1 Introduction

Deligne-Mumford [6] compactification \(\overline{\mathcal {M}}_g\) of the moduli space \({\mathcal {M}}_g\) of compact Riemann surfaces of genus \(g \ge 2\) can be seen from two points of view. On the one side, \(\overline{\mathcal {M}}_g\) is a projective algebraic variety, and as such it has also an underlying structure of an analytic variety of dimension \(3(g-1)\). On the other side, being the difference \(\overline{\mathcal {M}}_g{\setminus } \partial \,\overline{\mathcal {M}}_g\), the moduli space \({\mathcal {M}}_g\) is a quasi-projective variety. So apart of Zariski topology, it carries also the structure of analytic variety and the aim of our paper is to study the nature of connected components in the singular locus \({\mathcal {S}}_g\) using the Broughton [5] equisymmetric stratification of \({\mathcal {M}}_g\).

This stratification turned out to be very fruitful and, for instance, allowed Bartolini, Costa, Izquierdo and Porto to successfully study the connectivity of \({\mathcal {S}}_g\) in an extensive series of articles whose culmination and capstone was a description of those genera g for which \({\mathcal {S}}_g\) is connected. To be precise they showed in [1, 3] that \({\mathcal {S}}_g\) 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 \({\mathcal C}_g^{2,3}\). 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 \({\mathcal {M}}_g\) for \(g \le 25\). All this prompted Bartolini, Costa and Izquierdo, to ask in [3] whether there exist components other than \({\mathcal C}_g^ {2,3}\) containing more than one stratum, and the purpose of our paper is to answer in the affirmative, by showing that

Given an integer \(r\ge 6\) and a prime \(p> 5r+3\) for which \(pr+2\) is also prime, there is a connected component, composed by two strata in \(\mathcal {S}_g\) for \(g=rp(p-1)/2\).

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 \(\mathfrak {M}_g\) because of its role in the construction of the moduli space as the orbit space of its action on Teichmüller space \({\mathcal {T}}_g\) and its singular locus \({\mathcal {S}}_g\), viewed as the set of branch points of the projection \({\mathcal {T}}_g \rightarrow {\mathcal {M}}_g\). According to it, in principle, the equisymmetric stratum can be seen as the subset \({\mathcal {M}}_g^G\) of \({\mathcal {M}}_g\) 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 \(\mathfrak {M}_g\) induced by G. Having all of these, the results of Broughton [5] in question can be gathered into the following statement:

  • Nonempty equisymmetric strata \({\mathcal {M}}_g^{G}\) are smooth, connected, locally closed algebraic subvarieties of \({\mathcal {M}}_g\).

  • Zariski closures \(\overline{\mathcal {M}}_g^{G}\) of \({\mathcal {M}}_g\) are defined by relaxing conditions defining \({\mathcal {M}}_g^{G}\) assuming only that, in notations employed there, G is a subgroup of \(\textrm{Aut}(X)\).

  • \(\overline{\mathcal {M}}_g^G\) are closed, irreducible algebraic subvarieties of \({\mathcal {M}}_g\).

Following [7] and [8], we still would like to add that \(\overline{\mathcal {M}}_g^{G}\) is closed in the topology of \({\mathcal {M}}_g\) induced from the one of Teichmüller space \({\mathcal {T}}_g\) which in turn is homeomorphic to \(\mathbb {R}^{3(g-1)}\).

