Vertex-quasiprimitivity in regular maps Jozef Sir a n OU and - - PowerPoint PPT Presentation

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Vertex-quasiprimitivity in regular maps Jozef Sir a n OU and - - PowerPoint PPT Presentation

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

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Vertex-quasiprimitivity in regular maps

Jozef ˇ Sir´ aˇ n OU and STU 27th May 2015

Jozef ˇ Sir´ aˇ n OU and STU Vertex-quasiprimitivity in regular maps 27th May 2015 1 / 15

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The five Platonic maps M

Jozef ˇ Sir´ aˇ n OU and STU Vertex-quasiprimitivity in regular maps 27th May 2015 2 / 15

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The five Platonic maps M

Jozef ˇ Sir´ aˇ n OU and STU Vertex-quasiprimitivity in regular maps 27th May 2015 2 / 15

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The five Platonic maps M

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

Jozef ˇ Sir´ aˇ n OU and STU Vertex-quasiprimitivity in regular maps 27th May 2015 2 / 15

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SLIDE 5

The five Platonic maps M

Here, Aut+(M) and Aut(M) act regularly on darts and flags, respectively. Such maps (graph embeddings) are called orientably-regular and regular.

Jozef ˇ Sir´ aˇ n OU and STU Vertex-quasiprimitivity in regular maps 27th May 2015 2 / 15

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SLIDE 6

Is this embedding of K5 in a torus orientably-regular?

  • Jozef ˇ

Sir´ aˇ n OU and STU Vertex-quasiprimitivity in regular maps 27th May 2015 3 / 15

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SLIDE 7

Is this embedding of K5 in a torus orientably-regular?

  • Jozef ˇ

Sir´ aˇ n OU and STU Vertex-quasiprimitivity in regular maps 27th May 2015 3 / 15

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SLIDE 8

Is this embedding of K5 in a torus orientably-regular?

  • Presentation: Aut+(M) = r, s; r4 = s4 = (rs)2 = ... = 1

Jozef ˇ Sir´ aˇ n OU and STU Vertex-quasiprimitivity in regular maps 27th May 2015 3 / 15

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SLIDE 9

Is this embedding of K5 regular? Chirality

  • Jozef ˇ

Sir´ aˇ n OU and STU Vertex-quasiprimitivity in regular maps 27th May 2015 4 / 15

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SLIDE 10

Is this embedding of K5 regular? Chirality

  • Jozef ˇ

Sir´ aˇ n OU and STU Vertex-quasiprimitivity in regular maps 27th May 2015 4 / 15

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SLIDE 11

Regular and orientably-regular maps

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

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SLIDE 12

Regular and orientably-regular maps

A map is regular if its automorphism group is regular on flags.

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

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SLIDE 13

Regular and orientably-regular maps

A map is regular if its automorphism group is regular on flags.

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

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Regular and orientably-regular maps

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

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

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SLIDE 15

Regular and orientably-regular maps

A map is regular if its automorphism group is regular on flags. Aut(M) = x, y, z| x2 = y2 = z2 = (yz)k = (zx)m = (xy)2 = . . . = 1 Orientable regularity - the orientation-preserving map automorphism group is regular on arcs, or darts: Aut+(M) = r, s| rk = sm = (rs)2 = . . . = 1

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

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Regular and orientably-regular maps

A map is regular if its automorphism group is regular on flags. Aut(M) = x, y, z| x2 = y2 = z2 = (yz)k = (zx)m = (xy)2 = . . . = 1 Orientable regularity - the orientation-preserving map automorphism group is regular on arcs, or darts: Aut+(M) = r, s| rk = sm = (rs)2 = . . . = 1 Conversely, such group presentations determine (orientably-) regular maps.

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

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The famous Klein map of genus 3

Jozef ˇ Sir´ aˇ n OU and STU Vertex-quasiprimitivity in regular maps 27th May 2015 6 / 15

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The famous Klein map of genus 3

Jozef ˇ Sir´ aˇ n OU and STU Vertex-quasiprimitivity in regular maps 27th May 2015 6 / 15

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The Klein map of genus 3 – algebraically

Jozef ˇ Sir´ aˇ n OU and STU Vertex-quasiprimitivity in regular maps 27th May 2015 7 / 15

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The Klein map of genus 3 – algebraically

  • Regular, of type {7, 3}

Jozef ˇ Sir´ aˇ n OU and STU Vertex-quasiprimitivity in regular maps 27th May 2015 7 / 15

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SLIDE 21

The Klein map of genus 3 – algebraically

  • Regular, of type {7, 3}
  • Aut(M) = x, y, z; x2 =

y2 = z2 = (yz)3 = (zx)7 = (xy)2 = . . . = 1

Jozef ˇ Sir´ aˇ n OU and STU Vertex-quasiprimitivity in regular maps 27th May 2015 7 / 15

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SLIDE 22

The Klein map of genus 3 – algebraically

  • Regular, of type {7, 3}
  • Aut(M) = x, y, z; x2 =

y2 = z2 = (yz)3 = (zx)7 = (xy)2 = . . . = 1

  • 336 flags

Jozef ˇ Sir´ aˇ n OU and STU Vertex-quasiprimitivity in regular maps 27th May 2015 7 / 15

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SLIDE 23

The Klein map of genus 3 – algebraically

  • Regular, of type {7, 3}
  • Aut(M) = x, y, z; x2 =

y2 = z2 = (yz)3 = (zx)7 = (xy)2 = . . . = 1

  • 336 flags
  • Aut(M) ≃ PGL(2, 7)

Jozef ˇ Sir´ aˇ n OU and STU Vertex-quasiprimitivity in regular maps 27th May 2015 7 / 15

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SLIDE 24

Example of a non-orientable regular map

Jozef ˇ Sir´ aˇ n OU and STU Vertex-quasiprimitivity in regular maps 27th May 2015 8 / 15

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Example of a non-orientable regular map

The Petersen Graph on the projective plane, with its dual – K6:

Jozef ˇ Sir´ aˇ n OU and STU Vertex-quasiprimitivity in regular maps 27th May 2015 8 / 15

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SLIDE 26

Example of a non-orientable regular map

The Petersen Graph on the projective plane, with its dual – K6:

Jozef ˇ Sir´ aˇ n OU and STU Vertex-quasiprimitivity in regular maps 27th May 2015 8 / 15

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SLIDE 27

Example of a non-orientable regular map

The Petersen Graph on the projective plane, with its dual – K6:

  • Regular, of type {5, 3}

Jozef ˇ Sir´ aˇ n OU and STU Vertex-quasiprimitivity in regular maps 27th May 2015 8 / 15

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SLIDE 28

Example of a non-orientable regular map

The Petersen Graph on the projective plane, with its dual – K6:

