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Subdirect Products of M* Groups. Coy L. May and Jay Zimmerman We are interested in groups acting as motions of compact surfaces, with and without boundary. Restrictions on the Order. A compact surface with genus g 2 has at most 84(g – 1) automorphisms by Hurwitz Theorem. - PowerPoint PPT Presentation
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Subdirect Products of M* Groups
Coy L. May and Jay Zimmerman We are interested in groups acting
as motions of compact surfaces, with and without boundary.
Restrictions on the Order A compact surface with genus g 2 has at
most 84(g – 1) automorphisms by Hurwitz Theorem.
If only automorphisms which preserve the orientation of the surface are considered, then the bound becomes 42(g – 1).
Bordered Klein Surfaces A compact bordered Klein surface of
genus g 2 has at most 12(g – 1) automorphisms.
A bordered surface for which the bound is attained is said to have maximal symmetry and its group is called an M* group.
M* group properties Let Γ be the group generated by t, u and v,
with relators t2, u2, v2, (tu)2, (tv)3. A finite group G is an M* group if and
only if G is the image of Γ. If G is an M* group, the order of the
element uv is called an action index of G and is denoted q = o(uv).
Fundamental result G is the automorphism group of a bordered
Klein surface X with maximal symmetry and k boundary components,
where |G| = 2qk iff G is an M* group. Each component of the boundary X
is fixed by a dihedral subgroup of G of order 2q.
Canonical Subgroups of G
G+ = tu, uv and G' = tv, tutvtu. G' ≤ G+ ≤ G, where each subgroup
has index 1 or 2 in the larger group. X is orientable iff [G : G+] = 2. G/G' is the image of Z2 × Z2.
Subdirect Product Let G and H be M* groups. So : Γ G and : Γ H. Define : Γ G × H by (x) =
((x), (x)). L = Im() is a subdirect product
and an M* group.
Normal Subgroup of G Define G() = (ker()) and H()
= (ker()). G() is a normal subgroup of G.
H() is a normal subgroup of H.
G() × {1} = Im() (G × {1})
Index of the subdirect product
|G / G()| = [G × H : L] = |H / H()|.
G / G() Γ/(ker()ker()) H / H().
Obvious Consequences Suppose that H is a simple group. Then H() is either {1} or H.
If H() = 1, then G / G() H.
If H() = H, then L = G × H.
Action Indices Let G and H be M* groups with action
indices q and r and let d = gcd(q, r). For 1 d 5, then G / G() is the
image of Z2, D6, S4, Z2× S4 or Z2× A5, respectively.
If G or H is perfect and 1 d 4, then L = G × H.
G / G' H / H' Z2
Let G and H be M* groups If ker() ker()Γ', then
[G × H : L] 2. If ker() ker()Γ', then
G / G() is perfect.
G / G' H / H' Z2
Suppose that the only quotients of G and H that are isomorphic are abelian.
If ker() ker()Γ', then
[G × H : L] = 2. If ker() ker()Γ', then
L = G × H.
G / G' Z2 and H / H' Z4
[G × H : L] 2. Suppose that the only quotients
of G and H that are isomorphic are abelian.
[G × H : L] = 2.
G / G' H / H' Z4
[G × H : L] 4. Suppose that the only quotients
of G and H that are isomorphic are abelian.
[G × H : L] = 4.
Necessary Conditions The M* group L is a subdirect product
of two smaller M* groups iff L has normal subgroups J1 and J2 such that
[L : J1] > 6, [L : J2] > 6
and J1 J1 = 1.
Corollary Let L be an M* group with |L| >
12 and its Fitting subgroup F(L) divisible by two prime numbers.
Then L is a subdirect product of two smaller M* groups.
Conclusion These techniques can be used with many
different maximal actions, such as Hurwitz groups, odd order groups acting
maximally on Riemann surfaces, p-groups acting similarly.
Finally, I would like to draw some group actions on Riemann surfaces.
Burnside Burnside 1911 talked about
actions on compact surfaces. He even gave a picture of the
action of the Quaternion Group on a surface of Genus 2.
Quaternion Group Properties The surface has genus
2 and 16 region. Each vertex has
degree 8, corresponding to a rotation of order 4.
Image of Triangle Group, T(4,4,4).
Highly symmetric.
Dicyclic Group of Order 12
Quasiabelian Group of Order 16
Orientation Reversing Actions
Suppose that G acts on a surface with orientation reversing elements and G+ is the image of a triangle group.
Therefore, G is the image of either a Full Triangle group or of a Hybrid Triangle group.
The group, P48 of order 48.
P48 u, v | u3 = v2 = (uv)3(u-1v)3 = 1 P48 has symmetric genus 2.
It is the image of HT(3,4) which is a subgroup of FT(3,8,2).
The hyperbolic space region is distorted into a polygonal region.
Polygonal Representation of P48