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Riemann Surfaces: A playground for analysis,
topology, geometry, group theory,
and Galois theory
S. Allen Broughton
Department of Mathematics
Rose-Hulman Institute of Technology
March 9, 2016
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Introduction
Thank in advance!
Thanks to the MAI department and, especially,
Professor Izquierdo for hosting my stay in Linköping.
Thanks for your past hospitality during my previous visits.
Thanks to all of you for attending!
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Introduction
Overview -1
This talk gives a background for Riemann surfaces, especially
the joint research Professor Izquierdo and I will be undertaking.
Riemann surfaces touch upon many areas of mathematics. For
example:
Analysis - hyperelliptic, superelliptic, and abelian integrals,
Teichmüller and moduli space;
Topology - structure of Riemann surfaces, the mapping
class group, and moduli space;
Geometry - Fuchsian groups and hyperbolic geometry;
Group Theory - automorphisms of surfaces; and
Galois Theory - Belyi’s theorem and dessins d’enfants.
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Introduction
Overview
This talk gives is broken up into two main parts:
analysis and topology - the first two sections; and
geometry and group theory - the last two sections.
The transition between the two is to use geometry to abstract
analytical and topological ideas into group theory computations.
Needless to say, we will skip lots of detail and focus on special
cases that illustrate the main points.
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Analysis and topology
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Geometry and Group Theory
Galois Theory
Elliptic, super-elliptic and abelian integrals
Integrals from arclength - 1
In elementary calculus, one of the early integrals we teach
students is arclength on the circle
Z x
dy
√
arcsin(x) =
.
1 − u2
0
A bit later, when we look at arclength on the ellipse,
v 2 /a2 + u 2 = 1, we encounter
Z xp
1 + (a2 − 1)u 2
√
La (x) =
du.
1 − u2
0
This integral is not elementary and so is typically computed
numerically.
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Elliptic, super-elliptic and abelian integrals
Integrals from arclength - 2
Even later in our curriculum we look at the integrals as
functions of a complex variable.
Z z
du
√
arcsin(z) =
1 − u2
0
Z zp
1 + (a2 − 1)u 2
√
La (z) =
du
1 − u2
0
But there are issues: the integrand is multi-valued,
there are singularities in the integrand, and the value of the
integral depends on the homotopy class of the path of
integration from 0 to z.
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Elliptic, super-elliptic and abelian integrals
Integrals from arclength - 3
A method to resolve the issues is to use branch cuts so
that:
the integrand is single valued;
we avoid singularities of the integrand; and,
we avoid encircling singularities with loops.
We will pick the branch cuts so that the remainder of the
complex plane (cut plane) is simply connected.
With this choice, the value of the integral is independent of
the path.
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Elliptic, super-elliptic and abelian integrals
Integrals from arclength - 4
Here are the branch cuts and an integration path for
Z z
du
√
.
arcsin(z) =
1 − u2
0
Galois Theory
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Elliptic, super-elliptic and abelian integrals
Integrals from arclength - 5
√
For a = 2, here are the branch cuts and an integration path
for arclength of an ellipse:
Z z√
1 + u2
√
√
du.
L 2 (z) =
1 − u2
0
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Elliptic, super-elliptic and abelian integrals
Abelian integrals - 1
The preceding discussion can be extended to abelian integrals
of algebraic functions that were studied by Abel and others.
Given a rational function R(u, v ), we consider integrals of the
form
Z z
R(u, v )du
I(z) =
z0
such that:
v is a locally defined function that satisfies
F (u, v ) = r0 (u)v n + r1 (u)v n−1 + . . . + rn (u) = 0
for some rational functions ri (u); and,
the path of integration
γ(t), 0 ≤ t ≤ 1, γ(0) = z0 , γ(1) = z
does not pass through any singularities of the integrand.
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Elliptic, super-elliptic and abelian integrals
Abelian integrals - 2
Here are several classes of such integrals:
if
F (u, v ) = v 2 − f (u)
where f (x) is a polynomial of degree 3 or 4, then I(z) is
called elliptic;
if
F (u, v ) = v 2 − r (u)
where r (x) is a rational function of “higher degree”, then
I(z) is called hyper-elliptic; and,
If p is a prime
F (u, v ) = v p − r (u)
where r (u), is a rational function then I(z) is called
super-elliptic.
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Elliptic, super-elliptic and abelian integrals
Abelian integrals - 3
Frequently, F (u, v ) can be transformed into a nicer format.
