GENERALIZED QUASI-DIHEDRAL GROUP AS AUTOMORPHISM GROUP OF RIEMANN SURFACES RUB EN A. HIDALGO YERIKA MAR IN MONTILLA AND SA UL QUISPE

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GENERALIZED QUASI-DIHEDRAL GROUP AS AUTOMORPHISM GROUP

OF RIEMANN SURFACES

RUB ´

EN A. HIDALGO, YERIKA MAR´

IN MONTILLA AND SA ´

UL QUISPE

Abstract. In this paper, we discuss certain types of conformal/anticonformal actions of

the generalized quasi-dihedral group Gnof order 8n, for n≥2, on closed Riemann sur-

faces, pseudo-real Riemann surfaces and compact Klein surfaces, and in each of these

actions we study the uniqueness (up to homeomorphisms) action problem.

1. Introduction

Let Sbe a Riemann surface and Aut+(S) (respectively, Aut(S)) be its group of confor-

mal (respectively, conformal and anticonformal) automorphisms. In the generic situation,

Aut(S)=Aut+(S); otherwise, Aut+(S) is a subgroup of index two. In [46], Schwarz

proved that, if Sis a closed Riemann surface of genus g≥2, then Aut+(S) is ﬁnite and

later Hurwitz [26] obtained the upper bound |Aut+(S)| ≤ 84(g−1). Riemann surfaces

with non-trivial group of automorphisms deﬁne the branch locus Bgof the moduli space

Mg, of biholomorphism classes of closed Riemann surfaces of genus g≥2, known to be

a complex orbifold of dimension 3(g−1) [41]. If g≥4, then Bgcoincides with the locus

where Mgfails to be a topological manifold. The space Mghas a natural real structure

(this coming from complex conjugation). The ﬁxed points of such a real structure, its real

points, are the (isomorphism classes) closed Riemann surfaces admitting anticonformal

automorphisms. A closed Riemann surface that admits a reﬂection (i.e., an anticonformal

involution with ﬁxed points) as an automorphism is called real Riemann surface; otherwise,

it is called pseudo-real Riemann surface. Pseudo-real Riemann surfaces are examples of

Riemann surfaces which cannot be deﬁned over their ﬁeld of moduli [2]. In general, a ﬁ-

nite group might not be realized as the group of conformal/anticonformal automorphisms,

admitting anticonformal ones, of a pseudo-real Riemann surface (in [8], it was observed

that a necessary condition for that to happen is for the group to have order a multiple of 4).

Let us ﬁx a ﬁnite abstract group G. In [22], Greenberg proved that there are closed

Riemann surfaces Ssuch that Gcan be see as a subgroup of Aut+(S) (conformal action).

If, moreover, Gcontains an index two subgroup H, then it is possible to ﬁnd Ssuch that

G<Aut(S) and H=G∩Aut+(S) (conformal/anticonformal action). These facts permit

to deﬁne the strong symmetric genus σ0(G) (respectively, symmetric genus σ(G)) of G

as the minimal genus for a conformal (respectively, conformal/anticonformal) action of

G[14,26,48]. We note that σ(G)≤σ0(G) (if Ghas no index two subgroups, then

the equality holds). Groups for which σ0(G)=0 are given by the cyclic groups, the

dihedral groups and the Platonic solid symmetry groups. Those with σ0(G)=1 are also

known [23]. As a consequence of the Hurwitz’s bound, there are only ﬁnitely many (up

2010 Mathematics Subject Classiﬁcation. 30F10, 14H37, 14H57.

Key words and phrases. Riemann surfaces, Klein surfaces, Automorphisms, NEC groups, Dessins d’enfants.

Partially supported by Projects FONDECYT Regular N. 1220261 and 1190001. The second author has been

supported by ANID/Beca de Doctorado Nacional/21190335.

arXiv:2210.01577v1 [math.AG] 4 Oct 2022

2 RUB ´

EN A. HIDALGO, YERIKA MAR´

IN MONTILLA AND SA ´

UL QUISPE

to isomorphisms) groups of a given strong symmetric genus at least two. It is also known

that every integer at least two is the strong symmetric genus for some ﬁnite group [38]. A

conformal action of G, over a closed Riemann surface, is called purely-non-free if every

element acts with a non-empty set of ﬁxed points. In [5], it was observed that Gacts

purely-non-free on some Riemann surface. This permits to deﬁne the pure symmetric genus

σp(G) as the minimal genus on which Gacts purely-non-free (σ0(G)≤σp(G)). A compact

Klein surface Xis canonically doubly covered by a Riemann surface of some genus g≥2

(also called the algebraic genus of X). Its underlying topological surface is either (i) non-

orientable with empty boundary (closed Klein surface) or (ii) it has non-empty boundary

(and can be orientable or not). If kis the number of boundary components of X, then its

topological genus is γ=(g−k+1)/η, where η=2 if Xis an orientable surface and η=1

otherwise. It is known that Gacts as a group of automorphisms of bordered Klein surfaces

(i.e., k>0). The minimum algebraic genus ρ(G) of these surfaces is called the real genus

of G[34]. Also, it is known that Gacts as a group of automorphisms of closed Klein

surfaces (i.e., k=0). The minimum topological genus ˜σ(G) of these surfaces is called the

symmetric crosscap number of G[37].