  • Regular, of type {5, 3}
  • Aut(M) = x, y, z; x2 =

y2 = z2 = (yz)3 = (zx)5 = (xy)2 = . . . = 1

Jozef ˇ Sir´ aˇ n OU and STU Vertex-quasiprimitivity in regular maps 27th May 2015 8 / 15

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SLIDE 29

Example of a non-orientable regular map

The Petersen Graph on the projective plane, with its dual – K6:

  • Regular, of type {5, 3}
  • Aut(M) = x, y, z; x2 =

y2 = z2 = (yz)3 = (zx)5 = (xy)2 = . . . = 1

  • 60 flags

Jozef ˇ Sir´ aˇ n OU and STU Vertex-quasiprimitivity in regular maps 27th May 2015 8 / 15

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SLIDE 30

Example of a non-orientable regular map

The Petersen Graph on the projective plane, with its dual – K6:

  • Regular, of type {5, 3}
  • Aut(M) = x, y, z; x2 =

y2 = z2 = (yz)3 = (zx)5 = (xy)2 = . . . = 1

  • 60 flags
  • Aut(M) ≃ A5

Jozef ˇ Sir´ aˇ n OU and STU Vertex-quasiprimitivity in regular maps 27th May 2015 8 / 15

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SLIDE 31

Classification by automorphism groups

Jozef ˇ Sir´ aˇ n OU and STU Vertex-quasiprimitivity in regular maps 27th May 2015 9 / 15

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SLIDE 32

Classification by automorphism groups

Available results:

Jozef ˇ Sir´ aˇ n OU and STU Vertex-quasiprimitivity in regular maps 27th May 2015 9 / 15

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SLIDE 33

Classification by automorphism groups

Available results:

  • Abelian, dihedral: Folklore

Jozef ˇ Sir´ aˇ n OU and STU Vertex-quasiprimitivity in regular maps 27th May 2015 9 / 15

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SLIDE 34

Classification by automorphism groups

Available results:

  • Abelian, dihedral: Folklore
  • Sylow-cyclic: Classification of groups - H¨
  • lder 1895, Burnside 1905,

Zassenhaus 1936; maps - Conder, Potoˇ cnik, ˇ S 2010

Jozef ˇ Sir´ aˇ n OU and STU Vertex-quasiprimitivity in regular maps 27th May 2015 9 / 15

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SLIDE 35

Classification by automorphism groups

Available results:

  • Abelian, dihedral: Folklore
  • Sylow-cyclic: Classification of groups - H¨
  • lder 1895, Burnside 1905,

Zassenhaus 1936; maps - Conder, Potoˇ cnik, ˇ S 2010

  • Almost Sylow-cyclic: Solvable groups - Zassenhaus 1936; unsolvable -

Suzuki 1955, Wong 1966; maps - Conder, Potoˇ cnik, ˇ S 2010

Jozef ˇ Sir´ aˇ n OU and STU Vertex-quasiprimitivity in regular maps 27th May 2015 9 / 15

slide-36
SLIDE 36

Classification by automorphism groups

Available results:

  • Abelian, dihedral: Folklore
  • Sylow-cyclic: Classification of groups - H¨
  • lder 1895, Burnside 1905,

Zassenhaus 1936; maps - Conder, Potoˇ cnik, ˇ S 2010

  • Almost Sylow-cyclic: Solvable groups - Zassenhaus 1936; unsolvable -

Suzuki 1955, Wong 1966; maps - Conder, Potoˇ cnik, ˇ S 2010

  • Nilpotent maps - class 2 Malniˇ

c, Nedela, ˇ Skoviera 2012

Jozef ˇ Sir´ aˇ n OU and STU Vertex-quasiprimitivity in regular maps 27th May 2015 9 / 15

slide-37
SLIDE 37

Classification by automorphism groups

Available results:

  • Abelian, dihedral: Folklore
  • Sylow-cyclic: Classification of groups - H¨
  • lder 1895, Burnside 1905,

Zassenhaus 1936; maps - Conder, Potoˇ cnik, ˇ S 2010

  • Almost Sylow-cyclic: Solvable groups - Zassenhaus 1936; unsolvable -

Suzuki 1955, Wong 1966; maps - Conder, Potoˇ cnik, ˇ S 2010

  • Nilpotent maps - class 2 Malniˇ

c, Nedela, ˇ Skoviera 2012

  • Nilpotent maps - class 3 Du, Nedela, ˇ

Skoviera et al 201?

Jozef ˇ Sir´ aˇ n OU and STU Vertex-quasiprimitivity in regular maps 27th May 2015 9 / 15

slide-38
SLIDE 38

Classification by automorphism groups

Available results:

  • Abelian, dihedral: Folklore
  • Sylow-cyclic: Classification of groups - H¨
  • lder 1895, Burnside 1905,

Zassenhaus 1936; maps - Conder, Potoˇ cnik, ˇ S 2010

  • Almost Sylow-cyclic: Solvable groups - Zassenhaus 1936; unsolvable -

Suzuki 1955, Wong 1966; maps - Conder, Potoˇ cnik, ˇ S 2010

  • Nilpotent maps - class 2 Malniˇ

c, Nedela, ˇ Skoviera 2012

  • Nilpotent maps - class 3 Du, Nedela, ˇ

Skoviera et al 201?

  • Maps on PSL, PGL(2, q) - Sah 1969, Conder, Potoˇ

cnik, ˇ S 2010

Jozef ˇ Sir´ aˇ n OU and STU Vertex-quasiprimitivity in regular maps 27th May 2015 9 / 15

slide-39
SLIDE 39

Classification by automorphism groups

Available results:

  • Abelian, dihedral: Folklore
  • Sylow-cyclic: Classification of groups - H¨
  • lder 1895, Burnside 1905,

Zassenhaus 1936; maps - Conder, Potoˇ cnik, ˇ S 2010

  • Almost Sylow-cyclic: Solvable groups - Zassenhaus 1936; unsolvable -

Suzuki 1955, Wong 1966; maps - Conder, Potoˇ cnik, ˇ S 2010

  • Nilpotent maps - class 2 Malniˇ

c, Nedela, ˇ Skoviera 2012

  • Nilpotent maps - class 3 Du, Nedela, ˇ

Skoviera et al 201?

  • Maps on PSL, PGL(2, q) - Sah 1969, Conder, Potoˇ

cnik, ˇ S 2010

  • Maps of type {4, 5} on Suzuki groups - Jones, Silver 1993

Jozef ˇ Sir´ aˇ n OU and STU Vertex-quasiprimitivity in regular maps 27th May 2015 9 / 15

slide-40
SLIDE 40

Classification by automorphism groups

Available results:

  • Abelian, dihedral: Folklore
  • Sylow-cyclic: Classification of groups - H¨
  • lder 1895, Burnside 1905,

Zassenhaus 1936; maps - Conder, Potoˇ cnik, ˇ S 2010

  • Almost Sylow-cyclic: Solvable groups - Zassenhaus 1936; unsolvable -

Suzuki 1955, Wong 1966; maps - Conder, Potoˇ cnik, ˇ S 2010

  • Nilpotent maps - class 2 Malniˇ

c, Nedela, ˇ Skoviera 2012

  • Nilpotent maps - class 3 Du, Nedela, ˇ

Skoviera et al 201?

