On the structure of connected components of the moduli spaces of compact Riemann surfaces

Using the Broughton’s equisymmetric stratification of the moduli space of compact Riemann surfaces of genus g≥2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g \ge 2$$\end{document}, Bartolini, Costa and Izquierdo have shown that its singular locus Sg\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S_{g}$$\end{document} is disconnected except for g=3,4,7,13,17,19,59\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g = 3, 4, 7, 13, 17, 19, 59$$\end{document}. One of a lot of problems motivated by this result concerns the study of its connected components, both from the quantitative and qualitative points of view. In 2012, Bartolini and Izquierdo have shown that all strata corresponding to the actions of Z2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Z_{2}$$\end{document} and Z3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Z_{3}$$\end{document} are contained in a single connected component Cg2,3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^{2,3}_g$$\end{document}. The other components which were known to these authors were those composed by single strata and they asked whether there are components different from Cg2,3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^{2,3}_ g $$\end{document} being the union at least two equisymmetric strata. The aim of this paper is to give an affirmative answer to this question, by showing that there are connected components composed of precisely two strata in Sg\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S_{g}$$\end{document} for g=rp(p-1)/2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g = rp(p -1)/2$$\end{document} where r≥6\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r \ge 6$$\end{document} and p>5r+3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p > 5r + 3$$\end{document} is a prime for which pr+2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$pr + 2$$\end{document} is also a prime.

of analytic variety and the aim of our paper is to study the nature of connected components in the singular locus S g using the Broughton [5] equisymmetric stratification of M g .
This stratification turned out to be very fruitful and, for instance, allowed Bartolini, Costa, Izquierdo and Porto to successfully study the connectivity of S g in an extensive series of articles whose culmination and capstone was a description of those genera g for which S g is connected. To be precise they showed in [1,3] that S g is disconnected except for g = 3, 4, 7, 13, 17, 19, 59. One of many problems motivated by this result concerns the study of its connected components, both from the quantitative and qualitative points of view. In [4] Bartolini and Izquierdo have shown that all strata corresponding to the actions of Z 2 and Z 3 are contained in a single connected component C 2,3 g . The other components which were known to the authors of [3] were composed by single strata.
Looking for components of a different type was one of the topics of Bartolini's doctoral dissertation [2] in which he showed that there are no such components in M g for g ≤ 25. All this prompted Bartolini, Costa and Izquierdo, to ask in [3] whether there exist components other than C 2,3 g containing more than one stratum, and the purpose of our paper is to answer in the affirmative, by showing that Given an integer r ≥ 6 and a prime p > 5r + 3 for which pr + 2 is also prime, there is a connected component, composed by two strata in S g for g = r p( p − 1)/2.
The equisymmetric stratification mentioned above can be defined in some equivalent ways. The most illustrative one is parameterizing strata by conjugacy classes of finite subgroups of the mapping class group M g because of its role in the construction of the moduli space as the orbit space of its action on Teichmüller space T g and its singular locus S g , viewed as the set of branch points of the projection T g → M g . According to it, in principle, the equisymmetric stratum can be seen as the subset M G g of M g composed by the points parameterizing conformal classes of Riemann structures on a closed topological surface of genus g on which a given finite group G of homeomorphisms act as the full group of conformal automorphisms. Observe that the stratum defined in such a way actually depends on the conjugacy class of the subgroup [G] of the mapping class group M g induced by G. Having all of these, the results of Broughton [5] in question can be gathered into the following statement: • Nonempty equisymmetric strata M G g are smooth, connected, locally closed algebraic subvarieties of M g . • Zariski closures M G g of M g are defined by relaxing conditions defining M G g assuming only that, in notations employed there, G is a subgroup of Aut(X ).
• M G g are closed, irreducible algebraic subvarieties of M g . Following [7] and [8], we still would like to add that M G g is closed in the topology of M g induced from the one of Teichmüller space T g which in turn is homeomorphic to R 3(g−1) . These concepts will be explained with more detail in the next section but already at this stage, we can realize that both M G g and M G g cover M g . In addition M G g are pairwise disjoint. We say that two strata M G g and M G g are not separated if and only if Observe that two strata M G,θ g and M G ,θ g being separated, they may be contained in the same connected component. This happens if there is a sequence of strata such that M G i g and M G i+1 g are not separated for arbitrary 1 ≤ i ≤ n − 1. In principle, such a sequence may exist for G = Z 2 , Z 3 and G ∈ {Z 2 , Z 3 } what shows why the problem of searching components other of C 2,3 g is not easy.