In this paper, we study conformal/anticonformal actions of the generalized quasi-dihedral

group Gn, of order 8n, on closed Riemann surfaces, for n≥2. Below, we list some of the

main results of this paper. In Corollary 3.2, we observe that σ0(Gn) is equal to n. Such a

minimal conformal action happens for the quotient orbifold S/Gnof signature (0; 2,4,4n)

(this was previously observed in [35] for na power of two and in [38] for nodd). In The-

orem 3.1, we prove that, up to homeomorphisms, there is only one triangular action of Gn

for every n. These triangular action is produced in a familiar kind of hyperelliptic Riemann

surfaces (Wiman curves of type II [50]). They are described, in terms of the corresponding

monodromy group and the associated bipartite graph, in Remark 2. In Theorem 3.1, we

observe that the triangular action of Gnon a closed Riemann surface of genus nis purely-

non-free for neven, and it is not purely-non-free for nodd. On the other hand, in Corollary

3.2, we observe that σp(Gn) is equal to nfor neven, and 3nfor nodd. In Theorem 3.3, we

describe the isotypical decomposition of the Jacobian variety JS, induced by the triangular

action of Gn. As Gnhas index two subgroups (Lemma 2.1), it can be realized as the full

group of conformal/anticonformal automorphisms of suitable Riemann surfaces, such that

it admits anticonformal ones. As consequence of Proulx class [23], it is well known that

Gnacts with such a property in genus σ(Gn)=1. In Theorem 4.2, we observe that the next

minimal genus σhyp(Gn)≥2 over which Gnacts as a group of conformal/anticonformal

automorphisms, and admitting anticonformal ones, is nfor neven, and n−1 for nodd. This

permits to observe that for any non-negative integer g≥2, there is at least one group of

symmetric hyperbolic genus g(see, Corollary 4.3). In Theorem 4.5, we prove that this min-

imal action is unique, up to homeomorphisms. We also given integers k≥3 and n≥2, in

Theorem 5.2, we construct pseudo-real Riemann surfaces, of genus g=2nk−4n+1, whose

full group of conformal/anticonformal automorphisms is Gn(by results in [28], these Rie-

mann surfaces are non-hyperelliptic). The minimal genus of pseudo-real Riemann surfaces

with Gnas its full group of automorphisms, is 2n+1, this was previously proved in [15,

Proposition 3.2] (see, Proposition 5.3). In Corollary 5.4, we prove that this minimal action

is unique up homeomorphisms. For integers n≥3 odd, l≥2 and r≥1 odd, we observe

that there are pseudo-real Riemann surfaces of genus 4nl +6nr −8n+1 with Gnas group

of conformal automorphisms (Theorem 5.10). Moreover, in this case, in Theorem 5.11,

we prove that the minimal genus of pseudo-real Riemann surfaces, with Gnas group of

conformal automorphisms, is 6n+1 (this minimal action is not unique, see Remark 5).

GENERALIZED QUASI-DIHEDRAL GROUP AS AUTOMORPHISM GROUP OF RIEMANN SURFACES 3

For neven, in Theorem 5.9, we observe that the group Gncannot be realized as group of

conformal automorphisms of pseudo-real Riemann surfaces. However, in Theorem 5.7,

we constructed pseudo-real Riemann surfaces Sadmitting Gnas an index two subgroup of

Aut+(S). In Theorem 6.1, we observe that (a) ρ(Gn) is equal to 2n+1 and the action is

unique and (b) ˜σ(Gn) is equal to 2n+2 but the action is not unique for n,3. Part (a) was

previously observed in [36] for na power of two and in [17] for nodd.

Notation. Throughout this paper we denote by Cnthe cyclic group of order n, by D2mthe

dihedral group of order 2m, by DCnthe dicyclic group of order n.

2. Preliminaries

2.1. Generalized quasi-dihedral group. The generalized quasi-dihedral group of order

8nis

(1) Gn:=hx,y:x4n=y2=1,yxy =x2n−1i=C4no2n−1C2.

Using the relation yxk=x(2n−1)ky, for k>1, one may check the following properties.