  • Maps on PSL, PGL(2, q) - Sah 1969, Conder, Potoˇ

cnik, ˇ S 2010

  • Maps of type {4, 5} on Suzuki groups - Jones, Silver 1993
  • Maps of type {3, p}, p ≡ −1 mod 12, on Ree groups - Jones 1994

Jozef ˇ Sir´ aˇ n OU and STU Vertex-quasiprimitivity in regular maps 27th May 2015 9 / 15

slide-41
SLIDE 41

Classification by automorphism groups

Available results:

  • Abelian, dihedral: Folklore
  • Sylow-cyclic: Classification of groups - H¨
  • lder 1895, Burnside 1905,

Zassenhaus 1936; maps - Conder, Potoˇ cnik, ˇ S 2010

  • Almost Sylow-cyclic: Solvable groups - Zassenhaus 1936; unsolvable -

Suzuki 1955, Wong 1966; maps - Conder, Potoˇ cnik, ˇ S 2010

  • Nilpotent maps - class 2 Malniˇ

c, Nedela, ˇ Skoviera 2012

  • Nilpotent maps - class 3 Du, Nedela, ˇ

Skoviera et al 201?

  • Maps on PSL, PGL(2, q) - Sah 1969, Conder, Potoˇ

cnik, ˇ S 2010

  • Maps of type {4, 5} on Suzuki groups - Jones, Silver 1993
  • Maps of type {3, p}, p ≡ −1 mod 12, on Ree groups - Jones 1994

Classification also considered by underlying graphs and supporting surfaces.

Jozef ˇ Sir´ aˇ n OU and STU Vertex-quasiprimitivity in regular maps 27th May 2015 9 / 15

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SLIDE 42

Large regular maps from small ones by lifting

Jozef ˇ Sir´ aˇ n OU and STU Vertex-quasiprimitivity in regular maps 27th May 2015 10 / 15

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SLIDE 43

Large regular maps from small ones by lifting

M: orientably-regular, G = Aut+(M) = r, s| rk = sm = (rs)2 = . . . = 1

Jozef ˇ Sir´ aˇ n OU and STU Vertex-quasiprimitivity in regular maps 27th May 2015 10 / 15

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SLIDE 44

Large regular maps from small ones by lifting

M: orientably-regular, G = Aut+(M) = r, s| rk = sm = (rs)2 = . . . = 1 Taking K ⊳ G we may form the orientably-regular quotient map M/K with G/K = Aut+(M/K) = rK, sK| (rK)κ = (sK)µ = (rsK)2 = . . . = 1

Jozef ˇ Sir´ aˇ n OU and STU Vertex-quasiprimitivity in regular maps 27th May 2015 10 / 15

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SLIDE 45

Large regular maps from small ones by lifting

M: orientably-regular, G = Aut+(M) = r, s| rk = sm = (rs)2 = . . . = 1 Taking K ⊳ G we may form the orientably-regular quotient map M/K with G/K = Aut+(M/K) = rK, sK| (rK)κ = (sK)µ = (rsK)2 = . . . = 1 The projection G → G/K induces a branched covering M → M/K

  • f maps (and supporting surfaces);

Jozef ˇ Sir´ aˇ n OU and STU Vertex-quasiprimitivity in regular maps 27th May 2015 10 / 15

slide-46
SLIDE 46

Large regular maps from small ones by lifting

M: orientably-regular, G = Aut+(M) = r, s| rk = sm = (rs)2 = . . . = 1 Taking K ⊳ G we may form the orientably-regular quotient map M/K with G/K = Aut+(M/K) = rK, sK| (rK)κ = (sK)µ = (rsK)2 = . . . = 1 The projection G → G/K induces a branched covering M → M/K

  • f maps (and supporting surfaces); smooth iff κ = k and µ = m.

Jozef ˇ Sir´ aˇ n OU and STU Vertex-quasiprimitivity in regular maps 27th May 2015 10 / 15

slide-47
SLIDE 47

Large regular maps from small ones by lifting

M: orientably-regular, G = Aut+(M) = r, s| rk = sm = (rs)2 = . . . = 1 Taking K ⊳ G we may form the orientably-regular quotient map M/K with G/K = Aut+(M/K) = rK, sK| (rK)κ = (sK)µ = (rsK)2 = . . . = 1 The projection G → G/K induces a branched covering M → M/K

  • f maps (and supporting surfaces); smooth iff κ = k and µ = m.

Knowing M′ = M/K and K, we may reverse the process and lift M′ to M along K; the lift may not be unique.

Jozef ˇ Sir´ aˇ n OU and STU Vertex-quasiprimitivity in regular maps 27th May 2015 10 / 15

slide-48
SLIDE 48

Large regular maps from small ones by lifting

M: orientably-regular, G = Aut+(M) = r, s| rk = sm = (rs)2 = . . . = 1 Taking K ⊳ G we may form the orientably-regular quotient map M/K with G/K = Aut+(M/K) = rK, sK| (rK)κ = (sK)µ = (rsK)2 = . . . = 1 The projection G → G/K induces a branched covering M → M/K

  • f maps (and supporting surfaces); smooth iff κ = k and µ = m.

Knowing M′ = M/K and K, we may reverse the process and lift M′ to M along K; the lift may not be unique. Theory of covers and lifts: A. Malniˇ c;

Jozef ˇ Sir´ aˇ n OU and STU Vertex-quasiprimitivity in regular maps 27th May 2015 10 / 15

slide-49
SLIDE 49

Large regular maps from small ones by lifting

M: orientably-regular, G = Aut+(M) = r, s| rk = sm = (rs)2 = . . . = 1 Taking K ⊳ G we may form the orientably-regular quotient map M/K with G/K = Aut+(M/K) = rK, sK| (rK)κ = (sK)µ = (rsK)2 = . . . = 1 The projection G → G/K induces a branched covering M → M/K

  • f maps (and supporting surfaces); smooth iff κ = k and µ = m.

Knowing M′ = M/K and K, we may reverse the process and lift M′ to M along K; the lift may not be unique. Theory of covers and lifts: A. Malniˇ c; elementary Abelian case well understood: Malniˇ c, Maruˇ siˇ c, Potoˇ cnik 2004.