These concepts will be explained with more detail in the next section but already at this stage, we can realize that both \(\overline{\mathcal {M}}_g^G\) and \({\mathcal {M}}_g^G\) cover \({\mathcal {M}}_g\). In addition \({\mathcal {M}}_g^G\) are pairwise disjoint. We say that two strata \({\mathcal {M}}_g^{G}\) and \({\mathcal {M}}_g^{G'}\) are not separated if and only if

$$\begin{aligned} {\mathcal {M}}_g^{G} \cap \overline{{\mathcal {M}}}_g^{G'}\ne \emptyset \; {{ \mathrm or}} \; \overline{{\mathcal {M}}}_g^{G} \cap {\mathcal {M}}_g^{G'} \ne \emptyset \end{aligned}$$
(1)

Observe that two strata \( {\mathcal {M}}_g^{G, \theta } \) and \( {\mathcal {M}}_g^{G'\!, \theta '} \) being separated, they may be contained in the same connected component. This happens if there is a sequence of strata

$$\begin{aligned} {\mathcal {M}}_g^{G} = {\mathcal {M}}_g^{G_1}, {\mathcal {M}}_g^{G_2}, \ldots ,\, {\mathcal {M}}_g^{G_n}={\mathcal {M}}_g^{G'} \end{aligned}$$
(2)

such that \({\mathcal {M}}_g^{G_i}\) and \({\mathcal {M}}_g^{G_{i+1}} \) are not separated for arbitrary \( 1 \le i \le n-1 \). In principle, such a sequence may exist for \(G \ne Z_2,Z_3\) and \(G'\in \{Z_2, Z_3\} \) what shows why the problem of searching components other of \({\mathcal C}_g^{2,3}\) is not easy.

2 Preliminaries

2.1 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

$$\begin{aligned} \langle x_1, \ldots , x_r, a_1,b_1, \ldots , a_h, b_h \,:\, x_1^{m_1}, \ldots , x_r^{m_r}, x_1 \ldots x_r [a_1,b_1] \ldots [a_h, b_h] \rangle \end{aligned}$$
(3)

which will be denoted by the sequence

$$\begin{aligned} (h;m_1, \ldots , m_r) \end{aligned}$$
(4)

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

$$\begin{aligned} 2(h-1) + \sum _{i=1}^r\left( 1-\dfrac{1}{m_i}\right) \end{aligned}$$
(5)

is positive and in such case (5) is the normalized hyperbolic area of any fundamental region of an arbitrary Fuchsian group \(\Lambda \) with the signature (4). We shall denote that area by \(\mu (\Lambda )\).

Now, by the Riemann uniformization theorem, a compact Riemann surface \(X=X_g\) of genus \(g \ge 2\) can be considered as the orbit space \({\mathcal {H}}/\Gamma \) with respect to the action of a torsion-free 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 \(G\cong \Lambda /\Gamma \) for some Fuchsian group \(\Lambda \), say with signature (4). Conversely, given an F-group with presentation (3) and a torsion-free kernel epimorphism \(\theta : \Lambda \rightarrow G\), called smooth or surface-kernel, there is a realization of \(\Lambda \) as a group of isometries of the hyperbolic plane so that for \(\Gamma = \ker \theta \), \(G\cong \Lambda /\Gamma \) is a group of conformal automorphisms of a Riemann surface \(X={\mathcal {H}}/\Gamma \). The genus of X can be found from the Riemann-Hurwitz formula which in a general setting says that for an arbitrary Fuchsian group \(\Lambda \) and its subgroup \(\Delta \) we have

$$\begin{aligned}{}[\Lambda :\Delta ] =\dfrac{\mu (\Delta )}{\mu (\Lambda )}. \end{aligned}$$
(6)

Next, two smooth epimorphisms \(\theta _i: \Delta _i \rightarrow G_i\) define topologically equivalent actions if and only if the diagram

(7)

commutes for some isomorphisms \(\varphi \) and \(\psi \). The last means that the above equivalence classes of smooth epimorphisms from abstract groups \(\Lambda \) onto G correspond to conjugacy classes of finite subgroups of the mapping class group \(\mathfrak {M}_g\) of genus g. The above describes the topological aspects of the theory fully.

