Overview
QP-actions
Dessins d’Enfant
QP actions and dessins of PSL2 (q)
Quasi-platonic actions of PSL2 (q) and their
dessins
Preliminary report
S. Allen Broughton - Rose-Hulman Institute of Technology
East Lansing AMS meeting, March 14, 2015
Overview
QP-actions
Dessins d’Enfant
QP actions and dessins of PSL2 (q)
Overview
Quasi-Platonic surfaces and actions.
Dessins of QP actions and Galois action.
QP actions and the dessins of PSL2 (q)
Why PSL2 (q)? - simple group, many low genus actions,
easy calculations
Overview
QP-actions
Dessins d’Enfant
QP actions and dessins of PSL2 (q)
Conformal actions
Conformal actions
The finite group G acts conformally on the closed, orientable
Riemann surface S if there is a monomorphism:
: G → Aut(S),
the conformal automorphism group of S.
Example
Dihedral action on the sphere:
S = P 1 (C),
G = Dn = ha, b : a2 = bn = 1, aba = b−1 i,
G-action given by a : z → 1/z and b : z → ωz,
ω = exp( 2πi
n ).
Overview
QP-actions
Dessins d’Enfant
QP actions and dessins of PSL2 (q)
Conformal actions
Quasi-platonic actions - 1
Quasi-platonic actions satisfy:
The quotient surface has genus zero: S/G w P 1 (C).
The quotient map β : S → S/G is a meromorphic function:
β : S → P 1 (C)
which is ramified over three points, say {0, 1, ∞}.
β is called a Belyi function.
Example
Belyi function for the dihedral action:
β(z) =
z n + z −n + 2
4
Overview
QP-actions
Dessins d’Enfant
QP actions and dessins of PSL2 (q)
Conformal actions
Quasi-platonic actions - 2
For all w ∈ β −1 (0), w ∈ β −1 (1), w ∈ β −1 (∞) the local
degree of β at w has a common order l, m, n respectively.
The stabilizer Gw at w ∈ β −1 (0), w ∈ β −1 (1), w ∈ β −1 (∞)
is cyclic of order l, m, n respectively.
If S has genus σ then Riemann-Hurwitz equation becomes:
2σ − 2
1
1
1
=1− −
− .
|G|
l
m n
The triple (l, m, n) is called the signature of the action
: G → Aut(S).
Overview
QP-actions
Dessins d’Enfant
QP actions and dessins of PSL2 (q)
Covering groups
Covering by triangle groups - 1
An (l, m, n) triangle group is a Fuchsian group with
presentation
Tl,m,n = hA, B, C|Al = B m = C n = ABC = 1i
A, B, C are clockwise hyperbolic rotations through angles
2π 2π
of 2π
l , m , n respectively, at the vertices of a hyperbolic
2π 2π
triangle with angles 2π
l , m, n .
Overview
QP-actions
Dessins d’Enfant
QP actions and dessins of PSL2 (q)
Covering groups
Covering by triangle groups - 2
Given a quasi-platonic action of G on S, there is a triangle
group ∆, containing a torsion free Fuchsian group Π, such that:
Π E ∆, and S w H/Π and
G acts on S w H/Π via an epimorphism
η
Π ,→ ∆ G
such that η : ∆/Π ↔ G is the inverse of : G → Aut(S),
upon identifying S w H/Π.
η is called a surface kernel epimorphism.
Overview
QP-actions
Dessins d’Enfant
QP actions and dessins of PSL2 (q)
Covering groups
Covering by triangle groups - 2
Given ∆ = Tl,m,n = hA, B, Ci η : ∆ → G, let
a = η(A), b = η(B), c = η(C).
The triple (a, b, c) is called a generating (l, m, n)-triple of G.