Jozef ˇ Sir´ aˇ n OU and STU Vertex-quasiprimitivity in regular maps 27th May 2015 10 / 15

slide-50
SLIDE 50

Large regular maps from small ones by lifting

M: orientably-regular, G = Aut+(M) = r, s| rk = sm = (rs)2 = . . . = 1 Taking K ⊳ G we may form the orientably-regular quotient map M/K with G/K = Aut+(M/K) = rK, sK| (rK)κ = (sK)µ = (rsK)2 = . . . = 1 The projection G → G/K induces a branched covering M → M/K

  • f maps (and supporting surfaces); smooth iff κ = k and µ = m.

Knowing M′ = M/K and K, we may reverse the process and lift M′ to M along K; the lift may not be unique. Theory of covers and lifts: A. Malniˇ c; elementary Abelian case well understood: Malniˇ c, Maruˇ siˇ c, Potoˇ cnik 2004. Regular cyclic lifts of Platonic maps: Jones and Surowski 2000, generalised by Hu, Nedela and Wang 2014.

Jozef ˇ Sir´ aˇ n OU and STU Vertex-quasiprimitivity in regular maps 27th May 2015 10 / 15

slide-51
SLIDE 51

Large regular maps from small ones by lifting

M: orientably-regular, G = Aut+(M) = r, s| rk = sm = (rs)2 = . . . = 1 Taking K ⊳ G we may form the orientably-regular quotient map M/K with G/K = Aut+(M/K) = rK, sK| (rK)κ = (sK)µ = (rsK)2 = . . . = 1 The projection G → G/K induces a branched covering M → M/K

  • f maps (and supporting surfaces); smooth iff κ = k and µ = m.

Knowing M′ = M/K and K, we may reverse the process and lift M′ to M along K; the lift may not be unique. Theory of covers and lifts: A. Malniˇ c; elementary Abelian case well understood: Malniˇ c, Maruˇ siˇ c, Potoˇ cnik 2004. Regular cyclic lifts of Platonic maps: Jones and Surowski 2000, generalised by Hu, Nedela and Wang 2014. Examples of lifts of embeddings of the θ-graph:

Jozef ˇ Sir´ aˇ n OU and STU Vertex-quasiprimitivity in regular maps 27th May 2015 10 / 15

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SLIDE 52

Quasiprimitivity

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

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SLIDE 53

Quasiprimitivity

Take an orientably-regular map M; view G=Aut+(M) as a permutation group on the set Ω of vertices of M.

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

slide-54
SLIDE 54

Quasiprimitivity

Take an orientably-regular map M; view G=Aut+(M) as a permutation group on the set Ω of vertices of M. This works fine if G has trivial core

  • n Ω,

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

slide-55
SLIDE 55

Quasiprimitivity

Take an orientably-regular map M; view G=Aut+(M) as a permutation group on the set Ω of vertices of M. This works fine if G has trivial core

  • n Ω, where, for G transitive on Ω (which is our case),

core(G) = {h ∈ G; h(v) = v for each v ∈ Ω} =

(g∈G) g−1Hg ; H = StabG(u) for some u ∈ Ω.

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

slide-56
SLIDE 56

Quasiprimitivity

Take an orientably-regular map M; view G=Aut+(M) as a permutation group on the set Ω of vertices of M. This works fine if G has trivial core

  • n Ω, where, for G transitive on Ω (which is our case),

core(G) = {h ∈ G; h(v) = v for each v ∈ Ω} =

(g∈G) g−1Hg ; H = StabG(u) for some u ∈ Ω.

This is always the case when the underlying graph of M is simple.

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

slide-57
SLIDE 57

Quasiprimitivity

Take an orientably-regular map M; view G=Aut+(M) as a permutation group on the set Ω of vertices of M. This works fine if G has trivial core

  • n Ω, where, for G transitive on Ω (which is our case),

core(G) = {h ∈ G; h(v) = v for each v ∈ Ω} =

(g∈G) g−1Hg ; H = StabG(u) for some u ∈ Ω.

This is always the case when the underlying graph of M is simple. From now on: G=Aut+(M) acts faithfully as a permutation group on Ω.

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

slide-58
SLIDE 58

Quasiprimitivity

Take an orientably-regular map M; view G=Aut+(M) as a permutation group on the set Ω of vertices of M. This works fine if G has trivial core

  • n Ω, where, for G transitive on Ω (which is our case),

core(G) = {h ∈ G; h(v) = v for each v ∈ Ω} =

(g∈G) g−1Hg ; H = StabG(u) for some u ∈ Ω.

This is always the case when the underlying graph of M is simple. From now on: G=Aut+(M) acts faithfully as a permutation group on Ω. If there is a K ⊳ G such that M/K has at least two vertices, try lifts.

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

slide-59
SLIDE 59

Quasiprimitivity

Take an orientably-regular map M; view G=Aut+(M) as a permutation group on the set Ω of vertices of M. This works fine if G has trivial core

  • n Ω, where, for G transitive on Ω (which is our case),

core(G) = {h ∈ G; h(v) = v for each v ∈ Ω} =

(g∈G) g−1Hg ; H = StabG(u) for some u ∈ Ω.

This is always the case when the underlying graph of M is simple. From now on: G=Aut+(M) acts faithfully as a permutation group on Ω. If there is a K ⊳ G such that M/K has at least two vertices, try lifts. But what if there is no such K ⊳ G ?

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

slide-60
SLIDE 60

Quasiprimitivity

Take an orientably-regular map M; view G=Aut+(M) as a permutation group on the set Ω of vertices of M. This works fine if G has trivial core

  • n Ω, where, for G transitive on Ω (which is our case),

core(G) = {h ∈ G; h(v) = v for each v ∈ Ω} =

(g∈G) g−1Hg ; H = StabG(u) for some u ∈ Ω.

This is always the case when the underlying graph of M is simple. From now on: G=Aut+(M) acts faithfully as a permutation group on Ω. If there is a K ⊳ G such that M/K has at least two vertices, try lifts. But what if there is no such K ⊳ G ? Equivalently, what if every normal subgroup of our permutation group G is transitive on Ω ?

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

slide-61
SLIDE 61

Quasiprimitivity

Take an orientably-regular map M; view G=Aut+(M) as a permutation group on the set Ω of vertices of M. This works fine if G has trivial core

  • n Ω, where, for G transitive on Ω (which is our case),

core(G) = {h ∈ G; h(v) = v for each v ∈ Ω} =

(g∈G) g−1Hg ; H = StabG(u) for some u ∈ Ω.

This is always the case when the underlying graph of M is simple. From now on: G=Aut+(M) acts faithfully as a permutation group on Ω. If there is a K ⊳ G such that M/K has at least two vertices, try lifts. But what if there is no such K ⊳ G ? Equivalently, what if every normal subgroup of our permutation group G is transitive on Ω ? Such permutation groups are known as quasiprimitive.