2.2 On conformal realization of topological actions on compact surfaces.

The moduli space \({\mathcal {M}}_g\) of a compact Riemann surface of genus \(g \ge 2\), in the spirit we shall use in this paper, can be defined in a few steps. For the first let \(\Lambda \) be an F-group corresponding to some signature (4) and consider the set of monomorphisms

$$\begin{aligned} {\mathcal W}(\Lambda )=\{\rho : \Lambda \hookrightarrow {\textrm{Isom}}({\mathcal {H}}){\mathop {=}\limits ^{\textrm{thm}}} {\textrm{PSL}}(2,\mathbb {R})\; : \; \rho (\Lambda ) \; \text { is a Fuchsian group}\} \end{aligned}$$
(8)

called the Weil space of \(\Lambda \). Having it, we define the Teichmüller space \({\mathcal {T}}(\Lambda )\) of \(\Lambda \) as the factor space \({\mathcal W}(\Lambda )/\sim \), where \( \rho _1 \sim \rho _2\) if they are conjugate in the group of all isometries of the hyperbolic plane which means that

$$\begin{aligned} \text {there is an isometry} \; \delta \in {\textrm{Isom}}({\mathcal {H}}) \; \text {so that} \; \rho _1(\lambda ) = \delta \rho _2(\lambda )\delta ^{-1} \; \text {for all} \; \lambda \in \Lambda \end{aligned}$$
(9)

The space \({\mathcal {T}}(\Lambda )\) 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 \(\mathfrak {M}(\Lambda ) =\textrm{Aut}(\Lambda )/\textrm{Inn}(\Lambda )\) on it by

$$\begin{aligned} ([\varphi ],[\rho ]) \mapsto [\rho \circ \varphi ^{-1}] \end{aligned}$$

where the moduli space \({\mathcal {M}}(\Lambda )\) we have looked for is the quotient space. The mentioned space \({\mathcal {M}}_g\) is obtained if we take for \(\Lambda \), in the above construction, the fundamental group \(\Gamma = \langle a_1,b_1, \ldots , a_g,b_g \;: \; [a_1,b_1] \ldots [a_g,b_g] \rangle \) of a surface of genus g. \({\mathcal {M}}(\Lambda )\) is only a variety but it has the same dimension as \({\mathcal {T}}(\Lambda )\). The singular locus \({\mathcal {S}}_g\) of \({\mathcal {M}}_g\) (synonymously called branch locus) is composed of the points representing surfaces with nontrivial groups of conformal automorphisms.

2.3 How the above works

Summing up, the equisymmetric strata are parameterized by the classes of smooth epimorphisms \(\theta : \Lambda \rightarrow G\) from an F-group \(\Lambda \) onto a finite group G. Such a stratum \({\mathcal {M}}_g^{G,\theta }\) is composed by the set of points of the moduli space \({\mathcal M }_g\) corresponding to representatives of Teichmüller classes of Fuchsian group \(\Lambda \) so that

  • \(\Lambda \) is maximal in the class of all F-groups or

  • \(\Lambda \) is not maximal but none of its extensions contains \(\Gamma = \ker \theta \) as a normal subgroup.

The Zariski closure \(\overline{\mathcal {M}}_g^{G,\theta }\) 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 \(\theta _i:\Lambda _i\rightarrow G_i\) is nonempty if and only if there exists an F-group \(\Lambda _1'\) extending \(\Lambda _1\), containing \(\Gamma _1\) as a normal subgroup and \(\Lambda _2\) as a (non-necessarily normal) subgroup.

Summing up, we see that the approach described above allows, in an essential extent, methods of purely combinatorial group theory in a study of the nature of connected components of the singular locus.

2.4 On notations used in the paper

Overusing the language, we will write \({\mathcal {M}}_g^{G, \theta }\) as \({\mathcal {M}}_g^G \) or \({\mathcal {M}}_g^{\theta }\) if it will be known from elsewhere what \(\theta : \Delta \rightarrow G\) 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, \(\Lambda , \Delta , \Gamma \) etc. will stand for both F-groups and Fuchsian groups depending on the context.