The generating triple satisfies:
G = ha, b, ci
(1)
o(a) = l, o(b) = m, o(c) = n
(2)
abc = 1
(3)
Overview
QP-actions
Dessins d’Enfant
QP actions and dessins of PSL2 (q)
Equivalence of actions
Equivalent epimorphisms and actions
The surface-kernel epimorphisms of Tl,m,n , and hence
quasi-platonic G-actions, are in 1 − 1 correspondence to
the generating (l, m, n) triples of G.
Two G-actions 1 , 2 : G → Aut(S) are called algebraically
equivalent if 2 = 1 ◦ ω for some ω ∈ Aut(G).
The associated generating triples satisfy
(a2 , b2 , c2 ) = (ω(a1 ), ω(b1 ), ω(c1 )).
For this talk we gloss over the slightly finer notion of
conformal and topological equivalence of actions.
Overview
QP-actions
Dessins d’Enfant
QP actions and dessins of PSL2 (q)
Equivalence of actions
Algebraic classes of actions
For groups such as PSL2 (q), and for many signatures
(l, m, n) there are too many algebraic equivalence classes
of generating triples for a clean analysis.
We construct better partitions of the generating triples by
using an “approximate automorphism group” L satisfying
A = Aut(G) ⊇ L ⊇ Inn(G) = K .
Set g L = {ω(g) : ω ∈ L} and define
LG (a, b, c) = {(x, y , z) : x ∈ aL , y ∈ bL , x ∈ c L ) : xyz = 1}
L◦G (a, b, c) = {(x, y , z) ∈ LG (a, b, c) : G = hx, y , zi}
The two sets above often have nice enumeration formulas.
Each L◦G (a, b, c) is a union a small number of L classes,
upon each of which L acts freely.
Overview
QP-actions
Dessins d’Enfant
QP actions and dessins of PSL2 (q)
Tilings and dessins
Example: Icosahedral triangular tiling and dessins
Describe G-classes and quotient.
Show the three dessins: dodecahedral, icosahedral, and
rhombic.
Overview
QP-actions
Dessins d’Enfant
QP actions and dessins of PSL2 (q)
Tilings and dessins
The standard regular dessin d’enfant induced by G
The dessin d’enfant (dessin) of a Belyi function is
D = β −1 ([0, 1]).
G acts simply transitively on the edges of D.
D is a bipartite graph in S whose complement is a union of
congruent dihedrally symmetric polygons.
The polygons are regular polygons if a boundary edge has
a vertex of order 2. This is a clean dessin.
The tiling or map on S defined by the dessin is the origin of
terminology platonic and quasi-platonic action.
Overview
QP-actions
Dessins d’Enfant
QP actions and dessins of PSL2 (q)
Galois actions on dessins and QP actions
Belyi’s Theorem and Galois action
Belyi’s Theorem: A quasi-platonic surface is defined over a
number field.
An element θ ∈ Gal(C) determines a new surface S θ by
acting on the coefficients of a defining equation of S.
Applying θ pointwise induces a bijection
θS : S → S θ .
There is an action of Gal(Q) ⊂ Gal(C) on quasi-platonic
surfaces and hence their dessins.
Defining equations are hard to find so we look at the action
of Gal(C) on generating triples.
Overview
QP-actions
Dessins d’Enfant
QP actions and dessins of PSL2 (q)
Galois actions on dessins and QP actions
Galois action - 1
Given a G action : G → Aut(S), the action may be
transferred to S θ via θS
θ (g) = θS ◦ (g) ◦ θS−1
The quotient maps β : S → S/G = P 1 and
β θ : S θ → S θ /G = P 1 satisfy:
S
↓β
θ
S
−→
θ
1
Sθ
↓ βθ
P
P 1 (C) −→
P 1 (C)
Overview
QP-actions
Dessins d’Enfant
QP actions and dessins of PSL2 (q)
Galois actions on dessins and QP actions
Galois action - 2
The θ action is quasi-platonic and has the same signature
as the action.
θ
S
The bijection S −→
S θ maps β −1 (0, 1, ∞) = vertices of
tiling on S to (β θ )−1 (0, 1, ∞) = vertices of tiling on S θ .