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

slide-62
SLIDE 62

Quasiprimitivity

Take an orientably-regular map M; view G=Aut+(M) as a permutation group on the set Ω of vertices of M. This works fine if G has trivial core

  • n Ω, where, for G transitive on Ω (which is our case),

core(G) = {h ∈ G; h(v) = v for each v ∈ Ω} =

(g∈G) g−1Hg ; H = StabG(u) for some u ∈ Ω.

This is always the case when the underlying graph of M is simple. From now on: G=Aut+(M) acts faithfully as a permutation group on Ω. If there is a K ⊳ G such that M/K has at least two vertices, try lifts. But what if there is no such K ⊳ G ? Equivalently, what if every normal subgroup of our permutation group G is transitive on Ω ? Such permutation groups are known as quasiprimitive. The corresponding

  • rientably-regular and regular maps may be thought of as ‘irreducible’.

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

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SLIDE 63

The O’Nan-Scott-Praeger Theorem (1979,1993)

Jozef ˇ Sir´ aˇ n OU and STU Vertex-quasiprimitivity in regular maps 27th May 2015 12 / 15

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SLIDE 64

The O’Nan-Scott-Praeger Theorem (1979,1993)

Quasiprimitive permutation groups G on Ω (with all normal subgroups transitive) have N = soc(G) = T k for some simple group T, and:

Jozef ˇ Sir´ aˇ n OU and STU Vertex-quasiprimitivity in regular maps 27th May 2015 12 / 15

slide-65
SLIDE 65

The O’Nan-Scott-Praeger Theorem (1979,1993)

Quasiprimitive permutation groups G on Ω (with all normal subgroups transitive) have N = soc(G) = T k for some simple group T, and:

  • HA: Ω = Z k

p , G < AGL(k, p), StabG(u) irreducible, N = Z k p

Jozef ˇ Sir´ aˇ n OU and STU Vertex-quasiprimitivity in regular maps 27th May 2015 12 / 15

slide-66
SLIDE 66

The O’Nan-Scott-Praeger Theorem (1979,1993)

Quasiprimitive permutation groups G on Ω (with all normal subgroups transitive) have N = soc(G) = T k for some simple group T, and:

  • HA: Ω = Z k

p , G < AGL(k, p), StabG(u) irreducible, N = Z k p

  • HS: Ω = T, k = 2, N = T.Inn(T) < G < T.Aut(T)

Jozef ˇ Sir´ aˇ n OU and STU Vertex-quasiprimitivity in regular maps 27th May 2015 12 / 15

slide-67
SLIDE 67

The O’Nan-Scott-Praeger Theorem (1979,1993)

Quasiprimitive permutation groups G on Ω (with all normal subgroups transitive) have N = soc(G) = T k for some simple group T, and:

  • HA: Ω = Z k

p , G < AGL(k, p), StabG(u) irreducible, N = Z k p

  • HS: Ω = T, k = 2, N = T.Inn(T) < G < T.Aut(T)
  • HC: Ω = T ℓ, k = 2ℓ > 2, N = T ℓ.Inn(T ℓ) < G < T ℓ.Aut(T ℓ)

Jozef ˇ Sir´ aˇ n OU and STU Vertex-quasiprimitivity in regular maps 27th May 2015 12 / 15

slide-68
SLIDE 68

The O’Nan-Scott-Praeger Theorem (1979,1993)

Quasiprimitive permutation groups G on Ω (with all normal subgroups transitive) have N = soc(G) = T k for some simple group T, and:

  • HA: Ω = Z k

p , G < AGL(k, p), StabG(u) irreducible, N = Z k p

  • HS: Ω = T, k = 2, N = T.Inn(T) < G < T.Aut(T)
  • HC: Ω = T ℓ, k = 2ℓ > 2, N = T ℓ.Inn(T ℓ) < G < T ℓ.Aut(T ℓ)
  • AS: k = 1, N = T < G < Aut(T), T transitive on some set Ω

Jozef ˇ Sir´ aˇ n OU and STU Vertex-quasiprimitivity in regular maps 27th May 2015 12 / 15

slide-69
SLIDE 69

The O’Nan-Scott-Praeger Theorem (1979,1993)

Quasiprimitive permutation groups G on Ω (with all normal subgroups transitive) have N = soc(G) = T k for some simple group T, and:

  • HA: Ω = Z k

p , G < AGL(k, p), StabG(u) irreducible, N = Z k p

  • HS: Ω = T, k = 2, N = T.Inn(T) < G < T.Aut(T)
  • HC: Ω = T ℓ, k = 2ℓ > 2, N = T ℓ.Inn(T ℓ) < G < T ℓ.Aut(T ℓ)
  • AS: k = 1, N = T < G < Aut(T), T transitive on some set Ω
  • SD: Ω = N/H, H <dia N, N ⊳ G < N.Out(T) × Sk < SΩ, k ≥ 2

Jozef ˇ Sir´ aˇ n OU and STU Vertex-quasiprimitivity in regular maps 27th May 2015 12 / 15

slide-70
SLIDE 70

The O’Nan-Scott-Praeger Theorem (1979,1993)

Quasiprimitive permutation groups G on Ω (with all normal subgroups transitive) have N = soc(G) = T k for some simple group T, and:

  • HA: Ω = Z k

p , G < AGL(k, p), StabG(u) irreducible, N = Z k p

  • HS: Ω = T, k = 2, N = T.Inn(T) < G < T.Aut(T)
  • HC: Ω = T ℓ, k = 2ℓ > 2, N = T ℓ.Inn(T ℓ) < G < T ℓ.Aut(T ℓ)
  • AS: k = 1, N = T < G < Aut(T), T transitive on some set Ω
  • SD: Ω = N/H, H <dia N, N ⊳ G < N.Out(T) × Sk < SΩ, k ≥ 2
  • CD: Ω = N, G < H wr Sk, H quasiprimitive on T of SD-type, k ≥ 4

Jozef ˇ Sir´ aˇ n OU and STU Vertex-quasiprimitivity in regular maps 27th May 2015 12 / 15

slide-71
SLIDE 71

The O’Nan-Scott-Praeger Theorem (1979,1993)

Quasiprimitive permutation groups G on Ω (with all normal subgroups transitive) have N = soc(G) = T k for some simple group T, and:

  • HA: Ω = Z k

p , G < AGL(k, p), StabG(u) irreducible, N = Z k p

  • HS: Ω = T, k = 2, N = T.Inn(T) < G < T.Aut(T)
  • HC: Ω = T ℓ, k = 2ℓ > 2, N = T ℓ.Inn(T ℓ) < G < T ℓ.Aut(T ℓ)
  • AS: k = 1, N = T < G < Aut(T), T transitive on some set Ω
  • SD: Ω = N/H, H <dia N, N ⊳ G < N.Out(T) × Sk < SΩ, k ≥ 2
  • CD: Ω = N, G < H wr Sk, H quasiprimitive on T of SD-type, k ≥ 4
  • TW: Ω = N, T non-Abelian, G is a ‘twisted wreath product’

