Linköping Studies in Science and Technology.
Thesis No. 1413
On the Branch Loci of Moduli Spaces
of Riemann Surfaces of Low Genera
Gabriel Bartolini
Department of Mathematics
Linköping University, SE–581 83 Linköping, Sweden
Linköping 2009
Linköping Studies in Science and Technology. Thesis No. 1413
On the Branch Loci of Moduli Spaces of Riemann Surfaces of Low Genera Gabriel Bartolini
gabriel.bartoini@liu.se www.mai.liu.se Division of Applied Mathematics
Department of Mathematics Linköping University SE–581 83 Linköping
Sweden
ISBN 978-91-7393-532-6 ISSN 0280-7971 LIU-TEK-LIC-2009:21 Copyright c 2009 Gabriel Bartolini
Abstract
Compact Riemann surfaces of genus greater than 1 can be realized as quotient spaces of the hyperbolic plane by the action of Fuchsian groups. The Teichmüller space is the set of all complex structures of Riemann surfaces and the moduli space the set of con-formal equivalence classes of Riemann surfaces. For genus greater than two the branch locus of the covering of the moduli space by the Teichmüller space can be identified with the set of Riemann surfaces admitting non-trivial automorphisms. Here we give the orb-ifold structure of the branch locus of surfaces of genus 5 by studying the equisymmetric stratification of the branch locus. This gives the orbifold structure of the moduli space.
We also show that the strata corresponding to surfaces with automorphisms of order 2 and 3 belong to the same connected component for every genus. Further we show that the branch locus is connected with the exception of one isolated point for genera 5 and 6, it is connected for genus 7 and it is connected with the exception of two isolated points for genus 8.
Acknowledgments
First of all I would like to thank my supervisor Professor Milagros Izquierdo for introduc-ing me to the subject and takintroduc-ing time to discuss it with me. The LATEX-template used for
this thesis is due to Dr. Gustaf Hendeby, thank you. I would also like to thank the GAP community for helping me with the use of GAP. Thanks to the G. S. Magnuson foundation and the Knut and Alice Wallenberg foundation for the grants I have been given. Finally, I would like to thank my fellow PhD-students, my friends and my family for support and distraction during my time as a PhD-student so far.
Contents
Introduction 1 1 Preliminaries 3 1.1 Hyperbolic Geometry . . . 3 1.2 Fuchsian Groups . . . 4 1.3 Riemann Surfaces . . . 61.4 Automorphisms of Riemann Surfaces . . . 8
1.5 Teichmüller Theory . . . 11
2 The Equisymmetric Stratification of the Branch Locus of Riemann Surfaces of Genus 5 15 3 The Orbifold Structure of the Moduli Space 25 3.1 Surfaces with Automorphisms of Order a Power of 2 . . . 26
3.2 Surfaces with Automorphisms of Orders 2 and 3 . . . 35
3.3 Surfaces with Automorphisms of Order 5 or 11 . . . 42
Bibliography 47 4 Paper: On the Connectedness of the Branch Locus of the Moduli Space of Riemann Surfaces of Low Genus 49 1 Introduction . . . 51
2 Riemann surfaces and Fuchsian groups . . . 52
3 Properties of the strata corresponding to cyclic groups of order 2 and 3 . . 54
4 On the connectedness of the branch locus of the moduli space of Riemann surfaces of low genus . . . 57
References . . . 60
vi Contents
A List of Groups 63
Introduction
The first appearance of Riemann surfaces was in Riemann’s dissertationFoundations for a general theory of functions of a complex variable in 1851. Riemann used the surfaces as a tool to study many-valued complex functions. The first abstract definition was intro-duced by Weyl in 1913. Another perspective of Riemann surfaces is the uniformization theory of Poincaré, Klein and Koebe. The theory states that every Riemann surface ad-mits a Riemann metric of constant curvature. There are three types of geometries, elliptic geometry with curvature1, euclidean geometry with curvature 0 and hyperbolic geometry with curvature−1.
An Automorphism of a Riemann surface is a biholomorphic self-map. At the end of the 19th century different properties of automorphisms where studied by Klein, Poincaré, Hurwitz, Clebsch and others. Hurwitz [14] showed that the total number of automor-phisms for a surface of genusg ≥ 1 is bounded by 84(g − 1). Wiman [24] found that the order of a single automorphism is bounded by2(2g + 1).
More recently automorphisms of Riemann surfaces as been studied due to their re-lation to moduli spaces of Riemann surfaces. Harvey [13] alluded to the equisymmetric stratification of the branch locus and Broughton [5] proved it. Broughton also studied the structure of branch locus for Riemann surfaces of genus3. For genus 4 the structure was found by Costa and Milagros.
For genus5 the automorphism groups has been studied by Kimura and Kuribayashi classifying them up toGL(5, C)-conjugacy. Breuer [3] generalized those ideas and clas-sified in this way all the automorphism groups for2 ≤ g ≤ 48. Here we will study the automorphism groups of Riemann surfaces of genus5 up to topological equivalence and determine the complete structure of the equisymmetric stratification of the branch locus. This can be divided into the following steps.
1. Determine all pairs of signatures s and finite groups G such that G has an s-generating vector. This has been done [3, 17].
2 Introduction
2. Determine theAut(G) classes of the generating vectors. Then determine the B-orbits of the classes. This has been done with the use of GAP [11].
3. Remove non-maximal actions. This has been done with the use of Singerman’s list [23].
4. Compute actions determined by maximal subgroups ofG. This has been done with the use of GAP [11].
Outline of the thesis
Chapter 1
Here we introduce the basic concepts for hyperbolic geometry, Fuchsian groups, Riemann surfaces and Teichmüller theory. Further we introduce the concept of equisymmetric stratification.
Chapter 2
Here we consider surfaces of genus5. We give all classes of actions of finite groups and determine which are maximal.
Chapter 3
With the results in chapter 2 we study how the induced strata are related to each other. The calculations in chapter 2 and 3 has been aided by the use of GAP.
Chapter 4
In this paper we show that the strata corresponding to actions of order2 and 3 belong to the same connected component for arbitrary genera. Further we show that the branch locus is connected with the exception of one isolated point for genera 5 and 6, it is connected for genus 7 and it is connected with the exception of two isolated points for genus 8.
1
Preliminaries
Riemann surfaces can be studied in several different ways, for instance as manifolds, complex curves or quotient spaces. When studying automorphisms of Riemann surfaces of genus2 or greater it is convenient to represent them as quotient spaces of the hyperbolic plane by the action of Fuchsian groups. For details on those topics see [2, 15], for basic groups theory see [10] and for further details on Teichmüller theory see [21].
1.1
Hyperbolic Geometry
To work with Riemann surfaces as quotient spaces we need the concept of hyperbolic ge-ometry. Two common models of the hyperbolic plane which we will use arethe Poincaré half-plane modelU = {z ∈ ˆC|Im(z) > 0} with metric given by
ds = |dz| Im(z)
andthe Poincaré disc model ∆ = {z ∈ ˆC||z| < 1} with metric given by
ds = 2|dz| 1 − |z|2.
Thehyperbolic lines on the Poincaré models correspond to circles on the extended com-plex plane perpendicular to the boundary ofU or ∆ respectively. Onwards we will denote the hyperbolic plane, regardless of which model we use, byU. The group of conformal isometries ofU is the group of Möbius transformations
az + b
cz + d, a, b, c, d ∈ R, ad − bc = 1
4 1 Preliminaries
which is isomorphic to the groupP SL(2, R). The anti-conformal isometries are similarly given by
a¯z + b
c¯z + d, a, b, c, d ∈ R, ad − bc = −1.
The isometries can be divided into classes depending on the nature of their fixed points. Those classes can be characterized by their traces as follows. A conformal isometry is
parabolicif|a + d| = 2. It has one fixed point α ∈ R ∪ {∞} and is conjugate toz + 1 or z − 1.
hyperbolicif|a + d| > 2. It has one fixed set {α, β} ∈ R ∪ {∞} and is conjugate toλz (λ > 1).
ellipticif|a + d| < 2. It has one fixed point ξ ∈ U and is conjugate to a rotation ofU.
We are interested in subgroups actingproperly discontinuously on the hyperbolic plane. This means that for any compact subsetU we have that g(U ) ∩ U = ∅ except for a finite number of elementsg.
1.2
Fuchsian Groups
A group of conformal Möbius transformations is a Fuchsian group if it leaves a disc invariant on which it acts properly discontinuously. Here we will assume that this disc is U, then a Fuchsian group is a discrete subgroup of P SL(2, R). We are only interested in Fuchsian groupsΓ such that the quotient space U/Γ is a compact space. Then Γ has the following presentation D a1, b1, . . . , ag, bg, x1. . . xk|xm11 = · · · = xmkk= Y xi Y [ai, bi] = 1 E (1.1) wherexi is a elliptic element andai andbi are hyperbolic elements. If a groupΓ has presentation 1.1 then we say thatΓ hassignature
s(Γ) = (g; m1, . . . , mk) (1.2)
whereg is called thegenus of the topological surfaceU/Γ and mi, i = 1 . . . k are the ordersof the stabilizers of the cone points of the surface. The signature (1.2) gives us the algebraic structure ofΓ and the geometrical structure of U/Γ.
A Fuchsian group without elliptic elements is called asurface group. Later we will see why this is important.
Example 1.1
Let∆ be a hyperbolic triangle with angles π/m1, π/m2, π/m3 and letri,i = 1, 2, 3 be the reflections in the sides of∆. If Γ∗is the group generated byr
1, r2, r3thenΓ = Γ∗ ∩ P SL(2, R) is a Fuchsian group called atriangle group and is generated by the
1.2 Fuchsian Groups 5
elliptic elementsr1r2,r2r3,r3r1where(r1r2)m3 = (r2r3)m1 = (r3r1)m2 = Id. Γ has the following presentation
hx1, x2, x3|xm11 = x m2
2 = x m3
3 = x1x2x3= 1i .
Afundamental domainF of Fuchsian group is a closed subset of U such that (i) U =S
g∈Γg(F ).
(ii) ifp ∈ F and g(p) ∈ F , where g 6= Id, then p ∈ δF . (iii) µ(δF ) = 0, where µ is the hyperbolic measure.