Example
In the ellipse arclength example:
R(u, v ) = v , F (u, v ) = v 2 −
1 + (a2 − 1)u 2
,
1 − u2
set v1 = 1 − u 2 v , then we get an elliptic integral:
Z
z
Z
z
R(u, v )du =
z0
R1 (u, v1 )du
z0
R1 (u, v1 ) =
v1
1 − u2
F (u, v1 ) = v12 − 1 + (a2 − 1)u 2
1 − u2
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Transition to Riemann surfaces and topology
Transition to a Riemann surface - 1
We will be guided by the super-elliptic case:
r
Y
F (u, v ) = v − f (u) = v − (u − ai )ti ,
p
p
(1)
i=1
where ai , ti , and t = t1 + · · · + tr = deg(f ) satisfy these
conditions
the ai are distinct complex numbers;
0 < ti < p; and
p divides t.
The p
ai are called branch points.
v = p f (u) is a multi-valued, locally holomorphic function,
except at the branch points.
Two local solutions v1 , v2 are related by multiplying by a
p’th root of unity, ω: v2 (u) = ωv2 (u), which valid in a small
disc not meeting the branch points.
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Transition to Riemann surfaces and topology
Transition to a Riemann surface - 2
Make some branch cuts that satisfy:
the cuts are non-intersecting paths that connect branch
points to infinity,
the remainder of the complex plane (cut plane) is simply
connected.
Now, in our abelian integral we have
the integrand is single valued though discontinuous along
the branch cuts;
we avoid singularities in the integrand; and,
we avoid encircling any collection of branch points with a
loop.
The function v and, hence, the integrand can be analytically
continued across a branch cut, away from singularities.
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Transition to Riemann surfaces and topology
Transition to a Riemann surface - 3
We now have:
p copies of the cut plane with the same branch cuts;
p different integrals with different integrands related by
Rj (u, vj ) = Ri (u, ωi,j vi )
for a suitable root of unity ωi,j ; and,
for an arbitrary path that crosses the branch cuts, we have
to work with each piece individually in each of the cut
planes.
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Transition to Riemann surfaces and topology
Transition to a Riemann surface - 4
A simpler approach to abelian integrals is to work with the
closed Riemann surface S defined by F .
S is a compact, one dimensional complex manifold.
S is sphere with σ handles attached (σ = 2 pictured).
Surfaces of the same genus σ are homeomorphic.
Surfaces are conformally equivalent if there is a
biholomorphic map h : S1 → S2 .
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Transition to Riemann surfaces and topology
Transition to a Riemann surface - 5
Here is an outline of the “complex algebraic curve” construction
of the Riemann surface of an algebraic function defined by
F (u, v ) = 0:
define
S ◦ ⊂ C × C = {(u, v ) : F (u, v ) = 0} ;
compactify S ◦ by adding points at infinity; and,
resolve any singularities (non-manifold points).
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Transition to Riemann surfaces and topology
Transition to a Riemann surface - 6
Alternatively, here is a topological construction using branch
cuts:
Define the branch set, B, of F as the set of a ∈ C such that
r0 (a)v n + r1 (a)v n−1 + . . . + rn (a) = 0
has fewer than n roots.
Construct n different cut planes each with a different
branch of v defined on the cut plane.
Using analytic continuation, glue the cut planes along the
branch cuts so that v is continuous. Note that different
sides of a branch cut may get glued to different cut planes.
Compactify the resulting surface by adding points at infinity
and any points corresponding to branch points.
Resolve any singularities.
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Transition to Riemann surfaces and topology
Abelian integrals on a Riemann surface - 1
Here is where we stand.
We constructed a Riemann surface S with two important
meromorphic functions. Using the algebraic curve
construction we define:
b (u, v ) → u,
u : S → C,
b (u, v ) → v .
v : S → C,
The branch cut construction dissects our surface into n
polygons. The boundary arcs of the polygon connect
points lying over infinity to points lying over the branch
points. The function u lifts the base cut plane bijectively to
the polygons.
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Transition to Riemann surfaces and topology
Abelian integrals on a Riemann surface - 2
The field of meromorphic functions on S, C(S) = C(u, v ),
is a degree n extension of the rational function field C(u).
R(u, v ) is a meromorphic function on S and, so, the
abelian integral
Z
z
R(u, v )du
z0
is the same as the integral of the meromorphic differential
form R(u, v )du on a path lifted to S.