3 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:

Lemma 3.1

Let \(X={\mathcal {H}}/\Gamma \) be a Riemann surface with automorphisms group \(G \cong \Lambda /\Gamma \) and let \(x_1,\ldots ,x_r\) be a set of elliptic canonical generators of \(\Lambda \) whose periods are \(m_1,\ldots ,m_r\) respectively. Let \(\theta :\Lambda \rightarrow G\) be the canonical projection. Then the number \(F(\varphi )\) of points of X fixed by a nontrivial element \(\varphi \) of G is given by the formula

$$\begin{aligned} |{N}_G(\langle \varphi \rangle )|\;\sum \dfrac{1}{m_i}, \end{aligned}$$
(10)

where the operator N stands for the normalizer and the sum is taken over those i for which \(\varphi \) is conjugate to a power of \(\theta (x_i)\). \(\square \)

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 \(g\ge 2\), of prime order p, such that X/H is a Riemann sphere. If H is a proper subgroup of \(\textrm{Aut}(X)\) then H is a proper subgroup of its normalizer N.

Proof

Denote \({\textrm{Aut}}(X)\) by G and assume, to a contrary, that \(H=N\). Let \(\theta : \Delta \rightarrow H=Z_p\) be a smooth epimorphism defining an action of \(Z_p\). Then \(\Delta \) is a Fuchsian group with signature \((0;\,p, {\mathop {\ldots }\limits ^{r}},p)\) and \(X={\mathcal {H}}/\Gamma \) for \(\Gamma = \ker \theta \). Furthermore \(G= \Lambda /\Gamma \) for some Fuchsian group \(\Lambda \) containing \(\Delta \) as a subgroup, say of index m, and \(\Gamma \) 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 \(\theta ':\Lambda \rightarrow \Lambda /\Gamma \) extends \(\theta \). Since \(N=H\), \(\Lambda \) has the signature \((k\,;\,p, {\mathop {\ldots }\limits ^{s}}\,,p,m_1,\ldots , m_t)\), where \(m_i \ne p\). Observe that none of \(m_i\) is multiple of p since otherwise for a corresponding elliptic generators \(x_i\) of \(\Lambda \), \(a=\theta '(x_i) \in G- Z_p\) centralizes \(Z_p\). So by Lemma 3.1 each period p of \(\Lambda \) produces precisely one period of \(\Delta \). Hence \(s=r\) which is impossible since \(\mu (\Lambda ) < \mu (\Delta )\). \(\square \)

Complementary to the above Lemma is the following

Lemma 3.3

Let H be a group of automorphisms a compact Riemann surface X of genus \(g\ge 2\) 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 \(\textrm{Aut}(X)\). Then H is contained as a normal subgroup of a prime index q in a subgroup K of \(\textrm{Aut}(X)\).

Proof

By Lemma 3.2, H is a proper subgroup of its normalizer N in \(\textrm{Aut}(X)\) and so for arbitrary q dividing the order N/H we have its subgroup K/H of order q and hence the Lemma. \(\square \)

4 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 \({\mathcal C}_g^{2,3}\) and all other known components are those that are composed by isolated strata. The following Lemma describes when the simple stratum \({\mathcal {M}}_g^{G}\) form a connected component of \({\mathcal {S}}_g\).

Lemma 4.1

If a nonempty stratum \({\mathcal {M}}_g^G\) forms a connected component of \({\mathcal {S}}_g\), then G is a cyclic group of prime order p and it represents a finitely maximal subgroup of the mapping class group \(\mathfrak {M}_g\).