IfΓ has signature (1.2) then we can choose a fundamental domain F for Γ such that δF is a hyperbolic polygon. Further, two sidesα, α′of the polygon arecongruentif there exists g ∈ Γ such that g(α) = α′. Theng(F ) ∩ F = α′. Each elliptic elementx
ipairs sidesγi, γ′
isuch that the sides form an angle2π/mi,aipairs sidesα, α′andbipairs sidesβ, β′. The hyperbolic structure of the fundamental domain under the side pairings,F/∼, is the same as the hyperbolic structure ofU/Γ. We can also go the opposite way, formulated in the following theorem by Poincaré:
Theorem 1.1. LetF be a hyperbolic polygon with side pairings as above with non-zero
area then there exists a Fuchsian group withF as fundamental domain.
The hyperbolic area of a Fuchsian group corresponds to the hyperbolic area of any of its fundamental domains and is given by
µ(Γ) = 2π(2g − 2 +X i 1 − 1 mi ).
Now assume thatΓ′ is a subgroup ofΓ of index N . If F is a fundamental domain of Γ thenS gi(F ) is a fundamental domain of Γ′, where{gi} is a transversal for Γ′. From this the Riemann-Hurwitz formula follows:
[Γ : Γ′] = N = µ(Γ ′)
µ(Γ). (1.3)
The possible signatures and indices of subgroups of a given Fuchsian group is given by the following theorem due to Singerman:
Theorem 1.2. [22] LetΓ have signature (1.2) and presentation (1.1). Then Γ contains a
subgroupΓ′of indexn and signature (g′; m
11, . . . , mksk) if and only if
(i) there exist a transitive permutation representationθ : Γ → Σnsatisfyingθ(xi) has
exactlysicycles of length less thanmi, the lengths beingmi/mi1. . . mi/misi.
6 1 Preliminaries
1.3
Riemann Surfaces
ARiemann surfaceis a Haussdorf spaceX that is locally homeomorphic to the complex plane. This means that each point inX has an open neighbourhood Ui such that there exists a homeomorphismΦi: Ui→ Vi, whereViis an open subset of C. We call the pair (Φi, Ui) achart. Anatlas is a set of chartsA covering X and if Ui∩ Uj 6= ∅ then the transition function
Φi◦ Φ−1j : Φj(Ui∩ Uj) → Φi(Ui∩ Uj)
is analytic. Further we call two analytic atlasesA, Bcompatibleif all the transition func-tions of charts(Φ, U ) ∈ A, (Ψ, V ) ∈ B are analytic. Such atlases form an equivalence class called a complex structure. Different complex structures on the same topological surface yield different Riemann surfaces. With atlases we can define holomorphic maps between Riemann surfaces. LetX and Y be Riemann surfaces with atlases {(Φi, Ui)} and{(Ψj, Vj)} respectively. A map f : X → Y is calledholomorphic (or meromorphic) if the maps
Ψj◦ f ◦ Φ−1i : Φi(Ui∩ f−1(Vj)) → C
are analytic (or meromorphic), f is a homomorphism of Riemann surfaces. Further iff is bijective andf−1 is also holomorphic then we callf biholomorphic. Two Riemann surfaces areconformally equivalent if there exists a biholomorphic map between them. We do not distinguish between conformally equivalent surfaces.
Example 1.2
(i) C with an atlas consisting of the identity map.
(ii) ˆC with an atlas consisting of the identity map of C, Id, together with the map φ : ˆC\ {0} → C defined as φ(z) = 1/z and φ(∞) = 0. This is indeed an atlas sinceId ◦ φ−1= 1/z and φ ◦ Id−1 = 1/z are analytic on C ∩ ˆC\ {0} = C \ {0}. Acovering is a surjective continuous mapf : X → Y between two Riemann surfaces such that for any pointp ∈ Y the preimage of some neighbourhood V is a disjoint union of open sets ofX each mapping homeomorphically to V . The set f−1(p) ⊂ X is called thefiberofp. Each fiber has the same cardinality, if it is a finite number n we say that the covering isn-sheeted.
Theorem 1.3. [19] Letf : X → Y be a covering, with Y path-wise connected. For each p ∈ Y , the fundamental group π1(Y, p) acts transitively on the right in the fiber f−1(p).
The stabilizer of each pointx ∈ f−1(p) is H(x) = f
#π1(X, x). The permutations of a fiber f−1(p) by ∈ π
1(Y, p) forms a group called the mon-odromy groupoff at p. For Riemann surfaces all fibers are permuted in the same way so we might talk about the monodromy group off , in this case it is sometimes called the deck transformation group.
Now for an arbitrary surjective continuous mapf : X → Y there might exist points where we can not find neighborhoods with preimages consisting of disjoints union of open sets mapping homeomorphically to the neighborhood. Such points are calledbranch points. Abranched coveringis a surjective continuous mapf : X → Y such that the map
1.3 Riemann Surfaces 7
f : X∗ → Y∗ is a covering whereY∗is the surfaceY with the branch points removed andX∗is the surfaceX with the preimages of the branch points removed.
A branched covering: X → Y between Riemann surfaces is a morphism of Riemann surfaces.
Example 1.3
Consider the mapf : ˆC→ ˆC defined byz 7→ z3.f is a branched covering with branch points0 and ∞. We note that close to the branch points f is three-to-one.
Riemann Surfaces as Quotient Spaces
In order to study moduli spaces of Riemann surfaces we consider Riemann surfaces as quotient spaces of the hyperbolic plane. In fact every Riemann surface is conformally equivalent to a quotient space of a simply connected Riemann surface. This follows from the uniformization theorems.
Theorem 1.4. [15] (Uniformization theorem) Any simply connected Riemann surface is
isomorphic to one of the following spaces: (i) C
(ii) ˆC
(iii) U
Theorem 1.5. [15] Any Riemann surfaceX of genus g ≥ 2 is birationally equivalent to
the orbit spaceU/Γ for a Fuchsian group Γ with orbit genus g. Furthermore, U is the
universal covering ofX.
From the theorem it follows thatΓ is the lift of the fundamental group, π1(X, x), to the universal coveringU. Now assume that Γ′ is a subgroup ofΓ of index N . Then we have a branched coveringf : U/Γ′ → U/Γ. Let {g
i} be a transversal of Γ′ inΓ then each coset ofΓ induces a sheet of the covering f . We note that Γ permutes the sheets and generates the monodromy group of the covering. Themonodromy of the covering is the transitive permutation representationθ : Γ → ΣN given by the action ofΓ on the cosets ofΓ′. On the other hand, assume we have an epimorphismθ : Γ → G for some finite groupG. Let G′be a subgroup ofG of index N , then the images of the elements of Γ permutes the cosets ofG′ inducing a permutation representation ofΓ. By Theorem 1.2 Γ has a subgroup Γ′ of indexN and the induced permutation representation of Γ is the monodromy of the coveringU/Γ′ → U/Γ.
We prefer to work with Riemann surfaces as quotient spaces of surface groups due to properties we will see later on. We can consider an arbitrary Riemann surface of genus greater than1 as a quotient space of a surface group as we see in the following theorem:
Theorem 1.6. [15] Any compact Riemann surface X of genus g ≥ 2 is conformally
equivalent to the orbit space U/Γ for a Fuchsian surface group Γ with orbit genus g.
Furthermore,U is the universal covering of X, and the fundamental group Γ = AutX(U)
8 1 Preliminaries
By the use of surface Fuchsian groups we obtain several useful properties. The next theorem shows one which is important when working with Teichmüller spaces.
Theorem 1.7. [15] Let Γ1 and Γ2 be two surface Fuchsian groups. Then the
Rie-mann surfaces U/Γ1 and U/Γ2 are conformally equivalent if and only if there exists t ∈ P SL(2, R) such that tΓ1t−1 = Γ2.
1.4
Automorphisms of Riemann Surfaces
An automorphism of a Riemann surfaceX is a conformal homeomorphism f : X → X. We denote the group of automorphisms of a Riemann surfaceAut(X).
Example 1.4
(i) Aut(C) = {az + b|a, b ∈ C, a 6= 0}. (ii) Aut( ˆC) = P SL(2, C).
(iii) Aut(U) = P SL(2, R).
There is a well known upper bound of the cardinality of the automorphism group of a compact Riemann surface of genus greater than one:
Theorem 1.8. [14] LetX be a compact Riemann surface of genus g ≥ 2. Then |Aut(X)| ≤ 84(g − 1).
A Riemann surfaceX such that |Aut(X)| = 84(g − 1) is called aHurwitz surface andAut(X) aHurwitz group.
Example 1.5
The most well-known Hurwitz surface is the Klein quartic of genus3. It can be defined as a complex curve by the equation
x3y + y3z + z3x = 0.
The Automorphism group of the Klein quartic is isomorphic toP SL(2, 7) and |P SL(2, 7)| = 168 = 84(3 − 1).
LetX = U/Γ be a Riemann surface uniformized by a surface Fuchsian group Γ. Then a groupG is a group of automorphisms of X if and only if there exists a Fuchsian group ∆ and an epimorphismθ : ∆ → G such that ker(θ) = Γ [15]. In this case ∆ is the lift of the automorphism group ofX to the universal covering.
Given a Fuchsian group∆ with signature s = (g; m1, . . . , mk) and a finite group G, then the epimorphismθ : ∆ → G is uniquely associated to ans-generating vector ofG is a vector(a1, b1, . . . , ag, bg; c1, . . . , ck), where ai, bi, ci ∈ G and such that cihas order miand the long relation is satisfied. So by studying generating vectors we can determine ifG is a group of automorphisms of a surface of genus g.
Theorem 1.9. [4] (Riemann’s Existence Theorem) The groupG acts on the surface X, of
genusg, with branching data (g; m1, . . . , mk) if and only if the Riemann-Hurwitz formula
1.4 Automorphisms of Riemann Surfaces 9 Example 1.6
LetG be isomorphic to the cyclic group C6and consider the signatures = (0, 2, 2, 3, 6, 6). By Theorem 1.9C6 acts on a surface of genus5 since with the Riemann-Hurwitz for-mula (1.3) we find that 2g − 2 = 6(−2 + 1/2 + 1/2 + 2/3 + 5/6 + 5/6) = 8 and (a3, a3, a4, a, a) is an s-generating vector. Thus there exist a Fuchsian group ∆ with signatures(∆) = (0, 2, 2, 3, 6, 6) and a presentation
x1, x2, x3, x4, x5|x21= x22= x33= x64= x65= x1x2x3x4x5= 1
such that there exists an epimorphism θ : ∆ → C6 by θ(x1) = a3, θ(x2) = a3, θ(x3) = a4,θ(x4) = a and θ(x5) = a. Note that any elliptic element x of ∆ is conjugate to somexithusθ(x) = θ(y−1xiy) = θ(xi) 6= 1. So ker(θ) is a surface group of genus g. Also, θ is the monodromy of the covering X = U/Γ → U/∆ = X/G.
