGT 2016 Pilsen Regular maps with a given automorphism group, and on a given surface 17 / 18

slide-111
SLIDE 111

Regular maps, χ = −rn: The case G/O ≃ PSL(2, q)

  • Theorem. For any regular map M of Euler characteristic χ < 0 and

for any odd integer s there is a smooth cover ˜ M of M that has Euler characteristic s1−χχ.

  • Theorem. Let G be the automorphism group of a regular map of type

(k, m) with χ = −rn for an odd prime r. If G/O ≃ PSL(2, q) for some prime power q, then one of the following cases occur: r q (k, m) 3 5 (5, 5) 3 5 (3, 15) 7 13 (3,13) 13 13 (3, 7) Note: In each case there are infinitely many examples (by Thm above).

Jozef ˇ Sir´ aˇ n Open University and Slovak University of Technology AGT 2016 Pilsen Regular maps with a given automorphism group, and on a given surface 17 / 18

slide-112
SLIDE 112

Regular maps, χ = −rn: The case G/O ≃ PGL(2, q)

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

slide-113
SLIDE 113

Regular maps, χ = −rn: The case G/O ≃ PGL(2, q)

  • Theorem. Let G ∼

= Aut(M), regular, type (k, m), with χ = −rn for an

  • dd prime r. If G/O ≃ PGL(2, q), then G has a normal r-subgroup N

such that G/N ≃ (PSL(2, q) × Ct).2, with t odd, coprime to r, and:

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

slide-114
SLIDE 114

Regular maps, χ = −rn: The case G/O ≃ PGL(2, q)

  • Theorem. Let G ∼

= Aut(M), regular, type (k, m), with χ = −rn for an

  • dd prime r. If G/O ≃ PGL(2, q), then G has a normal r-subgroup N

such that G/N ≃ (PSL(2, q) × Ct).2, with t odd, coprime to r, and: r q (k, m) 3 5 (4, 5) 5 5 (4, 6), (20, 30) 3 9 (5tra, 8ra) 7 7 (3tra, 8ra) r p (tp, p−1) r p (tpra, (p−1)ra) r p (tpra, (p−1)ra)

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

slide-115
SLIDE 115

Regular maps, χ = −rn: The case G/O ≃ PGL(2, q)

  • Theorem. Let G ∼

= Aut(M), regular, type (k, m), with χ = −rn for an

  • dd prime r. If G/O ≃ PGL(2, q), then G has a normal r-subgroup N

such that G/N ≃ (PSL(2, q) × Ct).2, with t odd, coprime to r, and: r q (k, m) 3 5 (4, 5) 5 5 (4, 6), (20, 30) 3 9 (5tra, 8ra) 7 7 (3tra, 8ra) r p (tp, p−1) r p (tpra, (p−1)ra) r p (tpra, (p−1)ra) Note: ∞ examples for r = 3, p = 7,

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

slide-116
SLIDE 116

Regular maps, χ = −rn: The case G/O ≃ PGL(2, q)

  • Theorem. Let G ∼

= Aut(M), regular, type (k, m), with χ = −rn for an

  • dd prime r. If G/O ≃ PGL(2, q), then G has a normal r-subgroup N

such that G/N ≃ (PSL(2, q) × Ct).2, with t odd, coprime to r, and: r q (k, m) 3 5 (4, 5) 5 5 (4, 6), (20, 30) 3 9 (5tra, 8ra) 7 7 (3tra, 8ra) r p (tp, p−1) r p (tpra, (p−1)ra) r p (tpra, (p−1)ra) Note: ∞ examples for r = 3, p = 7, t = (3b + 8)/77, b ≡ 21 mod 30.

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

slide-117
SLIDE 117

Regular maps, χ = −rn: The case G/O ≃ PGL(2, q)

  • Theorem. Let G ∼

= Aut(M), regular, type (k, m), with χ = −rn for an

  • dd prime r. If G/O ≃ PGL(2, q), then G has a normal r-subgroup N

such that G/N ≃ (PSL(2, q) × Ct).2, with t odd, coprime to r, and: r q (k, m) 3 5 (4, 5) 5 5 (4, 6), (20, 30) 3 9 (5tra, 8ra) 7 7 (3tra, 8ra) r p (tp, p−1) r p (tpra, (p−1)ra) r p (tpra, (p−1)ra) Note: ∞ examples for r = 3, p = 7, t = (3b + 8)/77, b ≡ 21 mod 30. A wealth of results for orientably regular maps with χ = −2b: Gill [2014].

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

slide-118
SLIDE 118

Regular maps, χ = −rn: The case G/O ≃ PGL(2, q)

  • Theorem. Let G ∼

= Aut(M), regular, type (k, m), with χ = −rn for an

  • dd prime r. If G/O ≃ PGL(2, q), then G has a normal r-subgroup N

such that G/N ≃ (PSL(2, q) × Ct).2, with t odd, coprime to r, and: r q (k, m) 3 5 (4, 5) 5 5 (4, 6), (20, 30) 3 9 (5tra, 8ra) 7 7 (3tra, 8ra) r p (tp, p−1) r p (tpra, (p−1)ra) r p (tpra, (p−1)ra) Note: ∞ examples for r = 3, p = 7, t = (3b + 8)/77, b ≡ 21 mod 30. A wealth of results for orientably regular maps with χ = −2b: Gill [2014]. THANK YOU.

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