Lemma 2.1. Let n >2be an integer. Then

(1) The group Gnis a non-abelian group of order 8n, every element has a unique pre-

sentation of the form yjxi, where j and i are integers with j ∈ {0,1}and 0≤i≤4n,

and these have order

|xi|=4n

gcd(i,4n),|yxi|=









2,if i either even or zero,

4,if i odd.

(2) The index two subgroups of Gnare exactly the following ones

C4n=hxi,D4n=hx2,yi,DC4n=hx2,yxi.

(3) The number of involutions of the groups Gnand D4n,is 2n+1and, the number of

involutions of the groups C4nand DC4nis one.

(4) For n even there are exactly 2n+3conjugacy classes of Gn, with representatives

given in the following table

Rep. 1x x2· · · x2n−1x2ny yx

size 1 2 2 · · · 2 1 2n2n

(5) For n odd there are exactly 2n+6conjugacy classes of Gn, with representatives given

in the following table

Rep. 1x· · · xn−1xnxn+1xn+3· · · x2n−1x2nx2n+1· · · x3n−1x3ny yx yx2yx3

size 1 2 · · · 2 1 2 2 · · · 212· · · 2 1 n n n n

(6) The automorphisms of the group Gnare given by

ψu,2v(x)=xuand ψu,2v(y)=yx2v,

where u ∈ {1,...,4n−1}is such that gcd(u,4n)=1and v ∈ {0,1,...,2n−1}. The

group Aut(Gn)has order φ(4n)·2n, where φis the Euler function.

2.2. NEC groups. Let Lbe the group of isometries of the hyperbolic upper half-plane H

and L+be its index two subgroup of orientation-preserving elements. An NEC group is a

discrete subgroup ∆of Lsuch that the quotient space H/∆is a compact surface. An NEC

group contained in L+is called a Fuchsian group, and a proper NEC group otherwise. If

∆is a proper NEC group, then ∆+= ∆ ∩ L+is called its canonical Fuchsian subgroup.

Note that [∆:∆+]=2 and ∆+is the unique subgroup of index 2 in ∆contained in L+. In

4 RUB ´

EN A. HIDALGO, YERIKA MAR´

IN MONTILLA AND SA ´

UL QUISPE

general, the algebraic structure of an NEC group ∆is described by the so-called signature

s(∆) [32,49]:

(2) s(∆)=(h;±; [m1,· · · ,mr]; {(n11,· · · ,n1s1),· · · ,(nk1,· · · ,nksk)}),

where h,r,k,mi,ni j are integers with h,r,k≥0 and mi,ni j >1 for all i,j. Here his the

topological genus of the surface H/∆and “ +” means that H/∆is orientable, and “ −”

means that H/∆is non-orientable. The number kis the number of connected boundary

components of H/∆. We call mithe proper periods,ni j the periods, and (ni1,· · · ,nisi) the

period-cycles of s(∆). We will denote by [−], (−) and {−} the cases when r=0, si=0

and k=0, respectively. When there are no proper periods and there are no period-cycles

in s(∆) we say ∆is a surface group. The signature provides a presentation of ∆[32,49],

by generators:

(elliptic generators) βi∈∆+(i=1,· · · ,r);

(reﬂections) ci j ∈∆\∆+(i=1,· · · ,k;j=0,· · · ,si);

(boundary generators) ei∈∆+(i=1,· · · ,k);

(hyperbolic generators) ai,bi∈∆+(i=1,· · · ,h),if H/∆is orientable;

(glide reﬂections generators) di∈∆\∆+(i=1,· · · ,h),if H/∆is non-orientable;

and relations:

βmi

i=1 (i=1,· · · ,r);

eici0e−1

icisi=1 (i=1,· · · ,k);

i j−1=c2

i j =(ci j−1ci j)ni j =1 (i=1,· · · ,k;j=1,· · · ,si);

(long relation)

i=1

βi

i=1

[ai,bi]=1,if H/∆is orientable;

(long relation)

i=1

βi

i=1

i=1,if H/∆is non-orientable,

where 1 denotes the identity map idHin Hand [ai,bi]=aibia−1

ib−1

The hyperbolic area of ∆with signature (2) is the hyperbolic area of any fundamental

region for ∆, and is given by

(3) µ(∆)=2πηh+k−2+

i=11−1

mi+1

i=1

j=11−1

ni j ,

with η=2 or 1 depending on whether or not H/∆is orientable. An NEC group with

signature (2) actually exists if and only if the right-hand side of (3) is greater than 0. The

reduced area of an NEC group ∆, denoted by |∆|∗, is given by µ(∆)/2π. If Γis a subgroup

of ∆of ﬁnite index, then the Riemann-Hurwitz formula holds [∆:Γ]=µ(Γ)

µ(∆).