Note that in Theorem 1.9 we only need to consider a signature (g; m1, . . . , mk), that means the algebraic structure of some group. Further on, we will denote an abstract group with presentation (1.1) and signature (1.2) by∆(g; m1, . . . , mk).
Maximal Fuchsian Groups
We are interested in determining when an action of a finite group yields the full group of automorphisms of some Riemann surface. If it is the case we call it amaximalaction due to the following properties. A Fuchsian group∆ is called amaximal Fuchsian group if there is no other Fuchsian group∆′containing∆ with finite index and such that d(∆) = d(∆′). We also call a signature non-maximal if it is the signature of some non-maximal Fuchsian group. The full list of pairs of signaturess(∆), s(∆′) as above was obtained by Singerman in [23]. s(∆) s(∆′) |∆′: ∆| (2; −) (0; 2, 2, 2, 2, 2, 2) 2 (1; t, t) (0; 2, 2, 2, 2, t) 2 (1; t) (0; 2, 2, 2, 2t) 2 (0; t, t, t, t) (0; 2, 2, 2, t) 4 (0; t, t, u, u) (0; 2, 2, t, u) 2 (0; t, t, t) (0; 3, 3, t) 3 (0; t, t, t) (0; 2, 3, 2t) 6 (0; t, t, u) (0; 2, t, 2u) 2 (0; 7, 7, 7) (0; 2, 3, 7) 24 (0; 2, 7, 7) (0; 2, 3, 7) 9 (0; 3, 3, 7) (0; 2, 3, 7) 8 (0; 4, 8, 8) (0; 2, 3, 8) 12 (0; 3, 8, 8) (0; 2, 3, 8) 10 (0; 9, 9, 9) (0; 2, 3, 9) 12 (0; 4, 4, 5) (0; 2, 4, 5) 6 (0; n, 4n, 4n) (0; 2, 3, 4n) 6 (0; n, 2n, 2n) (0; 2, 4, 2n) 4 (0; 3, n, 3n) (0; 2, 3, 3n) 4 (0; 2, n, 2n) (0; 2, 3, 2n) 3 Table 1.4: Pairs of non-maximal signatures.
10 1 Preliminaries
When an action is induced by a non-maximal signature it can still be maximal as we see in the following proposition:
Proposition 1.1. LetΓ be a surface Fuchsian group and X = U/Γ a Riemann surface.
Assume that there exists a surface kernel epimorphismθ : ∆ → G such that ker(θ) = Γ
wheres(∆) is a non-maximal signature. Further assume that there exists another surface
kernel epimorphismθ′ : ∆′ → G′such thatker(θ′) = Γ where ∆′ ≥ ∆, G′ ≥ G and θ′−1(G) = ∆. Then G = ∆/Γ is the full group of automorphisms Aut(X) if and only
ifθ′|
∆ is not equivalent under automorphisms of∆ and G to θ for any such extensions θ′ : ∆′ → G′.
Example 1.7
Let∆ be a group with signature s(∆) = (0; 4, 4, 4, 4) such that there exists an epimor-phismθ : ∆ → C4× C2defined byθ(x1) = a, θ(x2) = a, θ(x3) = ab and θ(x4) = ab. With Riemann-Hurwitz formula we find thatθ induces a group of automorphisms of a surface of genus5. This group is indeed not the full group of automorphisms. Consider a group∆′ with signatures(∆′) = (0; 2, 2, 2, 4) such that there exists an epimorphism θ′ : ∆′ → (C
4× C2× C2) ⋊ C2defined byθ′(y1) = ba, θ′(y2) = b, θ′(y3) = c and θ′(y
4) = ac acting on a surface of genus 5. Consider the following commutative diagram
(x1, x2, x3, x4) −−−−→θ (a, a, ab, ab) y y (y3y2y4−1y2y4y2y4y2y3, y4, y2y4y2, y3y2y4y2y3) θ ′
−−−−→ (ac, ac, (ac)3(bc)2, (ac)3(bc)2) whereC4× C2≃ac, (ac)2(bc)2 ⊂ (C4× C2× C2) ⋊ C2. As seen in the diagram the actionθ can be extended to θ′.
To find out if an actionθ : ∆ → G extends to an action θ′ : ∆′ → G′ we consider the monodromy of the coveringU/θ′−1(G) → U/∆′. We show the calculations by an example.
Example 1.8
We return to the actions of Example 1.7. Let∆1 = θ′−1ac, (ac)2(bc)2 . To determine the monodromy group of the induced coveringU/∆1 → U/∆′ we consider the cosets of ac, (ac)2(bc)2 and by enumerating the cosets we have the following permutation representation of the monodromy:
θ′(y i) ac, (ac)2(bc)2 ba → (1 4)(2 3) b → (1 2)(3 4) c → (1 3)(2 4) ac → (1)(2)(3)(4)
1.5 Teichmüller Theory 11
1.5
Teichmüller Theory
On page 6 we constructed Riemann surfaces on a topological surfaceX and classified them up to conformal equivalence. We are interested in the space of those classes; the moduli space. However this space is not a manifold in its natural topology, it is an orb-ifold. To overcome this we consider Teichmüller spaces of Riemann surfaces. The Teich-müller space is constructed by classes of Riemann surfaces marked by quasi-conformal maps. However, when a Riemann surface is uniformized by a surface Fuchsian group we can construct the Teichmüller space by classes of surface Fuchsian groups. One benefit of the second approach is that we do not have to explicitly use quasi-conformal maps, see [21].
LetΓ be an abstract group with presentation (1.1) and let R(Γ) be the set of monomor-phismsr : Γ → P SL(2, R). R(Γ) has a natural topology given by
r 7→ (r(x1), . . . , r(xk), r(a1), . . . , r(bg)).
The groupP SL(2, R) acts on R(Γ) by conjugation; for t ∈ P SL(2, R) and x ∈ Γ we denote by t∗(r) → P SL(2, R) the isomorphism x → t−1r(x)t. The quotient space T (Γ) = R(Γ)/P SL(2, R)∗is called theextended Teichmüller space ofΓ and is home-omorphic to two copies of a complex space of dimensiond(Γ) = 3g − 3 + k. If Γ is a surface group then we denoteT (Γ) by Tg. Teichmüller spaces for different Fuchsian groups are related as follows.
Theorem 1.10. [12] Let Γ and Γ′ be two Fuchsian groups such that there exists a
monomorphismi : Γ → Γ′then the induced map
i∗: T (Γ′) → T (Γ), [r] 7→ [r ◦ i],
is an isometric embedding.
Let M (Γ) denote the group of outer automorphisms of Γ. M (Γ), which is also calledthe Teichmüller modular group or the mapping class group of Γ, acts on T (Γ) as [r] → [r ◦ α] where α ∈ M (Γ). The moduli space of Γ is the quotient space M(Γ) = T (Γ)/M (Γ). Similarly as before we denote by Mg the moduli space of a surface group of genus g. We are interested in the branch locus Bg of the covering Tg → Mg. As an application of Nielsen Realization Theorem [5, 13] one can iden-tify the branch locus of the action ofM (Γ) as the set Bg= {X ∈ Mg|Aut(X) 6= 1}, for g ≥ 3.
The Branch Locus
Consider the automorphism group Aut(X) of a Riemann surface X = U/Γ. G = Aut(X) determines a conjugacy class of subgroups, ¯G, of M (Γ) [5]. We call ¯G, or just G, thesymmetry typeofX. Two surfaces areequisymmetricis they have the same symme-try type, that is their automorphism groups are conjugate inMg. Conformally equivalent surfaces clearly have the same symmetry type so we can talk about the symmetry type of points in the moduli space. In terms of actions of finite groups on surfaces equisymmetry corresponds to topologically equivalent actions. Two actionsθ, θ′ ofG on a surface X
12 1 Preliminaries
aretopologically equivalent if there is anw ∈ Aut(G) and an h ∈ Hom+(X) such that θ′(g) = hθw(g)h−1. We can formulate this for Riemann surfaces as follows.
Lemma 1.1. [4] Two epimorphismsθ1, θ2 : ∆ → G define two topologically equivalent
actions ofG on X if there exists two automorphisms φ : ∆ → ∆ and w : G → G such
that the following diagram commutes:
∆ θ1 −−−−→ G φ x yw ∆ θ2 −−−−→ G
φ is the induced automorphism by the lifting h∗ofh to the universal orbifold covering. LetB be the subgroup of Aut(∆) composed of such automorphisms. While it is possible to find a finite generating set forB we will only produce the elements in an ad hoc fashion, using the following proposition.
Proposition 1.2. [4] Suppose thatγ = 0, or 1, let a = a1,b = b1and let other notation
be as above. Consider the automorphisms of∆ defined by the formulae below, where the
action of an automorphism on a generator of∆ is written down only if it actually moves
the generator. A product of these automorphisms defines an element ofB if and only if the
induced action preserves branching orders. I.a a → abn,n ∈ Z,
I.b b → ban,n ∈ Z,
II xj→ xj+1,xj+1→ x−1j+1xjxj+1
III.a a → xa, xj → yxjy−1, where x = b−1wz, y = zb−1w, w = x1. . . xj−1, z = xj+1. . . xr.
III.b b → xb, xj → yxjy−1, wherex = wza, y = zaw, w = x1. . . xj−1, z = xj+1. . . xr.
We define theG-equisymmetric stratum MG
g as the set of classes of surfaces with symmetry typeG. By MGg we denote the set of surfaces whose automorphism group contains a subgroup in the class defined byG. We will denote the set of all surfaces ad-mitting automorphisms of orderp by Mpg. We have the following theorem by Broughton:
Theorem 1.11. ([5]) LetMgbe the moduli space of Riemann surfaces of genusg, G a
finite subgroup of the corresponding modular groupMg. Then:
(1)MGg is a closed, irreducible algebraic subvariety ofMg.
(2)MG
g, if it is non-empty, is a smooth, connected, locally closed algebraic subvariety
ofMg, Zariski dense inM G g.
1.5 Teichmüller Theory 13
Theorem 1.12. [8] LetMCp,i
g be the stratum corresponding to an actioni of the cyclic
groupCpof prime orderp. Then Bg=
[
p∈P MCp,i
2
The Equisymmetric Stratification of
the Branch Locus of Riemann
Surfaces of Genus 5
The equisymmetric stratification of the branch locus,Bg, provides the orbifold structure of the moduli space,Mg. Costa and Izquierdo [9] found the orbifold structure ofM4. Kimura [16] found all the actions of finite groups on surfaces of genus 4. Magaard and others [20] studied the strata corresponding to large automorphism groups for genera up to10.