However, θS does not map the edges and faces of the
dessin on S to those on S θ .
Overview
QP-actions
Dessins d’Enfant
QP actions and dessins of PSL2 (q)
Galois actions on dessins and QP actions
Galois action - 3
If g ∈ G fixes w ∈ β −1 (0) by (g) then g acts at w as a
local rotation by ζ an l’th root of unity. Call ζ the rotation
number.
But g also fixes θ(w) ∈ (β θ )−1 (0) via θ (g) and g acts at
θ(w) as a local rotation by θ(ζ).
Similar remarks apply to w ∈ β −1 (1) and w ∈ β −1 (∞).
Let N = lcm(l, m, n) then θ acts on the Nth roots of unity,
and hence the rotation numbers, by ζ → ζ s for an s
relatively prime to N.
Overview
QP-actions
Dessins d’Enfant
QP actions and dessins of PSL2 (q)
Galois actions on dessins and QP actions
Galois action on generating triples
Theorem
Let notation be as above and let (a, b, c) be a generating
(l, m, n)-triple for the (G) action on S, and select t so that
st = 1 mod N. Then there are x, y , z ∈ G such that a generating
triple for the θ (G) action on S θ is
(a0 , b0 , c 0 ) = (xat x −1 , ybt y −1 , zc t z −1 ).
Remark
The Galois action on QP-actions maps the set L◦G (a, b, c) to
L◦G (at , bt , c t ). If L◦G (a, b, c) consists of several L-orbits then
more information is needed to specify the Galois action.
Overview
QP-actions
Dessins d’Enfant
QP actions and dessins of PSL2 (q)
Facts about PSL2 (q)
PSL2 (q) and its Automorphisms
For q = pe , p a prime, Fq is the finite field of order q.
PSL2 (q) = PSL2 (Fq ) the projective special linear group.
PGL2 (q) = PGL2 (Fq ) the projective general linear group.
PGL2 (q) acts on PSL2 (q) by matrix conjugation. Call these
automorphisms geometric automorphisms.
Gal(Fq ) acts on PSL2 (q) by action on the matrix entries.
Call these automorphisms Galois automorphisms. Gal(Fq )
is cyclic of order e.
Aut(PSL2 (q) = Gal(Fq ) n PGL2 (q)
For the remainder of this let G denote PSL2 (q) and PG denote
PGL2 (q)
Overview
QP-actions
Dessins d’Enfant
QP actions and dessins of PSL2 (q)
Facts about PSL2 (q)
Subgroups of PSL2 (q)
L.E. Dickson classified all the proper subgroups of PSL2 (q),
into three types.
Affine: Irrelevant for our discussion, but includes all cyclic
subgroups.
Projective: PSL(2, q 0 ) and PGL(2, q 0 ) for Fq 0 ⊂ Fq with
certain restrictions.
Exceptional - dihedral, tetrahedral, octahedral, and
icosahedral groups of the sphere. These are mostly
irrelevant.
Overview
QP-actions
Dessins d’Enfant
QP actions and dessins of PSL2 (q)
QP actions of PSL2 (q)
Lifting triples to the covering group - 1
There is an exact covering sequence.
h±1i ,→ SL2 (q) PSL2 (q)
For an (l, m, n)-triple (a, b, c) of PSL2 (q), let (A, B, C) be a
triple in SL2 (q), with ABC = I, projecting to (a, b, c).
Let
α = trace(A), β = trace(B), γ = trace(C)
then (A, B, C) lies in
Tr (α, β, γ) = {(A, B, C) ∈ (SL2 (q))3 : ABC = I,
trace(A) = α, trace(B) = β, trace(C) = γ}
Overview
QP-actions
Dessins d’Enfant
QP actions and dessins of PSL2 (q)
QP actions of PSL2 (q)
Lifting triples to the covering group - 2
The union of the (potential) eight sets Tr (±α, ±β, ±γ) map
onto PGG (a, b, c)
The four classes Tr (α, β, γ), Tr (α, −β, −γ), Tr (−α, β, −γ),
Tr (−α, −β, γ) have the same PG-invariant image in
PGG (a, b, c).