Proof

Let a Riemann surface X represent an element [X] of \({\mathcal {M}}_g^G\). Then three cases may occur. The first is that G has a proper nontrivial subgroup H for which \({\mathcal {M}}_g^H \ne \emptyset \) (see [13] for a sufficient condition for it). Then \([X]\in \overline{\mathcal {M}}_g^H\) and so \({\mathcal {M}}_g^G\) and \({\mathcal {M}}_g^H\) are two disjoint strata which belong to one connected component; we shall say in such a situation that \({\mathcal {M}}_g^G\) is not isolated from the below. The second possibility is that the action of G properly extends to K. Then any \([Y] \in {\mathcal {M}}_g^K\) belongs to \(\overline{\mathcal {M}}_g^G\) and so again \({\mathcal {M}}_g^G\) and \({\mathcal {M}}_g^K\) are contained in one connected component and now we shall say that \({\mathcal {M}}_g^G\) is not isolated from the above. Finally, it can also happen that none of the above cases occur for arbitrary \(X \in {\mathcal {M}}_g^G\). Then, by the previous cases, \(G=Z_p\) and G define a finitely maximal subgroup as claimed. \(\square \)

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 \(G= Z_p\rtimes Z_q\) 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

Lemma 4.2

Let \(\Lambda = (0;p, {\mathop {\ldots }\limits ^{r}}\,,p,p^2,p^2)\) and \(\Delta = (0;p, {\mathop {\ldots }\limits ^{pr+2}}\,,p)\) be Fuchsian groups with canonical sets of generators \(x_1, \ldots , x_{r+2}\) and \(y_1, \ldots , y_{pr+2}\) respectively. Then for \(w_1= x_{r+1}\) and \(w_2=x_{r+2}\) the map

$$\begin{aligned} \underbrace{y_{ir+1} , y_{ir+2} , \ldots ,y_{(i+1)r}}\; \mapsto \; \underbrace{x_1^{w_1^{i}}, x_2^{w_1^{i}},\ldots , x_r^{w_1^{i}}},\;\; \underbrace{ y_{pr+1},y_{pr+2}}\; \mapsto \;\underbrace{ w_1^p, w_2^{p},} \end{aligned}$$
(11)

for \(i=0, \ldots , p-1\), induces a normal embedding \(\Delta \hookrightarrow \Lambda \) of index p.

In the next lemma we define the actions corresponding to Fuchsian groups defined above, so that the one extends the other.

Lemma 4.3

Let \(G=Z_{p^2}= \langle b \rangle \) and let H be its subgroup generated by \(a=b^p\) and let \(p>r\). Let then \(\theta : \Delta \rightarrow H\) and \(\theta ': \Lambda \rightarrow G\) be smooth epimorphisms defined respectively by the assignments

$$\begin{aligned} \underbrace{y_{ir+1}, \ldots , y_{ir+(r-1)} , y_{(i+1)r}} \; \mapsto \; \underbrace{a, \ldots , a^{r-1}, a^{-r(r-1)/2}}, \;\; \underbrace{ y_{pr+1},y_{pr+2}}\; \mapsto \;\underbrace{ a, a^{-1},} \end{aligned}$$
(12)

for \(i=0, \ldots , p-1\), and

$$\begin{aligned} \underbrace{x_1,x_2, \ldots , x_r, x_{r+1}, x_{r+2}} \; \mapsto \; \underbrace{a,a^{2}, \ldots , a^{r-1},a^{-r(r-1)/2}, \,b,\, b^{-1}}. \end{aligned}$$
(13)

Then the diagram

(14)

commutes which precisely means that any conformal action of \(H=Z_p\) on a Riemann surface X, defined by \(\theta \), extends to an action of \(G=Z_{p^2}\) defined by \(\theta '\) on this surface.

Proof

It is straightforward to check that the diagram (14) commutes. \(\square \)

The conclusion of the above Lemma means, in particular, that the subgroup of \(\mathfrak {M}_g\) defined by \(\theta \) 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.