Here we calculate the orbifold structure ofM5. To determine the structure of the strata we need representations for the automorphism groups of Riemann surfaces of genus 5. We classify the actions by topological equivalence. Kuribayashi and Kimura [17] has classified them up toGL(5, C)-conjugacy. The classification in [17] coincides with the topological classification with the exception of one case, studied in Lemma 2.9 and Remark 2.1. The topological classification here has been done by calculating all actions with GAP [11] and reducing the number of equivalence classes accordingly. Examples of GAP programs appear in Appendix B. The same algorithm is used through Lemma 2.2 to Lemma 2.13. We exemplify this with detailed calculations in the proofs of Lemma 2.4 and Lemma 2.13.
Remember that a finite groupG is a group of automorphisms of a Riemann surface X = U/Γ, where Γ is Fuchsian surface group, if there exists a Fuchsian group ∆ and an epimorphism θ : ∆ → G such that Γ = ker(θ). Observe that the equisymmetric strata are associated with maximal actions of finite groups. Non-maximal actions extend to maximal actions as in the following diagram
16 2 The Equisymmetric Stratification of the Branch Locus of Riemann Surfaces of Genus 5 ∆′ θ′ −−−−→ G′ x x ∆ −−−−→ Gθ x x Γ −−−−→ Id
whereθ is a non-maximal action and θ′ is a maximal action. We prove non-maximality by finding extensions∆′ of ∆ and G′ of G such that θ′|
∆ = θ. To do this we use Theorem 1.2. We begin with a lemma calculating the possible signatures of Fuchsian groups inducing actions of finite groups.
Lemma 2.1. [17] Let ∆ be a Fuchsian group and G a finite group. If there exist an
epimorphismθ : ∆ → G such that ker(θ) is a surface group of genus 5. then ∆ has one
of the following signatures.
|G| signature |G| signature |G| signature
2 (0; 2,. . ., 2)12 10 (0; 2, 2, 2, 2, 5) (0; 2, 12, 12)∗ (1; 2,. . ., 2)8 (0; 2, 2, 10, 10)∗ 30 (0; 2, 6, 15) (2; 2, 2, 2, 2) (1; 5) 32 (0; 2, 2, 2, 4) (3; −) 11 (0; 11, 11, 11) (0; 4, 4, 4)∗ 3 (0; 3,. . ., 3)7 12 (0; 2, 2, 2, 2, 3) (0; 2, 8, 8)∗ (1; 3, 3, 3, 3) (0; 3, 3, 3, 3)∗ 40 (0; 2, 4, 20) 4 (0; 2,. . ., 2)8 (0; 2, 3, 4, 4) 48 (0; 2, 2, 2, 3) (0; 2,. . ., 2, 4, 4)5 (0; 2, 2, 6, 6)∗ (0; 3, 4, 4)∗ (0; 2, 2, 4, 4, 4, 4) (0; 6, 12, 12)∗ (0; 2, 6, 6)∗ (1; 2, 2, 2, 2) (1; 3)∗ (0; 2, 4, 12) (1; 2, 4, 4) 15 (0; 3, 15, 15)∗ 60 (0; 3, 3, 5)∗ (2; −)∗ 16 (0; 2, 2, 2, 2, 2) 64 (0; 2, 4, 8)∗ 5 (1; 5, 5)∗ (0; 2, 2, 4, 4)∗ 80 (0; 2, 5, 5)∗ 6 (0; 2, 2, 2, 2, 3, 3) (0; 4, 8, 8)∗ 96 (0; 3, 3, 4)∗ (0; 2, 3, 3, 3, 6) (1; 2)∗ (0; 2, 4, 6) (0; 2, 2, 3, 6, 6) 20 (0; 2, 2, 2, 10) 120 (0; 2, 3, 10) (0; 6, 6, 6, 6)∗ (0; 4, 4, 10)∗ 160 (0; 2, 4, 5) (1; 3, 3)∗ (0; 2, 20, 20)∗ 192 (0; 2, 3, 8) 8 (0; 2,. . ., 2)6 22 0; 2, 11, 22 (0; 2, 2, 2, 4, 4) 24 (0; 2, 2, 3, 3)∗ (0; 4, 4, 4, 4)∗ (0; 2, 2, 2, 6) (0; 2, 4, 8, 8) (0; 4, 4, 6)∗ (1; 2, 2)∗ (0; 3, 6, 6)∗ * Non-maximal
Proof: From Riemann’s Existence Theorem 1.9 we see that the existence of an
17
of∆ and the properties of the group G. The signatures must satisfy the Riemann-Hurwitz formula withs(Γ) = (5; −). This gives us the equation
8 = |G|(2g − 2 +X 1 − 1 mi ) (2.1)
and the signatures above are the solutions to this equation such that there existsG with an s-generating vector.
First we will list for each groupG the actions of G corresponding to a given signature by generating vectors with elements the images of the generators of a Fuchsian group with the given signature,(θ(a1), . . . , θ(bg); θ(x1), . . . , θ(xk)). We also show which actions are non-maximal and which are maximal, giving the symmetry of the surfaces in the stratum.
Lemma 2.2. The actions of order2 are
θ2,0: ∆(0; 2,. . ., 2) → C12 2 (a, a, a, a, a, a, a, a, a, a, a, a) θ2,1: ∆(1; 2,. . ., 2) → C8 2 (1, 1; a, a, a, a, a, a, a, a) θ2,2: ∆(2; 2, 2, 2, 2) → C2 (1, 1, 1, 1; a, a, a, a) θ2,3: ∆(3; −) → C2 (a, 1, 1, 1, 1, 1; −)
Lemma 2.3. The actions of order4 are
θ4,1: ∆(0; 2, 2, 4, 4, 4, 4) → C4 (a2, a2, a, a, a, a) θ4,2: ∆(0; 2, 2, 4, 4, 4, 4) → C4 (a2, a2, a, a, a−1, a−1) θ4,3: ∆(0; 2, 2, 2, 2, 2, 4, 4) → C4 (a2, a2, a2, a2, a2, a, a) θ4,4: ∆(1; 2, 4, 4) → C4 (1, 1; a2, a, a) θ4,5: ∆(1; 2, 2, 2, 2) → C4 (a, 1; a2, a2, a2, a2) θ4,6: ∆(2; −) → C4 (a, 1, 1, 1; −) θ4,7: ∆(0; 2,. . ., 2) → C8 2× C2 (a,. . ., a, b, b)6 θ4,8: ∆(0; 2,. . ., 2) → C8 2× C2 (a, a, a, a, b, b, b, b) θ4,9: ∆(0; 2,. . ., 2) → C8 2× C2 (a, a, a, a, b, b, ab, ab) θ4,10: ∆(1; 2, 2, 2, 2) → C2× C2 (1, 1; a, a, b, b) θ4,11: ∆(1; 2, 2, 2, 2) → C2× C2 (b, 1; a, a, a, a) θ4,12: ∆(2; −) → C2× C2 (a, b, 1, 1; −) θ4,13: ∆(2; −) → C2× C2 (a, 1, b, 1; −)
whereθ4,6,θ4,12andθ4,13are non-maximal actions.
Proof: Here we will prove non-maximality.
(1)θ4,6extends toθ8,17in Lemma 2.4 with the isomorphismC4≃ hai ⊂ D4.
(2)θ4,12extends toθ8,13in Lemma 2.4 with the isomorphismC2× C2≃ hb, abci ⊂ C2× C2× C2.
(3)θ4,13extends toθ8,14in Lemma 2.4 with the isomorphismC2× C2≃ hab, bci ⊂ C2× C2× C2.
18 2 The Equisymmetric Stratification of the Branch Locus of Riemann Surfaces of Genus 5 θ8,1 : ∆(0; 2, 4, 8, 8) → C8 (a4, a2, a, a) θ8,2 : ∆(0; 2, 4, 8, 8) → C8 (a4, a2, a3, a−1) θ8,3 : ∆(1; 2, 2) → C8 (a, 1; a4, a4) θ8,4 : ∆(0; 2, 2, 2, 4, 4) → C4× C2 (b, b, b, ba3, a) θ8,5 : ∆(0; 2, 2, 2, 4, 4) → C4× C2 (b, b, a2b, ab, a) θ8,6 : ∆(0; 2, 2, 2, 4, 4) → C4× C2 (b, b, a2, a, a) θ8,7 : ∆(0; 2, 2, 2, 4, 4) → C4× C2 (a2, a2b, b, a, a3) θ8,8 : ∆(0; 2, 2, 2, 4, 4) → C4× C2 (a2, a2, b, a3b, a) θ8,9 : ∆(0; 4, 4, 4, 4) → C4× C2 (a, a, ab, ab) θ8,10 : ∆(0; 4, 4, 4, 4) → C4× C2 (a, a3, ab, a3b) θ8,11 : ∆(1; 2, 2) → C4× C2 (a, 1; b, b) θ8,12 : ∆(1; 2, 2) → C4× C2 (a, b; a2, a2) θ8,13: ∆(0; 2,. . ., 2) → C6 2× C2× C2 (a, a, a, ab, bc, c) θ8,14: ∆(0; 2,. . ., 2) → C6 2× C2× C2 (a, a, b, b, c, c) θ8,15: ∆(0; 2,. . ., 2) → C6 2× C2× C2 (a, a, b, ab, bc, abc) θ8,16: ∆(1; 2, 2) → C2× C2× C2 (b, c; a) θ8,17: ∆(0; 2,. . ., 2) → D6 4 (s, s, s, s, sa, sa) θ8,18: ∆(0; 2,. . ., 2) → D6 4 (s, s, sa, sa, a2, a2) θ8,19: ∆(0; 2, 2, 2, 4, 4) → D4 (s, s, a2, a, a) θ8,20: ∆(1; 2, 2) → D4 (t, 1; s, s) θ8,21: ∆(0; 4, 4, 4, 4) → Q (i, i, j, j)
whereθ8,3,θ8,9,θ8,10,θ8,11,θ8,12,θ8,16,θ8,20andθ8,21are non-maximal actions.