The full power of periods of differential forms and the
Jacobian variety J(S) can be brought to bear on Abelian
integrals. Indeed, the Jacobian was invented for this very
purpose.
Due to a lack of time, we will skip over this fascinating
subject.
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Moduli
Moduli of integrals and surfaces
We briefly discuss moduli space, focusing on the super-elliptic
case.
On an ellipse the integral of arclength depends on a
parameter a.
Z xp
1 + (a2 − 1)u 2
√
La (x) =
du
1 − u2
0
The corresponding Riemann surfaces Sa also depends on
the parameter a.
The total arclength of the ellipse is (use Maple)
Z 1 p
p
1 + (a2 − 1)u 2
√
du = 4EllipticE
a2 − 1
2
1 − u2
−1
The arclength corresponds to an abelian integral over a
closed loop that exists in all the surfaces Sa , and so is a
measure of the conformal type of Sa .
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Moduli
Moduli space
For genus σ > 1 a typical surface is “determined” by 3σ − 3
complex parameters or moduli.
There are many candidates for the moduli.
The moduli space Mσ is the set of all conformal
equivalence classes of surfaces of genus σ.
The moduli can be used to coordinatize Mσ .
Mσ is a complex algebraic variety of dimension 3σ − 3,
with a complicated singularity set Bσ (discussed next
frame).
The moduli space is intensely and widely studied, but we
need to skip the discussion except in the next two frames.
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Moduli
Strata in the branch locus - 1
For large enough σ, the singularity set Bσ consists of those
surfaces with non-trivial automorphisms and is called the
branch locus.
The branch locus consists of various equisymmetric strata
corresponding to the finitely many possible automorphism
groups.
Considerable work on the branch locus (especially
super-elliptic curves) has been done by M. Izquierdo, A.
Costa, G. Bartolini, H. Parlier, as well as others.
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Strata in the branch locus - 2
Often we can construct families of surfaces with
automorphisms directly.
Families of super-elliptic surfaces are pervasive, with
equation and genus,
p
v =
r
Y
i=1
(u − ai )ti , σ =
(p − 1)(r − 2)
.
2
These families determine strata of dimension r − 3. Of
special interest is the hyperelliptic locus with p = 2.
The numerous images of the super-elliptic families form
major pieces of the branch locus.
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Strata in the branch locus - 2
Problem
Here are some research problems:
Describe as completely as possible the strata
corresponding to the super-elliptic surfaces.
Relax the condition that p is a prime.
Describe the strata for a general group of automorphisms.
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Surfaces as geometric quotients
The three geometries
Another approach to Riemann surfaces is through geometry
and group theory. This is an effective approach for:
constructing surfaces; and
working with automorphisms of surfaces.
There are three geometries to consider:
spherical - genus 0 - based on the Riemann sphere S2 ;
euclidian - genus 1 - based on the complex plane C; and,
hyperbolic - genus 2 or greater - based on the hyperbolic
plane H.
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Surfaces as geometric quotients
Torus as a quotient - 1
First we work out a simple example: a torus. Consider the
parallelogram P below and the two transformations:
T1 (x, y ) = (x + 1, y ), or T1 (z) = z + 1
T2 (x, y ) = (x + 1/2, y + 1), or T2 (z) = z + 1/2 + i
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Surfaces as geometric quotients
Torus as a quotient - 2
The two transformations generate a group Π = hT1 , T2 i of
conformal automorphisms of C such that:
the Π-transforms of P tile C;
T1 maps the left edge of P to the right edge of P; and,
T2 maps the bottom edge of P to the top edge of P.
The transforms T1 , T2 are called side pairing
transformations.
The group Π acts without fixed points on C.
The quotient space C/Π is a (non-standard) torus.
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Geometry and Group Theory
Surfaces as geometric quotients
Hyperbolic geometry
Explain H via the picture.
4−4−3 tiling
Galois Theory
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Surfaces as geometric quotients
Automorphisms of hyperbolic geometry
Using the Poincaré disc model for hyperbolic geometry, the
isometries are:
reflection or inversion in a circle - anti-conformal;
hyperbolic - translation with axis - conformal - two fixed
points on the boundary;
elliptic - rotation (typically of finite order) - conformal - one
fixed point in the disc the other outside the disc; and
parabolic - conformal - one fixed point on the boundary of
the disc.