On topological classification of conformal actions on Riemann surfaces
The above definition of equisymmetric stratification is pretty clear and easy to understand, but another question is an approach that allows the effective use of this concept. Let us define, for the needs of this section, an F-group or an abstract Fuchsian group to be an abstract group with the presentation which will be denoted by the sequence called its signature. Observe that each cocompact Fuchsian group is an F-group and conversely, it is well known that an F-group can be realized as a Fuchsian group if and only if is positive and in such case (5) is the normalized hyperbolic area of any fundamental region of an arbitrary Fuchsian group with the signature (4). We shall denote that area by μ( ). Now, by the Riemann uniformization theorem, a compact Riemann surface X = X g of genus g ≥ 2 can be considered as the orbit space H/ with respect to the action of a torsionfree Fuchsian group which is isomorphic to the fundamental group of X . Furthermore, an abstract group G can be seen as a group of automorphisms of a surface so represented if G ∼ = / for some Fuchsian group , say with signature (4). Conversely, given an F-group with presentation (3) and a torsion-free kernel epimorphism θ : → G, called smooth or surface-kernel, there is a realization of as a group of isometries of the hyperbolic plane so that for = ker θ , G ∼ = / is a group of conformal automorphisms of a Riemann surface X = H/ . The genus of X can be found from the Riemann-Hurwitz formula which in a general setting says that for an arbitrary Fuchsian group and its subgroup we have Next, two smooth epimorphisms θ i : i → G i define topologically equivalent actions if and only if the diagram commutes for some isomorphisms ϕ and ψ. The last means that the above equivalence classes of smooth epimorphisms from abstract groups onto G correspond to conjugacy classes of finite subgroups of the mapping class group M g of genus g. The above describes the topological aspects of the theory fully.

On conformal realization of topological actions on compact surfaces.
The moduli space M g of a compact Riemann surface of genus g ≥ 2, in the spirit we shall use in this paper, can be defined in a few steps. For the first let be an F-group corresponding to some signature (4) and consider the set of monomorphisms called the Weil space of . Having it, we define the Teichmüller space T ( ) of as the factor space W( )/ ∼, where ρ 1 ∼ ρ 2 if they are conjugate in the group of all isometries of the hyperbolic plane which means that The space T ( ) constructed in such a way is a manifold of dimension 3(h − 1) + r and we have a well-defined action of the mapping class group M( ) = Aut( )/Inn( ) on it by where the moduli space M( ) we have looked for is the quotient space. The mentioned space M g is obtained if we take for , in the above construction, the fundamental group = a 1 , b 1 , . . . , a g , b g : [a 1 , b 1 ] . . . [a g , b g ] of a surface of genus g. M( ) is only a variety but it has the same dimension as T ( ). The singular locus S g of M g (synonymously called branch locus) is composed of the points representing surfaces with nontrivial groups of conformal automorphisms.

How the above works
Summing up, the equisymmetric strata are parameterized by the classes of smooth epimorphisms θ : → G from an F-group onto a finite group G. Such a stratum M G,θ g is composed by the set of points of the moduli space M g corresponding to representatives of Teichmüller classes of Fuchsian group so that • is maximal in the class of all F-groups or • is not maximal but none of its extensions contains = ker θ as a normal subgroup.
The Zariski closure M G,θ g of such a stratum is obtained by omitting the above two conditions. Note that these conditions are purely algebraic. So having the class of F-groups well understood, they are effectively verifiable in concrete cases. For example, essential in the study of connectedness (cf. (1)), the intersection for two nonempty strata defined by smooth epimorphisms θ i : i → G i is nonempty if and only if there exists an F-group 1 extending

On notations used in the paper
Overusing the language, we will write M G,θ g as M G g or M θ g if it will be known from elsewhere what θ : → G we have in mind, and there is no confusion. Also whenever we have an extension K of a group G acting conformally on a Riemann surface X , we will understand that K is an extension of this action. Finally, as above, , , etc. will stand for both F-groups and Fuchsian groups depending on the context.

On conformal actions on compact Riemann surfaces
A useful tool for the study of extensions of topological actions is provided by the following theorem of Macbeath from [10] which we state as a lemma:

Then the number F(ϕ) of points of X fixed by a nontrivial element ϕ of G is given by the formula
where the operator N stands for the normalizer and the sum is taken over those i for which ϕ is conjugate to a power of θ(x i ).
The next lemma can be seen as a particular case of the more general result of Peterson, Russell and Wootton ( [11], Thm 6). Being a particular case, it requires less involved proof which we supply for the sake of completeness and a reader's convenience.