Proof: We show the detailed calculations to find actions ofQ as an example of the al-gorithm. There are six classes of epimorphisms up to the action of Aut(Q) ≃ S4. Those give the following generating vectors,(i, i, j, j), (i, j, −i, j), (i, j, j, i), (i, j, i, −j), (i, j, −j, −i) and (i, −i, −j, j). Now let Bi be the elements of type II in Proposition 1.2 defined byxi 7→ xj,xj 7→ x−1j xixj. We see thatB2(i, i, j, j) = (i, j, −jij, j) = (i, j, −i, j), B3(i, j, −i, j) = (i, j, j, −j−ij) = (i, j, j, i), B3(i, j, j, i) = (i, j, i, −iji) = (i, j, i, −j), B3(i, j, i, −j) = (i, j, −j, ji − j) = (i, j, −j, −i) and B2(i, j, −i, j) = (i, −i, ij − i, j) = (i, −i, −j, j). Thus there is one class of epimorphisms θ8,21 : ∆(0; 4, 4, 4, 4) → Q.
Now we will prove non-maximality starting by considering actions induced by the signature(0; 4, 4, 4, 4).
(1) Forθ8,9consider the isomorphismC4×C2≃ac, (ac)2(bc)2 ⊂ (C4×C2×C2)⋊ C2. Enumerating the cosets we have the permutation representation of the monodromy of the desired coveringU/∆ → U/∆′.
θ32,2(yi) ac, (bc)2 ba → (1 4)(2 3) b → (1 2)(3 4) c → (1 3)(2 4) ac → (1)(2)(3)(4)
(2) Similarly forθ8,10consider the isomorphismC4×C2≃ac, (ab)2 ⊂ (C2×C2× C2× C2) ⋊ C2. This gives the following coset permutation representation producing the desired coveringU/∆ → U/∆′.
19 θ32,1(yi) ac, (bc)2 cac → (1 3)(2 4) b → (1 2)(3 4) bc → (1 4)(2 3) ac → (1)(2)(3)(4) inducing the generating vector(ac, (ac)3, ac(ab)2, (ac)3(ab)2).
(3) Forθ8,21consider the isomorphismQ ≃ac, (ab)2 ⊂ (D4× C2) ⋊ C2. We have the following permutation representation of the coveringU/∆ → U/∆′is
abac → (1 4)(2 3)
a → (1 3)(2 4)
b → (1 3)(2 4)
ac → (1)(2)(3)(4) thuss(θ−132,3(ac, ab)2 ) = (0; 4, 4, 4, 4).
(4) With similar calculations to the ones above we see thatθ8,3 extends toθ16,4 by C8 ≃ hai ⊂ D8. θ8,11 andθ8,12 extends toθ16,5andθ16,6 respectively byC4× C2 ≃ ha, bi ⊂ D4× C2. θ8,16extends toθ16,3byC2× C2× C2 ≃ ha, bc, cdi ⊂ C24. θ8,20 extends toθ16,5byC4× C2≃ ha, sbi ⊂ D4× C2.
Lemma 2.5. The actions of order16 are
θ16,1 : ∆(0; 4, 8, 8) → C8× C2 (a6b, a, ab) θ16,2 : ∆(0; 2, 2, 4, 4) → C4× C2× C2 (b, c, a, a3bc) θ16,3 : ∆(0; 2, 2, 2, 2, 2) → C24 (a, b, c, d, abcd) θ16,4 : ∆(0; 2, 2, 2, 2, 2) → D8 (s, sa, sa7, sa2, (a)4) θ16,5 : ∆(0; 2, 2, 2, 2, 2) → D4× C2 (s, s, sa, sab, b) θ16,6 : ∆(0; 2, 2, 2, 2, 2) → D4× C2 (s, sa, sb, sab, (a)2) θ16,7 : ∆(0; 2, 2, 4, 4) → D4× C2 (s, sb, ab, (a)3) θ16,8: ∆(0; 2, 2, 4, 4) → C8⋊3C2 (s, s, as, (as)3) θ16,9: ∆(0; 2, 2, 4, 4) → (C4× C2) ⋊ C2 (b, c, ab, ac) θ16,10: ∆(0; 4, 8, 8) → C8⋊5C2 (asa, a, as) θ16,11: ∆(1; 2) → C8⋊5C2 (a, s; a4) θ16,12: ∆(1; 2) → C4⋊C4 (s, t; s2) θ16,13: ∆(0; 2, 2, 4, 4) → G4,4 (b, a2, a, ab) θ16,14: ∆(0; 2, 2, 4, 4) → G4,4 (a2b, b, a, a) θ16,15: ∆(0; 2, 2, 4, 4) → G4,4 (b, b, a, a−1) θ16,16: ∆(0; 2, 2, 4, 4) → G4,4 (aba, aba−1b, a, ab) θ16,17: ∆(1; 2) → G4,4 (a, b; aba−1b)
whereθ16,1,θ16,2,θ16,7,θ16,8,θ16,9,θ16,10,θ16,11,θ16,12,θ16,14,θ16,15, andθ16,17
are non-maximal actions.
Proof: Here we will prove non-maximality.
(1)θ16,1extends toθ192in Lemma 2.11 with the isomorphismC8×C2≃t, (st7s)2 ⊂ (((C4× C2) ⋊ C4) ⋊ C3) ⋊ C2.
(2) θ16,2 extends to θ32,2 in Lemma 2.6 with the isomorphism C4× C2× C2 ≃ ha, bcb, ci ⊂ (C4× C2× C2) ⋊ C2.
20 2 The Equisymmetric Stratification of the Branch Locus of Riemann Surfaces of Genus 5
(3)θ16,7 extends toθ32,1 with the isomorphismD4× C2≃ac, c, (ab)2 ⊂ (C2× C2× C2× C2) ⋊ C2.
(4)θ16,8extends toθ32,3with the isomorphismC8⋊3C2≃ hab, bci ⊂ (D4× C2) ⋊ C2.
(5)θ16,9 extends toθ32,3 with the isomorphism(C4× C2) ⋊ C2 ≃ (ab)2, a, c ⊂ (D4× C2) ⋊ C2.
(6)θ16,10extends toθ64,2in Lemma 2.6 with the isomorphismC8⋊5C2≃ ha, [a, s]i ⊂ ((C8⋊C2) ⋊ C2) ⋊ C2.
(7)θ16,11extends toθ32,3with the isomorphismC8⋊5C2≃ hab, ci ⊂ (D4×C2)⋊C2.
(8)θ16,12 extends toθ32,2 with the isomorphismC4⋊C4 ≃ ha, bci ⊂ (C4× C2× C2) ⋊ C2.
(9)θ16,14extends toθ32,2with the isomorphismG4,4 ≃ hac, bi ⊂ (C4× C2× C2) ⋊ C2.
(10)θ16,15extends toθ32,1with the isomorphismG4,4≃ hac, bi ⊂ (C2× C2× C2× C2) ⋊ C2.
(11)θ16,17extends toθ32,1with the isomorphismG4,4≃ hab, ci ⊂ (C2× C2× C2× C2) ⋊ C2.
Lemma 2.6. The actions of order32 and order 64 are
θ32,1: ∆(0; 2, 2, 2, 4) → C24⋊C2 (cac, b, bc, ac) θ32,2: ∆(0; 2, 2, 2, 4) → (C4× C2× C2) ⋊ C2 (ba, b, c, ac) θ32,3: ∆(0; 2, 2, 2, 4) → (D4× C2) ⋊ C2 (abac, a, b, ac) θ32,4: ∆(0; 4, 4, 4) → (C4× C2) ⋊ C4 (s, t, t3s3) θ32,5: ∆(0; 4, 4, 4) → ((C4× C2) ⋊ C2) ⋊ C2 (s, st, ts2) θ32,6: ∆(0; 2, 8, 8) → (C8× C2) ⋊ C2 (t, s, s−1t) θ32,7: ∆(0; 2, 8, 8) → (C8⋊C2) ⋊ C2 (t, s, s−1t) θ64,1: ∆(0; 2, 4, 8) → ((C8× C2) ⋊ C2) ⋊ C2 (t, ta−1, a) θ64,2: ∆(0; 2, 4, 8) → ((C8⋊C2) ⋊ C2) ⋊ C2 (t, ta−1, a)
whereθ32,4,θ32,5,θ32,6,θ32,7andθ64,1are non-maximal actions.
Proof: Here we will prove non-maximality.
(1)θ32,4 extends toθ192in Lemma 2.11 with the isomorphism(C4× C2) ⋊ C4 ≃ (t7s)2, (st7)2 ⊂ (((C
4× C2) ⋊ C4) ⋊ C3) ⋊ C2.
(2) θ32,5 extends toθ64,2 define above with the isomorphism((C4× C2) ⋊ C2) ≃ hat, si ⊂ ((C8⋊C2) ⋊ C2) ⋊ C2.
(3)θ32,6 extends toθ192in Lemma 2.11 with the isomorphism(C8× C2) ⋊ C2 ≃ t, (st)2t4 ⊂ (((C
4× C2) ⋊ C4) ⋊ C3) ⋊ C2.
(4) θ32,7 extends toθ64,2 defined above with the isomorphism(C8⋊C2) ⋊ C2 ≃ a2s, a ⊂ ((C
8⋊C2) ⋊ C2) ⋊ C2.
(5)θ64,1extends toθ192in Lemma 2.11 with the isomorphism((C8× C2) ⋊ C2) ⋊ C2≃st2s2t, st6s2 ⊂ (((C4× C2) ⋊ C4) ⋊ C3) ⋊ C2.
Lemma 2.7. The actions of order3 are
θ3,0 : ∆(0; 3, 3, 3, 3, 3, 3, 3) → C3 (a, a, a, a, a, a2, a2) θ3,1 : ∆(1; 3, 3, 3, 3) → C3 (1, 1; a, a, a2, a2)
21
Lemma 2.8. The actions of order6 are
θ6,1: ∆(0; 2, 2, 2, 2, 3, 3) → C6 (a3, a3, a3, a3, a2, a4) θ6,2: ∆(0; 2, 3, 3, 3, 6) → C6 (a3, a2, a2, a4, a) θ6,3: ∆(0; 2, 2, 3, 6, 6) → C6 (a3, a3, a4, a, a) θ6,4: ∆(0; 6, 6, 6, 6) → C6 (a, a, a5, a5) θ6,5: ∆(1; 3, 3) → C6 (a, 1; a2, a4) θ6,6: ∆(0; 2, 2, 2, 2, 3, 3) → D3 (s, s, s, s, a, a2) θ6,7: ∆(1; 3, 3) → D3 (s, 1; a, a2)
whereθ6,4,θ6,5andθ6,7are non-maximal actions.
Proof: Here we will prove non-maximality.
(1)θ6,4extends toθ24,6in Lemma 2.10 with the isomorphismC6≃a5b ⊂ D6×C2.