Jozef ˇ Sir´ aˇ n OU and STU Vertex-quasiprimitivity in regular maps 27th May 2015 12 / 15

slide-72
SLIDE 72

The O’Nan-Scott-Praeger Theorem (1979,1993)

Quasiprimitive permutation groups G on Ω (with all normal subgroups transitive) have N = soc(G) = T k for some simple group T, and:

  • HA: Ω = Z k

p , G < AGL(k, p), StabG(u) irreducible, N = Z k p

  • HS: Ω = T, k = 2, N = T.Inn(T) < G < T.Aut(T)
  • HC: Ω = T ℓ, k = 2ℓ > 2, N = T ℓ.Inn(T ℓ) < G < T ℓ.Aut(T ℓ)
  • AS: k = 1, N = T < G < Aut(T), T transitive on some set Ω
  • SD: Ω = N/H, H <dia N, N ⊳ G < N.Out(T) × Sk < SΩ, k ≥ 2
  • CD: Ω = N, G < H wr Sk, H quasiprimitive on T of SD-type, k ≥ 4
  • TW: Ω = N, T non-Abelian, G is a ‘twisted wreath product’
  • PA: Ω = N ⊳ G < H wr Sk, T<H<Aut(T), ‘product action’ of G on Ω.

Jozef ˇ Sir´ aˇ n OU and STU Vertex-quasiprimitivity in regular maps 27th May 2015 12 / 15

slide-73
SLIDE 73

The O’Nan-Scott-Praeger Theorem (1979,1993)

Quasiprimitive permutation groups G on Ω (with all normal subgroups transitive) have N = soc(G) = T k for some simple group T, and:

  • HA: Ω = Z k

p , G < AGL(k, p), StabG(u) irreducible, N = Z k p

  • HS: Ω = T, k = 2, N = T.Inn(T) < G < T.Aut(T)
  • HC: Ω = T ℓ, k = 2ℓ > 2, N = T ℓ.Inn(T ℓ) < G < T ℓ.Aut(T ℓ)
  • AS: k = 1, N = T < G < Aut(T), T transitive on some set Ω
  • SD: Ω = N/H, H <dia N, N ⊳ G < N.Out(T) × Sk < SΩ, k ≥ 2
  • CD: Ω = N, G < H wr Sk, H quasiprimitive on T of SD-type, k ≥ 4
  • TW: Ω = N, T non-Abelian, G is a ‘twisted wreath product’
  • PA: Ω = N ⊳ G < H wr Sk, T<H<Aut(T), ‘product action’ of G on Ω.

Lemma [Li, ˇ S, Wang] If G is the automorphism group of a regular or an

  • rientably-regular map, quasiprimitive on V , then G is HA, AS, TW or PA.

Jozef ˇ Sir´ aˇ n OU and STU Vertex-quasiprimitivity in regular maps 27th May 2015 12 / 15

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SLIDE 74

The ‘Holomorph-Abelian’ case: A sample result

Jozef ˇ Sir´ aˇ n OU and STU Vertex-quasiprimitivity in regular maps 27th May 2015 13 / 15

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SLIDE 75

The ‘Holomorph-Abelian’ case: A sample result

This is the only case in which we have a characterisation:

Jozef ˇ Sir´ aˇ n OU and STU Vertex-quasiprimitivity in regular maps 27th May 2015 13 / 15

slide-76
SLIDE 76

The ‘Holomorph-Abelian’ case: A sample result

This is the only case in which we have a characterisation: Theorem [Li, ˇ S, Wang, w. i. p.] Let N = Z d

p for an odd prime p and let

G = N ⋊ H, where H = h ≃ Zk is an irreducible subgroup of AGL(d, p)

Jozef ˇ Sir´ aˇ n OU and STU Vertex-quasiprimitivity in regular maps 27th May 2015 13 / 15

slide-77
SLIDE 77

The ‘Holomorph-Abelian’ case: A sample result

This is the only case in which we have a characterisation: Theorem [Li, ˇ S, Wang, w. i. p.] Let N = Z d

p for an odd prime p and let

G = N ⋊ H, where H = h ≃ Zk is an irreducible subgroup of AGL(d, p) (that is, k is an even primitive divisor of pd − 1)

Jozef ˇ Sir´ aˇ n OU and STU Vertex-quasiprimitivity in regular maps 27th May 2015 13 / 15

slide-78
SLIDE 78

The ‘Holomorph-Abelian’ case: A sample result

This is the only case in which we have a characterisation: Theorem [Li, ˇ S, Wang, w. i. p.] Let N = Z d

p for an odd prime p and let

G = N ⋊ H, where H = h ≃ Zk is an irreducible subgroup of AGL(d, p) (that is, k is an even primitive divisor of pd − 1) such that hk/2 inverts N. Let 1 = g ∈ N and let S = {ghi; i ∈ Zk}, with natural cyclic ordering.

Jozef ˇ Sir´ aˇ n OU and STU Vertex-quasiprimitivity in regular maps 27th May 2015 13 / 15

slide-79
SLIDE 79

The ‘Holomorph-Abelian’ case: A sample result

This is the only case in which we have a characterisation: Theorem [Li, ˇ S, Wang, w. i. p.] Let N = Z d

p for an odd prime p and let

G = N ⋊ H, where H = h ≃ Zk is an irreducible subgroup of AGL(d, p) (that is, k is an even primitive divisor of pd − 1) such that hk/2 inverts N. Let 1 = g ∈ N and let S = {ghi; i ∈ Zk}, with natural cyclic ordering. Then, the Cayley graph Cay(N, S) embeds as an orientably-regular Cayley map;

Jozef ˇ Sir´ aˇ n OU and STU Vertex-quasiprimitivity in regular maps 27th May 2015 13 / 15

slide-80
SLIDE 80

The ‘Holomorph-Abelian’ case: A sample result

This is the only case in which we have a characterisation: Theorem [Li, ˇ S, Wang, w. i. p.] Let N = Z d

p for an odd prime p and let

G = N ⋊ H, where H = h ≃ Zk is an irreducible subgroup of AGL(d, p) (that is, k is an even primitive divisor of pd − 1) such that hk/2 inverts N. Let 1 = g ∈ N and let S = {ghi; i ∈ Zk}, with natural cyclic ordering. Then, the Cayley graph Cay(N, S) embeds as an orientably-regular Cayley map; the map is regular if and only if d = 2e and k divides pe + 1.