Every conformal automorphism is a linear fractional
transformation and has the form
az + b
T (z) =
.
bz + a
Every conformal automorphism is product of two
reflections.
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Surfaces as geometric quotients
Higher genus surfaces as a quotient - 1
The central octagon is a fundamental polygon for a genus 2
surface.
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Surfaces as geometric quotients
Higher genus surfaces as a quotient - 2
For a surface S of genus σ ≥ 2, we have the following:
a convex, hyperbolic, fundamental polygon P with 4σ sides;
2σ side pairing transformations T1 , . . . , T2σ of P;
the group Π = hT1 , . . . , T2σ i acts without fixed points on H;
the Π-transforms of P tile H; and,
the quotient space H/Π is conformally equivalent to S.
The group Π is called a Fuchsian surface group.
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Automorphism groups as quotients
Automorphism groups of surfaces
An automorphism of a Riemann surface is a conformal
homeomorphism of the surface.
The automorphism group Aut(S) is finite and
| Aut(S)| ≤ 84(σ − 1).
For a super-elliptic surface, Aut(S) contains a cyclic group
of order p of this format:
(u, v ) → (u, ωv ), ω p = 1
since
F (u, ωv ) = (ωv )p − f (u) = v p − f (u) = F (u, v )
Otherwise, automorphisms of a generic Riemann surface
are hard to find.
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Automorphism groups as quotients
Start with the automorphism group! - 1
To find a surface with automorphisms, we do the following:
find a Fuchsian group Γ with a nice generating system;
determine a suitable, finite index, normal, surface group
(no elliptic elements)
Π C Γ;
then, the finite group Γ/Π acts naturally as a group of
automorphisms on the surface S ' H/Π.
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Automorphism groups as quotients
Start with the automorphism group! - 2
Here is how to construct a Γ and a Π.
Pick an r -sided convex polygon P in H such that
successive counter-clockwise edges ei , ei+1 meet an angle
of nπi .
Let γi = ri ri+1 be the product of the reflections in ei and
ei+1 . The element γi is an elliptic rotation of order ni fixing
the vertex ei ∩ ei+1 .
Let
Γ = hγ1 , . . . , γr i .
a group generated by rotations.
P is half a fundamental polygon for Γ.
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Automorphism groups as quotients
Start with the automorphism group! - 3
Γ has this presentation:
*
Γ=
γ1 , . . . , γ r :
r
Y
+
γi =
γ1n1
= ··· =
γrnr
=1
i=1
Continuing:
Pick an epimorphism η : Γ → G that preserves the order of
elliptic elements. This is called a surface kernel
epimorphism.
Π = ker(η) is the desired surface group, and G ' Γ/Π acts
as a group of automorphisms of S ' H/Π. The inverse
map : G ,→ Aut(S) of η : Γ/Π ↔ G is an action of G on S
determined by the surface kernel epimorphism.
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Automorphism groups as quotients
Start with the automorphism group! - 4
Let ci = η(γi ). The r -tuple (c1 , . . . , cr ) satisfies:
G = hc1 , . . . , cr i ,
o(ci ) = ni ,
c1 · · · cr
= 1.
Such an r -tuple is called a generating (n1 , . . . , nr )-vector of
G. The ci are rotations on S of order ni with one or more
fixed points.
Surface kernel epimorphisms η : Γ → G are in 1-1
correspondence with generating vectors.
Generating vectors are easy to classify. So, we have
completely solved the problem of finding automorphism
groups, but have not solved the moduli problem of surfaces
with automorphisms.
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Automorphism groups as quotients
Superelliptic strata - 1
Select a generating vector. By varying the fundamental
polygon P, we get families of conformally in-equivalent
surfaces with the same automorphism group and, hence, a
stratum in the branch locus.
Different generating vectors may yield the same stratum.
The ambiguity is easily handled, but we don’t discuss that
here, due to lack of time.
We return to the super-elliptic case. Let Cp = hgi be the
cyclic group of order p. Each ci = g ti for some ti . With a
proper choice of g, these are the same ti as in the
definition of super-elliptic. We may use (t1 , . . . , tr ) to
denote a generating vector.