(2)θ6,5extends toθ12,4with the isomorphismC6≃ hai ⊂ D6.
(3)θ6,7extends toθ12,5with the isomorphismD3≃a2, sa ⊂ D6.
Lemma 2.9. The actions of order12 are
θ12,1: ∆(0; 6, 12, 12) → C12 (a10, a, a) θ12,2: ∆(0; 2, 2, 6, 6) → C6× C2 (b, b, a, a−1) θ12,3: ∆(0; 2, 2, 6, 6) → C6× C2 (b, a3, a, ba2) θ12,4: ∆(0; 2, 2, 2, 2, 3) → D6 (s, s, sa3, sa, a2) θ12,5: ∆(0; 2, 2, 2, 2, 3) → D6 (a3, a3, s, sa4, a2) θ12,6: ∆(0; 2, 2, 6, 6) → D6 (sa2, s, a, a) θ12,7: ∆(1; 3) → D6 (s, a; a2) θ12,8: ∆(0; 3, 3, 3, 3) → A4 (a, a2, b, b2) θ12,9: ∆(0; 3, 3, 3, 3) → A4 (a, a, ab, b2) θ12,10: ∆(0; 2, 3, 4, 4) → C3⋊C4 (s2, t, s, st2) θ12,11: ∆(1; 3) → C3⋊C4 (t, s; t)
whereθ12,1,θ12,2,θ12,3,θ12,6,θ12,7,θ12,8,θ12,9andθ12,11are non-maximal actions. Remark 2.1. We note that there are two non-equivalent actions induced by the signature (0; 3, 3, 3, 3). Indeed by considering all actions B on (a, a2, b, b2), 218 in total, we note that none of the resulting generating vectors contain a pair of equal elements. These groups have the same representation up toGL(5, C)-conjugacy, see [17]. See appendix B for further information.
Proof: Here we will prove non-maximality.
(1)θ12,1extends toθ48,1in Lemma 2.11 with the isomorphismC12≃ hai ⊂ (C12× C2) ⋊ C2.
(2)θ12,2extends toθ24,6with the isomorphismC6× C2≃ ha, bi ⊂ D6× C2.
(3)θ12,3extends toθ24,7with the isomorphismC6× C2≃b, (ab)2 ⊂ (C6× C2) ⋊ C2.
(4)θ12,6extends toθ24,6with the isomorphismD6≃ hab, sbi ⊂ D6× C2.
(5)θ12,7extends toθ24,6with the isomorphismD6≃a2b, sa3 ⊂ D6× C2.
(6)θ12,8extends toθ48,5 in Lemma 2.11 with the isomorphismA4 ≃b3ab3, a ⊂ S4× C2.
22 2 The Equisymmetric Stratification of the Branch Locus of Riemann Surfaces of Genus 5
(7)θ12,9extends toθ24,2with the isomorphismA4≃(ab)2, b2 ⊂ S4× C2.
(8)θ12,11extends toθ24,7with the isomorphismC3⋊C4≃ab3, b2 ⊂ (C6× C2) ⋊ C2.
Lemma 2.10. The actions of order24 are
θ24,1: ∆(0; 2, 12, 12) → C12× C2 (b, a, a−1b) θ24,2: ∆(0; 2, 2, 3, 3) → A4× C2 (ab3, b3, b2, b4a) θ24,3: ∆(0; 2, 2, 3, 3) → A4× C2 (ab3, ab3, b2, b4) θ24,4: ∆(0; 3, 6, 6) → A4× C2 (b4, ba, ab) θ24,5: ∆(0; 2, 2, 3, 3) → S4 (ab, ab, a, a−1) θ24,6: ∆(0; 2, 2, 2, 6) → D6× C2 (s, a3, sa4b, a5b) θ24,7: ∆(0; 2, 2, 2, 6) → (C6× C2) ⋊ C2 (a, b3, ab4, aba) θ24,8: ∆(0; 4, 4, 6) → (C3⋊C4) × C2 (s, st, s2t5)
whereθ24,1,θ24,3,θ24,4,θ24,5, andθ24,8are non-maximal actions.
Proof: Here we will prove non-maximality.
(1) θ24,1 extends toθ48,1 with the isomorphismC12× C2 ≃ a, (ab)2 ⊂ (C12× C2) ⋊ C2.
(2)θ24,3extends toθ48,5with the isomorphismA4× C2≃ab2a2, ac ⊂ S4× C2.
(3)θ24,4extends toθ96,1with the isomorphismA4× C2≃ h[a, b], ai ⊂ A4× C2× C2
(4)θ24,5extends toθ48,5with the isomorphismS4≃ ha, bi ⊂ S4× C2.
(5)θ24,1extends toθ48,1with the isomorphism(C3⋊C4) × C2≃ab, a2 ⊂ (C12× C2) ⋊ C2.
Lemma 2.11. The actions of order48, 96 and 192 are
θ48,1: ∆(0; 2, 4, 12) → (C12× C2) ⋊ C2 (b, ba, a11) θ48,2: ∆(0; 2, 6, 6) → A4× C2× C2 (b, a, a5b) θ48,3: ∆(0; 3, 4, 4) → A4⋊C4 (a, b, b3a2) θ48,4: ∆(0; 3, 4, 4) → A4⋊C4 (a, b3, ba2) θ48,5: ∆(0; 2, 2, 2, 3) → S4× C2 (ab, b2c, ca2b3, a) θ96,1: ∆(0; 2, 4, 6) → (A4× C2× C2) ⋊ C2 (b, bac, ca5) θ96,2: ∆(0; 3, 3, 4) → ((C4× C2) ⋊ C4) ⋊ C3 (s, s2t7, t) θ192: ∆(0; 2, 3, 8) → (((C4× C2) ⋊ C4) ⋊ C3) ⋊ C2 (st5, s, t5)
whereθ48,2,θ48,3andθ96,2are non-maximal actions.
Proof: Here we will prove non-maximality.
(1)θ48,2extends toθ96,1sinceA4× C2× C2⊂ (A4× C2× C2) ⋊ C2
(2)θ48,3extends toθ96,1with the isomorphismA4⋊C4≃a4, a3cb ⊂ (A4× C2× C2) ⋊ C2.
(3)θ96,2extends toθ192.
23 θ5: ∆(1; 5, 5) → C5 (1, 1; a, a4) θ10,1 : ∆(0; 2, 2, 10, 10) → C10 (a5, a5, a, a9) θ10,2 : ∆(0; 2, 2, 2, 2, 5) → D5 (s, s, s, sa5, a) θ10,3 : ∆(1; 5) → D5 (s, a; a2) θ15: ∆(0; 3, 15, 15) → C15 (a5, a4, a8) θ20,1 : ∆(0; 2, 20, 20) → C20 (a10, a, a9) θ20,2 : ∆(0; 2, 2, 2, 10) → D10 (a5, s, sa2, a3) θ20,3 : ∆(0; 4, 4, 10) → C5⋊C4 (ab, a, a2b) θ30: ∆(0; 2, 6, 15) → D5× C3 (b3, ba, bab) θ40: ∆(0; 2, 4, 20) → D5× C4 (b, ba, a−1) θ60: ∆(0; 3, 3, 5) → A5 (b, bab, b2a4b) θ80: ∆(0; 2, 5, 5) → (C24) ⋊ C5 (b, a, a−1b) θ120: ∆(0; 2, 3, 10) → A5× C2 (ba2c, b, a2c) θ160: ∆(0; 2, 4, 5) → ((C24) ⋊ C5) ⋊ C2 (ab, b−1, a−1)
whereθ5,θ10,1,θ10,3,θ15,θ20,1,θ20,3,θ60andθ80are non-maximal actions.
Proof: For θ10,3 consider the subgroup a2, sa
⊂ D10, clearly θ20,2 induces θ10,3. Using Table 1.4 it is easy to see thatθ5, θ10,1, θ15, θ20,1, θ20,3,θ60 andθ80 are non-maximal actions due to the fact there is only one action for each group and the actions θ30,θ40,θ120andθ160do exist.
Lemma 2.13. The actions of order11 and 22 are
θ11,1: ∆(0; 11, 11, 11) → C11 (a, a2, a8) θ11,2: ∆(0; 11, 11, 11) → C11 (a, a, a9) θ22: ∆(0; 2, 11, 22) → C22 (a11, a10, a)
whereθ11,2is a non-maximal action.
Proof: Here we will prove non-maximality.
There are two classes of actions ofC11with representativesθ11,1: ∆ → C11, defined by (a, a2, a8) and θ
11,2 : ∆ → C11, defined by(a, a, a9). Now θ11,2 extends to φ : ∆(0; 2, 11, 22) → C22 defined by(b11, b10, b). By Theorem 1.2 φ−1b2 is a group with signature (0; 11, 11, 11) and the images of the elliptic generators by φ (with the isomorphismb2→ a) are a, a and a9inducingθ
11,2, thusM 11,2 5 ≡ M 22 5 . By Theorem 1.2s(φ−12 b11 ) = (0; 2,. . ., 2) thus12 M 22 5 ⊂ M C2,0
5 . The epimorphismθ11,1 yield a maximal action ofC11inM5producing an isolated pointM
11,1 5 .
Now each maximal action corresponds to an equisymmetric strata [5]. Letting the strata corresponding to a maximal actionθi,jbe denotedMi,j5 we have from Lemma 2.2 through Lemma 2.13 above the following theorem.
24 2 The Equisymmetric Stratification of the Branch Locus of Riemann Surfaces of Genus 5
Strata dim. Strata dim. Strata dim. Strata dim.