Jozef ˇ Sir´ aˇ n OU and STU Vertex-quasiprimitivity in regular maps 27th May 2015 13 / 15

slide-81
SLIDE 81

The ‘Holomorph-Abelian’ case: A sample result

This is the only case in which we have a characterisation: Theorem [Li, ˇ S, Wang, w. i. p.] Let N = Z d

p for an odd prime p and let

G = N ⋊ H, where H = h ≃ Zk is an irreducible subgroup of AGL(d, p) (that is, k is an even primitive divisor of pd − 1) such that hk/2 inverts N. Let 1 = g ∈ N and let S = {ghi; i ∈ Zk}, with natural cyclic ordering. Then, the Cayley graph Cay(N, S) embeds as an orientably-regular Cayley map; the map is regular if and only if d = 2e and k divides pe + 1. Moreover, all (orientably) regular maps with a vertex-quasiprimitive automorphism group of type HA for odd p arise this way.

Jozef ˇ Sir´ aˇ n OU and STU Vertex-quasiprimitivity in regular maps 27th May 2015 13 / 15

slide-82
SLIDE 82

The ‘Holomorph-Abelian’ case: A sample result

This is the only case in which we have a characterisation: Theorem [Li, ˇ S, Wang, w. i. p.] Let N = Z d

p for an odd prime p and let

G = N ⋊ H, where H = h ≃ Zk is an irreducible subgroup of AGL(d, p) (that is, k is an even primitive divisor of pd − 1) such that hk/2 inverts N. Let 1 = g ∈ N and let S = {ghi; i ∈ Zk}, with natural cyclic ordering. Then, the Cayley graph Cay(N, S) embeds as an orientably-regular Cayley map; the map is regular if and only if d = 2e and k divides pe + 1. Moreover, all (orientably) regular maps with a vertex-quasiprimitive automorphism group of type HA for odd p arise this way. Similar results for non-orientable regular maps; modifications for p = 2.

Jozef ˇ Sir´ aˇ n OU and STU Vertex-quasiprimitivity in regular maps 27th May 2015 13 / 15

slide-83
SLIDE 83

The ‘Twisted Wreath Product’ case

Jozef ˇ Sir´ aˇ n OU and STU Vertex-quasiprimitivity in regular maps 27th May 2015 14 / 15

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SLIDE 84

The ‘Twisted Wreath Product’ case

Let a group B have a transitive action on {1, . . . , k}, with S = StabB(1).

Jozef ˇ Sir´ aˇ n OU and STU Vertex-quasiprimitivity in regular maps 27th May 2015 14 / 15

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SLIDE 85

The ‘Twisted Wreath Product’ case

Let a group B have a transitive action on {1, . . . , k}, with S = StabB(1). Let ϕ : S → Aut(T) be such that coreB(ϕ−1(Inn(T)) = {1B}.

Jozef ˇ Sir´ aˇ n OU and STU Vertex-quasiprimitivity in regular maps 27th May 2015 14 / 15

slide-86
SLIDE 86

The ‘Twisted Wreath Product’ case

Let a group B have a transitive action on {1, . . . , k}, with S = StabB(1). Let ϕ : S → Aut(T) be such that coreB(ϕ−1(Inn(T)) = {1B}. Define A := {f : B → T : f (bs) = f (b)ϕ(s) for all b ∈ B, s ∈ S}.

Jozef ˇ Sir´ aˇ n OU and STU Vertex-quasiprimitivity in regular maps 27th May 2015 14 / 15

slide-87
SLIDE 87

The ‘Twisted Wreath Product’ case

Let a group B have a transitive action on {1, . . . , k}, with S = StabB(1). Let ϕ : S → Aut(T) be such that coreB(ϕ−1(Inn(T)) = {1B}. Define A := {f : B → T : f (bs) = f (b)ϕ(s) for all b ∈ B, s ∈ S}. Then A is a group under pointwise multiplication and A ∼ = T k.

Jozef ˇ Sir´ aˇ n OU and STU Vertex-quasiprimitivity in regular maps 27th May 2015 14 / 15

slide-88
SLIDE 88

The ‘Twisted Wreath Product’ case

Let a group B have a transitive action on {1, . . . , k}, with S = StabB(1). Let ϕ : S → Aut(T) be such that coreB(ϕ−1(Inn(T)) = {1B}. Define A := {f : B → T : f (bs) = f (b)ϕ(s) for all b ∈ B, s ∈ S}. Then A is a group under pointwise multiplication and A ∼ = T k. Further, let B act on A by ψ : f b(x) := f (bx) for all b, x ∈ B.

Jozef ˇ Sir´ aˇ n OU and STU Vertex-quasiprimitivity in regular maps 27th May 2015 14 / 15

slide-89
SLIDE 89

The ‘Twisted Wreath Product’ case

Let a group B have a transitive action on {1, . . . , k}, with S = StabB(1). Let ϕ : S → Aut(T) be such that coreB(ϕ−1(Inn(T)) = {1B}. Define A := {f : B → T : f (bs) = f (b)ϕ(s) for all b ∈ B, s ∈ S}. Then A is a group under pointwise multiplication and A ∼ = T k. Further, let B act on A by ψ : f b(x) := f (bx) for all b, x ∈ B. The twisted wreath product of T and B is T twrϕ B := A ⋊ψ B;

Jozef ˇ Sir´ aˇ n OU and STU Vertex-quasiprimitivity in regular maps 27th May 2015 14 / 15

slide-90
SLIDE 90

The ‘Twisted Wreath Product’ case

Let a group B have a transitive action on {1, . . . , k}, with S = StabB(1). Let ϕ : S → Aut(T) be such that coreB(ϕ−1(Inn(T)) = {1B}. Define A := {f : B → T : f (bs) = f (b)ϕ(s) for all b ∈ B, s ∈ S}. Then A is a group under pointwise multiplication and A ∼ = T k. Further, let B act on A by ψ : f b(x) := f (bx) for all b, x ∈ B. The twisted wreath product of T and B is T twrϕ B := A ⋊ψ B; it has a transitive action on Ω=A given by f (g,b) = gf b for f , g ∈ A and b ∈ B.