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Automorphism groups as quotients
Superelliptic strata! - 2
Example
Here is a table of strata for a small number of branch points r ,
stratum dimension t = r − 3, prime p, and genus σ. Normalized
generating vectors (t1 , . . . , tr ) for different strata are given.
r = 3, t = 0
r = 4, t = 1
r = 3, t = 0
r = 4, t = 1
p=3
σ = 1, (1, 1, 1)
σ = 2, (1, 1, 2, 2)
p=5
σ = 2, (1, 1, 2)
σ = 4, (1, 1, 1, 2),
(1, 1, 4, 4), (1, 2, 3, 4)
p=7
σ = 3, (1, 1, 5), (1, 2, 4)
σ = 6, (1, 1, 1, 4),
(1, 1, 6, 6)
p = 11
σ = 5, (1, 1, 9)
σ = 10, (1, 1, 1, 8),
(1, 1, 10, 10)
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Belyi’s theorem and dessins d’enfant
Belyi’s theorem and dessins d’enfant
Suppose that a surface action : G ,→ Aut(S) has a generating
vector (c1 , c2 , c3 ), so that there are three branch points. Then,
the following hold (pictures next frame):
The branch locus stratum is a single point.
Belyi’s theorem: S can be defined over a number field!
The triangle tiling on H descends to a triangle tiling on S.
The triangle tiling may be aggregated into a quasi-platonic
tiling on S.
The edges of the quasi-platonic tiling form a (regular)
dessin d’enfant, an embedded bipartite graph, with simply
transitive G-action.
The surface S and G-action are called quasi-platonic.
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Belyi’s theorem and dessins d’enfant
Icosahedral triangular tiling and dessins
Describe the (2, 3, 5) tiling.
Show the three quasi-platonic tilings and dessins:
dodecahedral, icosahedral, and rhombic.
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Belyi’s theorem and dessins d’enfant
Belyi’s theorem and Galois action
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 .
The map
(g) → θ (g) = θ(g)θ−1
transfers the action of G on S to an θ action on S θ .
So, there is an action of Gal(Q) ⊂ Gal(C) on quasi-platonic
surfaces, their dessins, and the G-actions.
Defining equations are hard to find so we look at the action
of Gal(Q) on generating triples.
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Belyi’s theorem and dessins d’enfant
Galois action on generating vectors and dessins - 1
Theorem (Branch cycle argument)
Let (c1 , c2 , c3 ) be a generating (n1 , n2 , n3 )-vector for the action
of G on S. Let N = lcm(n1 , n2 , n3 ) and suppose that θ ∈ Gal(C)
acts on the N’th roots of unity by ζ → ζ s . Select t so that
st = 1 mod N.
Then, there are x, y , z ∈ G such that a generating vector
(c10 , c20 , c30 ) for the θ of action of G on S θ is given by
(c10 , c20 , c30 ) = (xc1t x −1 , yc2t y −1 , zc3t z −1 ).
Theorem (Gonzales Diez & Jaikin-Zapirain)
The absolute Galois group acts faithfully on regular dessins.
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Belyi’s theorem and dessins d’enfant
Galois action on generating vectors and dessins - 2
Problem
For interesting groups, describe the Galois action completely in
terms of the structure of G.
This has been completely worked out in the case of PSL2 (q),
independently, by Broughton and Clark-Voight.
Some interesting groups are simple groups and their covers.
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References
G. Bartolini, A. Costa, M. Izquierdo, On isolated strata of
p-gonal Riemann surfaces in the branch locus of moduli
spaces, Albanian J. Math. 6, no, 1, (2012) 11–19.
G. Bartolini, A. Costa, M. Izquierdo, On automorphism
groups of cyclic p-gonal Riemann surfaces, J. Symbolic
Comput. 57, (2013) 61–69.
A. Costa, M. Izquierdo, H. Parlier, Connecting p-gonal loci
in the compactification of moduli space,
arXiv:1305.0284v2.
S.A. Broughton, The equisymmetric stratification of the
moduli space and the Krull dimension of the mapping class
group, Topology and its Applications 37 (1990), 101–113.
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References continued
S.A Broughton, Quasi-platonic PSL2 (q)-actions on closed
Riemann surfaces, Albanian J. Math. 9, (2015), no. 1,
31-61.
P.L. Clark and J. Voight, Algebraic Curves Uniformized by
Congruence Subgroups of Triangle Groups, preprint
(2015).
G. Gonzales Diez & A. Jaikin-Zapirain, The absolute
Galois group acts faithfully on regular dessins and on
Beauville surfaces, preprint (2013).
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Questions
Any questions?
The slides of this talk will be available at http://www.rosehulman.edu/˜brought/Epubs/Linkoping/LinkopingSabb.html
Done