M2,05 9 M 6,1 5 3 M 8,19 5 2 M 24,7 5 1 M2,15 8 M 6,2 5 2 M 10,2 5 2 M305 0 M2,25 7 M6,35 2 M11,15 0 M32,15 1 M2,35 6 M6,65 3 M12,45 2 M32,25 1 M3,05 4 M8,15 1 M12,55 2 M32,35 1 M3,15 4 M 8,2 5 1 M 12,10 5 1 M405 0 M4,15 3 M 8,4 5 2 M 16,3 5 2 M 48,1 5 0 M4,25 3 M8,55 2 M16,45 2 M48,45 0 M4,35 4 M8,65 2 M16,55 2 M48,55 1 M4,45 3 M8,75 2 M16,65 2 M64,25 0 M4,55 4 M 8,8 5 2 M 16,13 5 1 M1205 0 M4,75 5 M8,135 3 M16,165 1 M96,15 0 M4,85 5 M8,145 3 M20,25 1 M160 5 0 M4,95 5 M8,155 3 M22 5 0 M1965 0 M4,105 4 M 8,17 5 3 M 24,2 5 1 M4,115 4 M 8,18 5 3 M 24,6 5 1
3
The Orbifold Structure of the Moduli
Space
Here we will compute the structure of the equisymmetric stratification of the branch lo-cus. Remember thatG = Aut(X) determines a conjugacy class of subgroups of M (Γ), defining a symmetry type ofX [5]. MGg is the set of classes of surfaces with full auto-morphism group inducing the symmetryG and MGg is the set of surfaces such that the automorphisms group contains a subgroup in the class defined byG. If G contains a sub-groupG′ thenMG
g ⊂ M G′
g . The action ofG determines a Fuchsian group ∆′ and an epimorphismθ′: ∆′→ G. Letting ∆′= θ′−1(G′) and examining the monodromy of the covering
U/∆ → U/∆′
we can find the induced actionθ : ∆ → H by the use of Theorem 1.2. Now as we have a one to one correspondence of the cosets ofH and the cosets of ∆ it is enough to study the permutations in the finite groupG. As in Chapter 2, all the group symbolic calculations in finding the induced actions have been done with GAP.
We begin with a theorem giving the structure of the branch locus in terms of strata corresponding to actions of prime order.
Theorem 3.1. [1] The branch locus is contained in
M2,05 ∪ M2,15 ∪ M2,25 ∪ M52,3∪ M3,05 ∪ M3,15 ∪ M11,15 .
Proof: We use Theorem 1.12 to prove the statement.
(1)M2,05 , M2,15 ,M2,25 andM2,35 correspond to epimorphismsθ : ∆ → C2 with signatures s(∆0) = (0; 2,. . ., 2), s(∆12 1) = (1; 2,. . ., 2), s(∆8 2) = (2; 2, 2, 2, 2) and s(∆3) = (3; −) respectively.
(2) The strataM3,05 and M3,15 correspond to epimorphisms θ : ∆ → C3 where s(∆0) = (0; 3,. . ., 3) and s(∆7 1) = (1; 3, 3, 3, 3) respectively.
(3)M55is induced by non-maximal epimorphismsθ : ∆ → C5,s(∆) = (1; 5, 5). They extend to surface kernel epimorphismφ : ∆′→ D
5=a, s|a5= s2= (sa)2= 1 ,
26 3 The Orbifold Structure of the Moduli Space
s(∆′) = (0; 2, 2, 2, 2, 5), defined by (s, s, s, sa, a−1). We see that s(φ−1hai) = (1; 5, 5) ands(φ−1hsi) = (2; 2, 2, 2, 2). Thus M5
5≡ M D5,θ
5 ⊂ M 2,2 5 .
(4) Signature(0; 11, 11, 11). There are two classes of actions of C11 with represen-tatives θ11,1 : ∆ → C11, defined by (a, a2, a−3) and θ11,2 : ∆ → C11, defined by (a, a, a−2). Now θ
11,2extends toφ : ∆(0; 2, 11, 22) → C22defined by(b11, b10, b). By Theorem 1.2φ−1b2 is a group with signature (0; 11, 11, 11) and the images of the ellip-tic generators byφ (with the isomorphism b2→ a) are a, a and a−2. SoM11,2
5 ≡ M 22 5 . By Theorem 1.2s(φ−12 b11 ) = (0; 2,. . ., 2), thus M12 22 5 ⊂ M C2,0 5 . The epimorphism θ11,1yield a maximal action ofC11inM5producing an isolated pointM
11,1 5 .
3.1
Surfaces with Automorphisms of Order a Power
of 2
We begin with the strata corresponding to actions of2-groups. We show the inclusion relations of the strata in decreasing order of the number of automorphisms of the surfaces. From [1] it follows that those strata belong to the same connected component of the branch locus.
Theorem 3.2.
M64,25 ⊂ M32,15 ∩ M16,165 ∩ M8,25 ∩ M8,145 ∩ M4,25 ∩ M4,55
Proof: Maximal subgroups of((C8⋊C2)⋊C2)⋊C2are;t, s, a2s ≃ (C2×C2×C2× C2) ⋊ C2,hat, si ≃ ((C4× C2) ⋊ C2) ⋊ C2anda2s, a ≃ ((C8⋊C2) ⋊ C2). Now θ64,2 induces actionsθ32,1,θ32,5andθ32,7, where actionsθ32,5andθ32,7are non-maximal.
(1) Consider the actionθ32,5: ∆(0; 4, 4, 4) → ((C4× C2) ⋊ C2) ⋊ C2. Maximal sub-groups of((C4× C2) ⋊ C2) ⋊ C2ares, (st)2 , st, s2 ≃ G4,4ands2t, t, [t, s−1] ≃ D4× C2. The permutation representations of the monodromy of the coverings induced are given by the following table:
s, (st)2 st, s2 s2t, t, [t, s−1] s → (1)(2) (1 2) (1 2) st → (1 2) (1)(2) (1 2) ts2→ (1 2) (1 2) (1)(2)
From the table above we see thats(θ32,5−1 s, (st)2 ) = (0; 2, 2, 4, 4) and ((st)2, (ts2)2, tst, s) ≃ (s(st)2s, s(st)2s−1(st)2, s, s(st)2) which corresponds to the action θ
16,16: ∆(0; 2, 2, 4, 4) → G4,4as in lemma 2.5. We also see thats(θ−132,5st, s2 ) = (0; 2, 2, 4, 4) and (s2, (ts2)2, ts, st) ≃ ((st)s2(st), (st)s2(st)−1s2, st, (st)s2) which also corresponds to the action θ16,16. Finally s(θ32,5−1 s2t, t, [t, s−1] ) = (0; 2, 2, 4, 4), inducing the action θ16,7 : ∆(0; 2, 2, 4, 4) → D4× C2which is non-maximal.
(2) Consider the action θ16,7 : ∆(0; 2, 2, 4, 4) → D4× C2. Maximal Subgroups ofD4× C2areha, bi ≃ C4× C2,a2, s, b , a2, sa, b ≃ C2× C2× C2andha, si, hab, si, ha, sbi, hab, sbi ≃ D4. The permutation representations of the monodromy of the coverings induced are given by the following table:
3.1 Surfaces with Automorphisms of Order a Power of 2 27 ha, bi a2, s, b a2, sa, b s → (1 2) (1)(2) (1 2) sb → (1 2) (1)(2) (1 2) ab → (1)(2) (1 2) (1 2) a3→ (1)(2) (1 2) (1 2)
Nows(θ−116,7ha, bi) = (0; 4, 4, 4, 4) and (ab, a3b, a3, a) corresponds to the action θ8,10 : ∆(0; 4, 4, 4, 4) → C4× C2in Lemma 2.4. Furthers(θ16,7−1 a2, s, b ) = (0; 2, 2, 2, 2, 2, 2) and(s, s, sb, sb, a2, a2) corresponds to the action θ
8,14: ∆(0; 2,. . ., 2) → C6 2× C2× C2. Finally s(θ16,7−1 a2, sa, b ) = (1; 2, 2), inducing the action θ8,16 : ∆(1; 2, 2) → C2× C2× C2.θ8,10andθ8,16are non-maximal, forθ8,16see Theorem 3.6. For the rest of the subgroups the permutation representations of the monodromy of the coverings induced are given by the following table:
ha, si hab, si ha, sbi hab, sbi s → (1)(2) (1)(2) (1 2) (1 2) sb → (1 2) (1 2) (1)(2) (1)(2) ab → (1 2) (1)(2) (1 2) (1)(2) a3→ (1)(2) (1 2) (1)(2) (1 2)
From the table above we see thats(θ−116,7ha, si) = s(θ−116,7hab, si) = s(θ16,7−1 ha, sbi) = s(θ−116,7hab, sbi) = (0; 2, 2, 2, 4, 4), inducing the action θ8,19: ∆(0; 2, 2, 2, 4, 4) → D4.
(3) Consider the actionθ8,10 : ∆(0; 4, 4, 4, 4) → C4× C2. Maximal subgroups of C4× C2arehai , habi ≃ C4anda2, b , habi ≃ C2× C2. The permutation representa-tions of the monodromy of the coverings induced are given by the following table:
hai habi a2, b
a → (1)(2) (1 2) (1 2)
a3→ (1)(2) (1 2) (1 2) ab → (1 2) (1)(2) (1 2) a3b → (1 2) (1)(2) (1 2)
By Theorem 1.2 we see thats(θ−18,10hai) = s(θ8,10−1 habi) = (0; 2, 2, 4, 4, 4, 4), inducing the actionθ4,2. Also,s(θ−18,10a2, b ) = (1; 2, 2, 2, 2), inducing the action θ4,5.
(4) Consider the actionθ32,7 : ∆(0; 2, 8, 8) → (C8⋊C2) ⋊ C2. Maximal subgroups of((C8⋊C2) ⋊ C2) are hs, [s, t]i , hst, [s, t]i ≃ C8⋊5C2ands2, t, [s, t] ≃ D4× C2. The coset permutations are given in the following table:
hs, [s, t]i hst, [s, t]i s2, t, [s, t]
t → (1 2) (1 2) (1)(2)
s → (1)(2) (1 2) (1 2)
s−1t → (1 2) (1)(2) (1 2)
Nows(θ−132,7hs, [s, t]i) = s(θ32,7−1 hst, [s, t]i) = (0; 4, 8, 8) inducing the action θ16,10 : ∆(0; 4, 8, 8) → C8⋊5C2ands(θ−132,7s2, t, [s, t] ) = (0; 2, 2, 4, 4) inducing the action θ16,7∆(0; 2, 2, 4, 4) → D4× C2in Lemma 2.5, whereθ16,7andθ16,10are non-maximal, see Theorem 3.4.
28 3 The Orbifold Structure of the Moduli Space
(5) Consider the actionθ16,10. Maximal subgroups ofC8⋊5C2arehai , hasi ≃ C8 anda2, s ≃ C
4× C2. (a4, a6, a5, a) corresponds to the action θ8,2,(a2, a3, sa2, sa6) corresponds to the actionθ8,10.