Likewise for the remaining four classes.
So PGG (a, b, c) could contain two distinct PG-classes of
generating triples, but not more. These are called fused
classes.
Overview
QP-actions
Dessins d’Enfant
QP actions and dessins of PSL2 (q)
QP actions of PSL2 (q)
Lifting triples to the covering group - 2
For (A, B, C) ∈ Tr (α, β, γ) set
0 −1
β−x
A=
,B =
1 α
z
y
x
Then
x 2 + αxy + y 2 − βx − γy + 1 = 0
giving a solution of the form (x, y , 1) of
Q(x, y , z) = x 2 + y 2 + z 2 + αxy − βxz − γyz = 0
The discriminant of the form above is
DQ(α, β, γ) = α2 + β 2 + γ 2 − αβγ − 4.
Overview
QP-actions
Dessins d’Enfant
QP actions and dessins of PSL2 (q)
QP actions of PSL2 (q)
Macbeath’s results
Macbeath proves the following, suitably paraphrased for our
context
The subgroup hA, B, Ci is affine (hence proper) if and only
if (α, β, γ) is singular, i.e.,
DQ(α, β, γ) = α2 + β 2 + γ 2 − αβγ − 4 = 0.
If a triple (α, β, γ) is non-singular then
|Tr (α, β, γ)| = |PGL2 (q)|
Overview
QP-actions
Dessins d’Enfant
QP actions and dessins of PSL2 (q)
QP actions of PSL2 (q)
Admissible trace triples
Definition
Let (A, B, C) ∈ SL2 (q)3 and (a, b, c) ∈ PSL2 (q)3 be the
projection. Let (α, β, γ) be the traces of (A, B, C) and (l, m, n)
be the orders of (a, b, c). Then (α, β, γ) is admissible if
(α, β, γ) is non-singular i.e. DQ(α, β, γ) 6= 0.
(a, b, c) is hyperbolic i.e., (l, m, n) 6∈
{(2, 2, d), (2, 3, 3), (2, 3, 4), (2, 3, 5), (2, 3, 6), (3, 3, 3)}
Overview
QP-actions
Dessins d’Enfant
QP actions and dessins of PSL2 (q)
QP actions of PSL2 (q)
Admissible trace triples - generated subgroups
The table below tells us when an admissible triple may or may
not yield a generating triple.
Subgroup type
Projective
Exceptional
Subgroup
PSL2 (q)
PSL2 (q 0 ), PGL2 (q 0 )
Octahedral
Exceptional
Icosahedral
Note
Fp (α, β, γ) = Fq
Fp (α, β, γ) = Fq 0 ⊂ Fq
(l, m, n) = (3, 3, 4),
(3, 4, 4), (4, 4, 4)
(l, m, n) = (2, 5, 5),
(3, 3, 5), (3, 5, 5), (5, 5, 5)
Overview
QP-actions
Dessins d’Enfant
QP actions and dessins of PSL2 (q)
results
Specific results
Show tables of results
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QP-actions
Dessins d’Enfant
QP actions and dessins of PSL2 (q)
done
Any Questions?
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QP-actions
Dessins d’Enfant
QP actions and dessins of PSL2 (q)
done
References
L.E. Dickson, Linear groups: With an exposition of the
Galois field theory. (1901).
A.M. Macbeath, Generators of the Linear Fractional
Groups, Proc. Symp. Pure Math. Vol. XII, Amer. Math.
Soc. (1969), pp. 14–32.
H. Glover & D. Sjerve, Representing PSL2 (p) on a Surface
of Least Genus, L’Enseignement Mathématique, Vol. 31
(1985), pp. 305–325.
H. Glover & D. Sjerve, The Genus of PSL2 (q), J. reine
angew. Math, Vol. 380 (1987), pp. 59–86.