Lemma 3.2 Let H be a group of automorphisms of a compact Riemann surface X of genus g ≥ 2, of prime order p, such that X /H is a Riemann sphere. If H is a proper subgroup of Aut(X ) then H is a proper subgroup of its normalizer N .
Proof Denote Aut(X ) by G and assume, to a contrary, that H = N . Let θ : → H = Z p be a smooth epimorphism defining an action of Z p . Then is a Fuchsian group with signature (0; p, r . . ., p) and X = H/ for = ker θ . Furthermore G = / for some Fuchsian group containing as a subgroup, say of index m, and as a normal subgroup. By our assumption p 2 does not divide |G| since in a p-group each subgroup is subnormal. Then the canonical projection θ : → / extends θ . Since N = H , has the signature (k ; p, s . . . , p, m 1 , . . . , m t ), where m i = p. Observe that none of m i is multiple of p since otherwise for a corresponding elliptic generators x i of , a = θ (x i ) ∈ G − Z p centralizes Z p . So by Lemma 3.1 each period p of produces precisely one period of . Hence s = r which is impossible since μ( ) < μ( ).

Lemma 3.3 Let H be a group of automorphisms a compact Riemann surface X of genus g ≥ 2 of a prime order p, with r fixed points and with the orbit space X /H being the Riemann sphere. Furthermore, assume that H is a proper subgroup of Aut(X ). Then H is contained as a normal subgroup of a prime index q in a subgroup K of Aut(X ).
Proof By Lemma 3.2, H is a proper subgroup of its normalizer N in Aut(X ) and so for arbitrary q dividing the order N /H we have its subgroup K /H of order q and hence the Lemma.

Connected components of the singular locus of moduli space composed by two equisymmetric strata
Recall that all strata corresponding to the actions of cyclic groups of order 2 or 3 are contained in a single connected component that we have denoted in the introduction by C 2,3 g and all other known components are those that are composed by isolated strata. The following Lemma describes when the simple stratum M G g form a connected component of S g .

Lemma 4.1
If a nonempty stratum M G g forms a connected component of S g , then G is a cyclic group of prime order p and it represents a finitely maximal subgroup of the mapping class group M g .
Proof Let a Riemann surface X represent an element [X ] of M G g . Then three cases may occur. The first is that G has a proper nontrivial subgroup H for which M H g = ∅ (see [13] for a sufficient condition for it). Then [X ] ∈ M H g and so M G g and M H g are two disjoint strata which belong to one connected component; we shall say in such a situation that M G g is not isolated from the below. The second possibility is that the action of G properly extends to K . Then any [Y ] ∈ M K g belongs to M G g and so again M G g and M K g are contained in one connected component and now we shall say that M G g is not isolated from the above. Finally, it can also happen that none of the above cases occur for arbitrary X ∈ M G g . Then, by the previous cases, G = Z p and G define a finitely maximal subgroup as claimed.
So if we want to construct a connected component, composed of precisely two strata then, in principle, these strata must correspond to two actions H < G where H = Z p and G = Z p 2 or G = Z p Z q where we allow the possibility of p = q and semidirect product to be direct. Here we shall define an action of Z p which extends to the action of Z p 2 . The first step is made in the lemma
In the next lemma we define the actions corresponding to Fuchsian groups defined above, so that the one extends the other.
Then the diagram commutes which precisely means that any conformal action of H = Z p on a Riemann surface X , defined by θ , extends to an action of G = Z p 2 defined by θ on this surface.
Proof It is straightforward to check that the diagram (14) commutes.
The conclusion of the above Lemma means, in particular, that the subgroup of M g defined by θ given in (12) is not finitely maximal. In the next two Lemmas we shall show that the action G = Z p 2 given there is its only extension of a prime index.
where k = 0, s + u = 2 and v = qt + u. Furthermore if y 1 , . . . , y s , z 1 , . . . , z t , w 1 , . . . , w u is a canonical system of generators for˜ , then is a canonical system of generators for .
Proof The signature of the group˜ given in (15) follows from the fact that its only periods can be q, p and qp. Then, by Lemma 3.1, v = qt +u which by the Riemann-Hurwitz formula gives k = 0 and s + u = 2. For the second part of the proof, it is enough to prove that the elements x 1 , x 2 , . . . , x v generate a normal subgroup of˜ and their product is trivial what is a straightforward checking.

Lemma 4.5
The action of H = Z p defined by a smooth epimorphism θ given in (12), where r ≥ 6 and p > r is a prime so that pr + 2 is also a prime, has no other extension of a prime index q than the one given in (13) of Lemma 4.3.
Proof Assume that such extension H exists and let˜ be a corresponding Fuchsian group containing as a normal subgroup of prime index q and denote pr +2 by v to relate notation to Lemma 4.4 which shall be involved in the proof. The canonical elliptic generators for and for˜ together with an embedding →˜ defined in their terms are described in Lemma 4.4, where in particular we have that pr + 2 = qt + u. Observe first that H is cyclic since otherwise u = 0 and so q = pr + 2, since pr + 2 is a prime. But this is impossible since in such case Z q is a normal subgroup of H by the Sylow theorem and so H would be cyclic in