Lemma 4.4

Let \(\Delta \) be a Fuchsian group with the signature \((0;p, {\mathop {\ldots }\limits ^{{v}}}\,, p)\) contained in a Fuchsian group \({\tilde{\Delta }}\) as a normal subgroup of prime index q. Then \({\tilde{\Delta }}\) has a signature

$$\begin{aligned} (k;q, {\mathop {\ldots }\limits ^{{s}}}\,, q, p, {\mathop {\ldots }\limits ^{{t}}}\,, p, qp, {\mathop {\ldots }\limits ^{{u}}}\,, qp), \end{aligned}$$
(15)

where \(k=0, \; {s} + {u}=2\) and \({v}=q{t} + {u}\). Furthermore if

$$\begin{aligned} y_1, \ldots , y_{s}, z_1, \ldots , z_{t}, w_1, \ldots , w_{u} \end{aligned}$$

is a canonical system of generators for \({\tilde{\Delta }}\), then

$$\begin{aligned} x_1, \ldots , x_{v} = \left\{ \begin{array}{llll} {\mathrm{(a)}}\; &{} \underbrace{z_1,\ldots , z_{t}},\underbrace{z_1^{y_2},\ldots , z_{t}^{y_2}}, \ldots , \underbrace{z_1^{y_2^{q-1}},\ldots , z_{t}^{y_2^{q-1}}} &{} if &{} {u}=0,\\ {\mathrm{(b)}}\; &{} \underbrace{z_1,\ldots , z_{t}} \underbrace{,z_1^{y_1},\ldots , z_{t}^{y_1}}, \ldots , \underbrace{z_1^{y_1^{q-1}},\ldots , z_{t}^{y_1^{q-1}}}, w_1^q&{} if &{} {u}=1,\\ {\mathrm{(c)}} \;&{} \underbrace{z_1,\ldots , z_{t}},\underbrace{z_1^{w_1},\ldots , z_{t}^{w_1}}, \ldots , \underbrace{z_1^{w_1^{q-1}}\!\!,\ldots , z_{t}^{w_1^{q-1}}}, w_1^q,w_2^q &{} if &{} {u}=2, \end{array}\right. \nonumber \\ \end{aligned}$$
(16)

is a canonical system of generators for \(\Delta \).

Proof

The signature of the group \(\tilde{\Delta }\) 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 \(x_1,x_2,\ldots ,x_{v}\) generate a normal subgroup of \({\tilde{\Delta }}\) and their product is trivial what is a straightforward checking. \(\square \)

Lemma 4.5

The action of \(H=Z_p\) defined by a smooth epimorphism \(\theta \) given in (12), where \(r\ge 6\) 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 \({\widetilde{H}}\) exists and let \({\tilde{\Delta }}\) be a corresponding Fuchsian group containing \(\Delta \) 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 \(\Delta \) and for \(\tilde{\Delta }\) together with an embedding \(\Delta \hookrightarrow \tilde{\Delta }\) defined in their terms are described in Lemma 4.4, where in particular we have that \(pr+2=qt+u\). Observe first that \(\widetilde{H}\) 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 \(\widetilde{H}\) by the Sylow theorem and so \({\widetilde{H}}\) would be cyclic in contrary to our assumption. Let b be the generator of \(\widetilde{H}\) and let \(a=b^p\). Now for \(u=0\), we get \(t=1\) which is impossible since then \(\tilde{\Delta }\) has signature (0; q, q, p) which does not allow a smooth epimorphism onto \(Z_{pq}\).

So it remains to consider the cases \(u=1\) and \(u=2\). Recall that here \(\theta \) maps, after possible permutation, the canonical elliptic generators into

$$\begin{aligned} \big (\underbrace{a, \ldots , a}_p\;,\ldots , \underbrace{a^{r-1}, \ldots , a^{r-1}}_p, \underbrace{a^{-r(r-1)/2}, \ldots , a^{-r(r-1)/2}}_p, \, a,a^{-1} \big ) \end{aligned}$$
(17)

while, for an extension induced by (b) in (16), after possible permutation of these generators, they would be mapped into

$$\begin{aligned} \big (\underbrace{a^{e_1}, \ldots , a^{e_1}}_q,\ldots ,\underbrace{a^{e_t},\ldots ,a^{e_t}}_q,a^{\beta }\big ) \end{aligned}$$
(18)

for some \(e_i\) and this is a contradiction. Indeed since \(a^{-1}\ne a^i\) for \(i\le r-1\) we see that there are at least \(r-3\) elements \(1\le \alpha _1<\ldots <\alpha _{r-3}\le p-1\) so that \(a^{\alpha _i}\) appears in (17) precisely p times. On the other hand a power \(a^{\gamma }\) may appear precisely in (18) at most once. So (18) and (17) do not define equivalent actions since \(r-3\ge 3\).