Theorem 3.3. M32,15 ⊂ M 16,3 5 ∩ M 16,5 5 ∩ M 8,14 5 ∩ M 8,19 5 ∩ M 4,2 5 ∩ M 4,5 5
Proof: Maximal subgroups of(C2×C2×C2×C2)⋊C2are;b, c, (ab)2, (ac)2 ≃ C2× C2× C2× C2,a, b, (ac)2 , ac, c, (ab)2 , a, bc, (ac)2 ≃ D4× C2andhab, ci,hac, bi, hac, bci ≃ G4,4. For subgroupsG of (C2× C2× C2× C2) ⋊ C2the monodromies of the coveringsU/θ−132,1(G) → U/∆(0; 2, 2, 2, 4) are given by:
a, b, (ac)2 a, c, (ab)2 a, bc, (ac)2 cac → (1)(2) (1)(2) (1)(2) b → (1)(2) (1 2) (1 2) bc → (1 2) (1 2) (1)(2) ac → (1 2) (1)(2) (1 2)
From the table above we see thats(θ32,1−1 a, b, (ac)2 ) = (0; 2, 2, 2, 2, 2) and (b, b, a, cac, (ac)2) = (b, b, a, a(ac)2, (ac)2) corresponds to the action θ
16,5 : ∆(0; 2, 2, 2, 2, 2) → D4× C2. Also, s(θ32,1−1 ac, c, (ab)2 ) = (0; 2, 2, 4, 4) inducing the non-maximal ac-tion θ16,7 : ∆(0; 2, 2, 4, 4) → D4 × C2, see Theorem 3.2 for calculations. Finally s(θ−132,1a, bc, (ac)2 ) = (0; 2, 2, 2, 2, 2) and the generating vector (bc, bc, a, cac, (ac)2) = (bc, bc, a, a(ac)2, (ac)2) corresponds to the action θ
16,3 : ∆(0; 2, 2, 2, 2, 2) → C2× C2× C2× C2. The permutations of the cosets of the rest of the subgroups are given in the following table:
hab, ci hac, bi hac, bci
cac → (1 2) (1 2) (1 2)
b → (1 2) (1)(2) (1 2)
bc → (1 2) (1 2) (1)(2)
ac → (1 2) (1)(2) (1)(2)
Nows(θ32,1−1 hab, ci) = (1; 2), inducing the action θ16,17 : ∆(1; 2) → G4,4. We also see thats(θ−132,7hac, bi) = (0; 2, 2, 4, 4) and the generating vector (b, b, ac, ca) corresponds to the actionθ16,15 : ∆(0; 2, 2, 4, 4) → G4,4, similarlys(θ32,1−1 hac, bci) = (0; 2, 2, 4, 4) and the generating vector(bc, bc, ac, ca) corresponds to the action θ16,15. Bothθ16,15and θ16,17are non-maximal.
(1) Consider the actionsθ16,15andθ16,17. Maximal subgroups ofG4,4areha, [a, b]i , ab, a2 ≃ C
4× C2anda2, b, [a, b] ≃ C2× C2× C2. The permutations of the cosets by the actionsθ16,15andθ16,17are given in the following table:
3.1 Surfaces with Automorphisms of Order a Power of 2 29
ha, [a, b]i ab, a2
a2, b, [a, b] b → (1 2) (1 2) (1)(2) b → (1 2) (1 2) (1)(2) a → (1)(2) (1 2) (1 2) a3→ (1)(2) (1 2) (1 2) [a, b] (1)(2) (1)(2) (1)(2)
From the table above we see thats(θ16,15−1 ha, [a, b]i) = (0; 4, 4, 4, 4) and the generating vector (a, a3, bab, ba3b) corresponds to the action θ
8,10 : ∆(0; 4, 4, 4, 4) → C4× C2 which is non-maximal, for calculations, see Theorem 3.2. s(θ−116,15ab, a2 ) = (1; 2, 2) and(a2, a2) corresponds to the non-maximal action θ
8,12 : ∆(1; 2, 2) → C4× C2, see Theorem 3.9. Also,s(θ−116,15a2, b, [a, b] ) = (0; 2, 2, 2, 2, 2, 2) and (b, b, a2, a2, aba, aba) corresponds to the actionθ8,14. Forθ16,17we see from the table thats(θ16,17−1 ha, [a, b]i) = s(θ−116,17ab, a2 ) = s(θ−1
16,17a2, b, [a, b] ) = (1; 2, 2) and ([a, b], [a, b]) correspond to the actionsθ8,11: ∆(1; 2, 2) → C4× C2andθ8,16 : ∆(1; 2, 2) → C2× C2× C2.θ8,11 andθ8,16are non-maximal, see Theorems 3.8 and 3.6.
Theorem 3.4. M32,25 ⊂ M 16,5 5 ∩ M 16,6 5 ∩ M 8,6 5 ∩ M 8,14 5 ∩ M 8,15 5 ∩ M 4,2 5 ∩ M 4,5 5
Proof: Maximal subgroups of (C4× C2× C2) ⋊ C2are; a, c, (bc)2 ≃ C4× C2× C2,a, b, (bc)2 , bc, c, a2 , bac, c, a2 ≃ D4× C2, ha, bci ≃ C4⋊C4,hac, bi and hac, bai ≃ G4,4. For subgroupsG of (C4× C2× C2) ⋊ C2the monodromies of the coveringsU/θ−132,2(G) → U/∆(0; 2, 2, 2, 4) are given by:
a, c, (bc)2
ha, bci hac, bi hac, bai
ba → (1 2) (1 2) (1 2) (1)(2)
b → (1 2) (1 2) (1)(2) (1 2)
c → (1)(2) (1 2) (1 2) (1 2)
ac → (1)(2) (1 2) (1)(2) (1)(2)
From the table above we see that s(θ32,2a, c, (bc)2 ) = (0; 2, 2, 4, 4), inducing the action θ16,2. s(θ32,2ha, bci) = (1; 2), inducing the action θ16,12. s(θ32,2hac, bi) = (0; 2, 2, 4, 4) and the generating vectors (a−1ba, b, ac, ac) = ((ac)2b, b, ac, ac) corre-sponds to the actionθ16,14. s(θ32,2hac, bai) = (0; 2, 2, 4, 4) and the generating vector (a−1ba2, ba, ac, ac) = ((ac)2ba, ba, ac, ac) corresponds to the action θ16,14.θ16,2,θ
16,12 andθ16,14are non-maximal. For the last three subgroups we have:
a, b, (bc)2 bc, c, a2 bac, c, a2 ba → (1)(2) (1 2) (1)(2) b → (1)(2) (1)(2) (1 2) c → (1 2) (1)(2) (1)(2) ac → (1 2) (1 2) (1 2)
We have the signatures(θ32,2a, b, (bc)2 ) = (0; 2, 2, 2, 2, 2), by Theorem 1.2, and the generating vector (b, ba, cbc, cbac, a2) = (b, ba, b(bc)2, ba(bc)2, a2) corresponds to the
30 3 The Orbifold Structure of the Moduli Space
actionθ16,6: ∆(0; 2, 2, 2, 2, 2) → D4× C2. We do also see thats(θ32,2bc, c, (bc)2 ) = (0; 2, 2, 2, 2, 2) and the generating vector (b, b, c, a−1ca, a2) = (b, b, c, ca2, a2) corre-sponds to the actionθ16,5 : ∆(0; 2, 2, 2, 2, 2) → D4× C2. s(θ32,2bac, c, (bc)2 ) = (0; 2, 2, 2, 2, 2), which also induce the action θ16,5.
(1) Consider the actionθ16,2: ∆(0; 2, 2, 4, 4) → C4× C2× C2. Maximal subgroups of C4× C2× C2 are: ha, bi , ha, ci , ha, bci , hac, bi , hab, ci , hab, bci ≃ C4 × C2 and a2, b, c ≃ C
2× C2× C2. θ16,2is represented by the generating vector(b, c, a, a3bc) and since the group is abelian it is easy to see that the subgroups induce the actionθ8,6 : ∆(0; 2, 2, 2, 4, 4) → C4× C2, the non-maximal actionθ8,9 : ∆(0; 4, 4, 4, 4) → C4× C2, the non-maximal actionθ8,12 : ∆(1; 2, 2) → C4× C2, see Theorem 3.9, and the action θ8,14.
(2) Consider the actionθ8,9. Maximal subgroups ofC4× C2arehai , habi ≃ C4and a2, b ≃ C
2× C2. We see thats(θ8,9−1hai) = s(θ8,9−1habi) = (0; 2, 2, 4, 4, 4, 4), inducing the actionθ4,2and thats(θ8,9−1a2, b ) = (1; 2, 2, 2, 2), inducing the action θ4,5.
(3) Consider the actionθ16,12: ∆(1; 2) → C4⋊C4. Maximal subgroups ofC4⋊C4 ares, t2 , t, s2 , st, s2 ≃ C4× C2. (s2, s2) corresponds to the action θ8,12,θ8,11 andθ8,11 respectively. θ8,11 andθ8,12 are non-maximal, see Theorems 3.8 and 3.9 for calculations.
(4) Consider the actionθ16,14 : ∆(0; 2, 2, 4, 4) → G4,4. Maximal subgroups ofG4,4 areha, [a, b]i ,ab, a2 ≃ C
4× C2anda2, b, [a, b] ≃ C2× C2× C2. The permutations of the cosets are:
ha, [a, b]i ab, a2
a2, b, [a, b]
a2b → (1 2) (1 2) (1)(2)
b → (1 2) (1 2) (1)(2)
a → (1)(2) (1 2) (1 2)
a → (1)(2) (1 2) (1 2)
From the table we see thats(θ−116,14ha, [a, b]i) = (0; 4, 4, 4, 4) and (a, a, bab, bab) corre-sponds to the actionθ8,9 : ∆(0; 4, 4, 4, 4) → C4× C2. We also see thats(θ16,14−1 ab, a2 ) = (1; 2, 2) and (a2, a2) corresponds to the non-maximal action θ
8,11 : ∆(1; 2, 2) → C4× C2, see Theorem 3.8 for calculations. Finallys(θ−116,14a2, b, [a, b] ) = (0; 2, 2, 2, 2, 2, 2) and(a2, a2, b, a2b, aba3, a3ba3) corresponds to the action θ
8,15 : ∆(0; 2,. . ., 2) → C6 2× C2× C2. Theorem 3.5. M32,35 ⊂ M 16,6 5 ∩ M 8,7 5 ∩ M 8,18 5 ∩ M 8,19 5 ∩ M 4,2 5 ∩ M 4,5 5
Proof: Maximal Subgroups of(D4×C2)⋊C2are;hab, ai , habc, ai ≃ D8,(ab)2, b, c ≃ D4× C2,hacb, bi , hab, bci ≃ C8⋊3C2,(ab)2, a, c ≃ (C4× C2) ⋊ C2andhab, ci ≃ C8×5C2. For subgroupsG of (D4 × C2) ⋊ C2 the monodromies of the coverings U/θ−1