Jozef ˇ Sir´ aˇ n OU and STU Vertex-quasiprimitivity in regular maps 27th May 2015 14 / 15

slide-91
SLIDE 91

The ‘Twisted Wreath Product’ case

Let a group B have a transitive action on {1, . . . , k}, with S = StabB(1). Let ϕ : S → Aut(T) be such that coreB(ϕ−1(Inn(T)) = {1B}. Define A := {f : B → T : f (bs) = f (b)ϕ(s) for all b ∈ B, s ∈ S}. Then A is a group under pointwise multiplication and A ∼ = T k. Further, let B act on A by ψ : f b(x) := f (bx) for all b, x ∈ B. The twisted wreath product of T and B is T twrϕ B := A ⋊ψ B; it has a transitive action on Ω=A given by f (g,b) = gf b for f , g ∈ A and b ∈ B. A sample of results:

Jozef ˇ Sir´ aˇ n OU and STU Vertex-quasiprimitivity in regular maps 27th May 2015 14 / 15

slide-92
SLIDE 92

The ‘Twisted Wreath Product’ case

Let a group B have a transitive action on {1, . . . , k}, with S = StabB(1). Let ϕ : S → Aut(T) be such that coreB(ϕ−1(Inn(T)) = {1B}. Define A := {f : B → T : f (bs) = f (b)ϕ(s) for all b ∈ B, s ∈ S}. Then A is a group under pointwise multiplication and A ∼ = T k. Further, let B act on A by ψ : f b(x) := f (bx) for all b, x ∈ B. The twisted wreath product of T and B is T twrϕ B := A ⋊ψ B; it has a transitive action on Ω=A given by f (g,b) = gf b for f , g ∈ A and b ∈ B. A sample of results:

  • Every quasiprimitive regular map of type TW is orientable.

Jozef ˇ Sir´ aˇ n OU and STU Vertex-quasiprimitivity in regular maps 27th May 2015 14 / 15

slide-93
SLIDE 93

The ‘Twisted Wreath Product’ case

Let a group B have a transitive action on {1, . . . , k}, with S = StabB(1). Let ϕ : S → Aut(T) be such that coreB(ϕ−1(Inn(T)) = {1B}. Define A := {f : B → T : f (bs) = f (b)ϕ(s) for all b ∈ B, s ∈ S}. Then A is a group under pointwise multiplication and A ∼ = T k. Further, let B act on A by ψ : f b(x) := f (bx) for all b, x ∈ B. The twisted wreath product of T and B is T twrϕ B := A ⋊ψ B; it has a transitive action on Ω=A given by f (g,b) = gf b for f , g ∈ A and b ∈ B. A sample of results:

  • Every quasiprimitive regular map of type TW is orientable.
  • Infinite families of orientably-regular (chiral, if desired) map M with

Aut+(M) ∼ = T twrϕ B, where T = PSL(2, p), B = Z2k, ϕ(z) = z mod k.

Jozef ˇ Sir´ aˇ n OU and STU Vertex-quasiprimitivity in regular maps 27th May 2015 14 / 15

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SLIDE 94

The ‘Product Action’ and ‘Almost Simple’ cases

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

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SLIDE 95

The ‘Product Action’ and ‘Almost Simple’ cases

  • The ‘Product Action’ type (description omitted) splits into the ‘Straight

Diagonal’, ‘Twisted Diagonal’ and ‘Non-Diagonal’ subcases.

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

slide-96
SLIDE 96

The ‘Product Action’ and ‘Almost Simple’ cases

  • The ‘Product Action’ type (description omitted) splits into the ‘Straight

Diagonal’, ‘Twisted Diagonal’ and ‘Non-Diagonal’ subcases. We have infinite families of examples of (orientably-) regular maps in the first two categories (for chiral examples we need to invoke the family of Suzuki simple groups), and a proof that no example exists in the third category.

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

slide-97
SLIDE 97

The ‘Product Action’ and ‘Almost Simple’ cases

  • The ‘Product Action’ type (description omitted) splits into the ‘Straight

Diagonal’, ‘Twisted Diagonal’ and ‘Non-Diagonal’ subcases. We have infinite families of examples of (orientably-) regular maps in the first two categories (for chiral examples we need to invoke the family of Suzuki simple groups), and a proof that no example exists in the third category.

  • The ‘Almost Simple’ type: k=1, N=T<G<Aut(T), T transitive on Ω

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

slide-98
SLIDE 98

The ‘Product Action’ and ‘Almost Simple’ cases

  • The ‘Product Action’ type (description omitted) splits into the ‘Straight

Diagonal’, ‘Twisted Diagonal’ and ‘Non-Diagonal’ subcases. We have infinite families of examples of (orientably-) regular maps in the first two categories (for chiral examples we need to invoke the family of Suzuki simple groups), and a proof that no example exists in the third category.

  • The ‘Almost Simple’ type: k=1, N=T<G<Aut(T), T transitive on Ω

A consequence of a deep result of Malle, Saxl, and Weigel 1991: Every finite simple group is isomorphic to the automorphism group of an

  • rientably regular map.

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

slide-99
SLIDE 99

The ‘Product Action’ and ‘Almost Simple’ cases

  • The ‘Product Action’ type (description omitted) splits into the ‘Straight

Diagonal’, ‘Twisted Diagonal’ and ‘Non-Diagonal’ subcases. We have infinite families of examples of (orientably-) regular maps in the first two categories (for chiral examples we need to invoke the family of Suzuki simple groups), and a proof that no example exists in the third category.

  • The ‘Almost Simple’ type: k=1, N=T<G<Aut(T), T transitive on Ω

A consequence of a deep result of Malle, Saxl, and Weigel 1991: Every finite simple group is isomorphic to the automorphism group of an

  • rientably regular map.

With some exceptions, the map can be assumed to be trivalent;

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

slide-100
SLIDE 100

The ‘Product Action’ and ‘Almost Simple’ cases

  • The ‘Product Action’ type (description omitted) splits into the ‘Straight

Diagonal’, ‘Twisted Diagonal’ and ‘Non-Diagonal’ subcases. We have infinite families of examples of (orientably-) regular maps in the first two categories (for chiral examples we need to invoke the family of Suzuki simple groups), and a proof that no example exists in the third category.

  • The ‘Almost Simple’ type: k=1, N=T<G<Aut(T), T transitive on Ω

A consequence of a deep result of Malle, Saxl, and Weigel 1991: Every finite simple group is isomorphic to the automorphism group of an

  • rientably regular map.

With some exceptions, the map can be assumed to be trivalent; with a larger set of exceptions the result extends to regular maps.

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

slide-101
SLIDE 101

The ‘Product Action’ and ‘Almost Simple’ cases

  • The ‘Product Action’ type (description omitted) splits into the ‘Straight

Diagonal’, ‘Twisted Diagonal’ and ‘Non-Diagonal’ subcases. We have infinite families of examples of (orientably-) regular maps in the first two categories (for chiral examples we need to invoke the family of Suzuki simple groups), and a proof that no example exists in the third category.

  • The ‘Almost Simple’ type: k=1, N=T<G<Aut(T), T transitive on Ω

A consequence of a deep result of Malle, Saxl, and Weigel 1991: Every finite simple group is isomorphic to the automorphism group of an

  • rientably regular map.

With some exceptions, the map can be assumed to be trivalent; with a larger set of exceptions the result extends to regular maps. THANK YOU.

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