The case \(u=2\) can be considered similarly. Here we have

$$\begin{aligned} \big (\underbrace{a^{e_1},\ldots ,a^{e_1}}_q,\ldots ,\underbrace{a^{e_t},\ldots ,a^{e_t}}_q,a^{\beta },a^{\beta '}\big ) \end{aligned}$$
(19)

for some \(e_i\) and this is also a contradiction. Here, unless \(q=p\), a power \(a^{\gamma }\) may appear precisely p times in (19) at most twice. So (19) and (17) do not define equivalent actions since \(r-3\ge 3\). For the case \(q=p\) we get precisely the embedding (13), what completes the proof of the Lemma. \(\square \)

We are now in a position to prove the following concluding theorem

Theorem 4.6

Given an integer \(r\ge 6\) and a prime \(p>5r+3\) for which \(pr+2\) is also prime, there exists a connected component in \({\mathcal {S}}_g\) for \(g= rp(p-1)/2\) composed by precisely two equisymmetric strata of dimensions \(pr-1\) and \(r-1\) corresponding to some actions of \(Z_p\) and \(Z_{p^2}\).

Proof

Let \(\Gamma \, \unlhd \, \Delta \,\unlhd \Lambda \) be Fuchsian groups giving the actions \(H=Z_p = \Delta /\Gamma \), \(G=Z_{p^2}= \Lambda /\Gamma \) on a Riemann surface \(X={\mathcal {H}}/\Gamma \), where the actions of H and G are defined in Lemma 4.3 and the embedding \(H\hookrightarrow G \) is described in Lemma 4.2. Let also \({\mathcal {M}}_g^K\) be an arbitrary stratum of the singular locus of \({\mathcal {M}}_g\) distinct from \({\mathcal {M}}_g^{H, \theta }\) and \({\mathcal {M}}_g^{G, \theta '}\). Assume, to get a contradiction, that \({\mathcal {M}}_g^K\) and \({\mathcal {M}}_g^{G, \theta '}\) are not separated. Then \(\overline{\mathcal {M}}_g^K \cap {\mathcal {M}}_g^{G, \theta '} = \emptyset \) since otherwise \(K \le G\) and therefore \(K=G\). But then \(\mathcal {M}_g^K\cap \overline{\mathcal {M}}_g^{G,\theta '}\ne \emptyset \) which means that there exists a Riemann surface Y with a group of automorphisms G being a normal Sylow \(p-\)subgroup in K by ([9], Thm 1.2) due to our assumption \(p>5r+3\). But since H is a characteristic subgroup of G, it is also a normal subgroup of K, so there is an extension of H of prime index \(q\ne p\) which contradicts Lemma 4.5. Now assume that \(\mathcal {M}_g^{H,\theta }\) and \(\mathcal {M}_g^K\) are not separated. Then \(\overline{\mathcal {M}}_g^{H,\theta }\cap \mathcal {M}_g^K\ne \emptyset \) which means that there exists a Riemann surface S with full group of automorphisms K and \(H\le K\). If H is not a Sylow \(p-\)subgroup of K then \(\mathcal {M}_g^{K}\cap \overline{\mathcal {M}}_g^{G,\theta '}\ne \emptyset \) which was already ruled out. In the other case, we use Lemma 3.3 to get a contradiction with Lemma 4.5 as above. \(\square \)