2.3. Topologically equivalent conformal/anticonformal actions. Let Sbe a closed Rie-

mann surface Sof genus g≥2. By the uniformization theorem, up to biholomorphisms,

S=H/K, where Kis a Fuchsian surface group. We say that a ﬁnite group G acts as a group

of conformal (respectively, conformal/anticonformal) automorphisms of Sif it can be real-

izable as a subgroup of Aut+(S) (respectively, Aut(S)). This is equivalent to the existence

of a Fuchsian (respectively, an NEC) group ∆, containing Kas a normal subgroup, and of

an epimorphism θ:∆→Gwhose kernel is K, we say that θprovides a conformal (respec-

tively, conformal/anticonformal) action of Gin S. Two conformal/anticonformal actions

GENERALIZED QUASI-DIHEDRAL GROUP AS AUTOMORPHISM GROUP OF RIEMANN SURFACES 5

θ1,θ2are topologically equivalent if there is an ω∈Aut(G) and an h∈Hom+(S) such that

θ2(g)=hθ1(ω(g))h−1for all g∈G. This is equivalent to the existence of automorphisms

φ∈Aut(∆) and ω∈Aut(G) such that θ2=ω◦θ1◦φ−1.

If Gacts conformally, then S/G=H/∆is a closed Riemann surface of genus hand

it has exactly ncone points of respective cone orders k1,...,kn; the tuple (h;k1,...,kn) is

also called the signature of S/G. If h=0 and n=3, then we talk of a triangular action.

2.4. Dessins d’enfants. Adessin d’enfant corresponds to a bipartite map Gon a closed

orientable surface X. These objects were studied as early as the nineteenth century and re-

discovered by Grothendieck in the twentieth century in his ambitious research outline [24]

(see also, the recent books [19,29]). The dessin d’enfant deﬁnes (up to isomorphisms)

a unique Riemann surface structure Son X, together with a holomorphic branched cover

β:S→b

Cwhose branch values are containing in the set {∞,0,1}(βis called a Belyi map,

Sis a Belyi curve and (S, β) is a Belyi pair). Conversely, to every Belyi pair (S, β) there

is associated a dessin d’enfant in S(β−1(0) and β−1(1) provide, respectively, the white and

black vertices, and the edges being β−1([0,1])). The dessin is called regular if its group

of automorphisms G(i.e, the deck group of β) acts transitively on the set of edges (equiv-

alently, βis a Galois branched covering). In this case, Gdeﬁnes a triangular action and,

in particular, Gis generated by two elements (in fact, every ﬁnite group generated by two

elements appears as the group of automorphisms of some regular dessin). Examples of

such type of groups are the generalized quasi-dihedral groups Gn(see, Section 2.1). There

is a bijection between (i) equivalence classes of regular dessins d’enfants with group of

automorphisms Gand (ii) G-conjugacy classes of pairs of generators of G.

3. Triangular conformal actions of Gn

In this section, we describe the triangular conformal actions of Gn. First, we observe

the well know family Snof hyperelliptic Riemann surface admitting a triangular conformal

action of Gn.

3.1. The strong and pure symmetric genus of Gn.If n≥2, then the Riemann surface

Snof genus n, deﬁned by the algebraic curve w2=z(z2n−1), is called the Wiman curve

of type II [50]. Some conformal automorphisms of Snare given by x(z,w)=(ρ2nz, ρ4nw),

y(z,w)=(ρ2n/z,iρ4nw/zn+1) with (ρ2n)n=−1 and hx,yiGn. The quotient orbifold

Sn/hx,yihas signature (0; 2,4,4n). If n≥3, then this is a maximal signature [47], so

Aut+(Sn)=hx,yi. If n=2, then Snhas as an extra conformal automorphism of order

three, given by t(z,w)=(i(1−z)/(1+z),2(1+i)w/(z+1)3). In this case, Aut+(Sn)=hx,y,ti

(a group of order 48).

The bipartite graph, associated to the regular dessin d’enfant on Sninduced by Gn, is the

graph K2

2,2n, which is obtained from the complete bipartite graph (see, [16, pp. 17]) K2,2nin

which each of its edges is replaced by two edges. In order to observe this, consider a regular

branch cover β:Sn→b

C, whose deck group is Gn. We may assume that β=Q◦P, where

P:Sn→b

Chas deck group C4n=hxi, and Q(x)=x2, whose deck group is Gn/hxiC2

(see also, Remark 2in where the corresponding monodromy group is described).

Theorem 3.1. Let S be a closed Riemann surface such that Aut+(S)=Gn, for n ≥2, and

such that S/Gnhas triangular signature. Then

(a) S/Gnhas signature (0; 2,4,4n)and S is isomorphic to S n.

(b) If n ≥2even, the action of Gnis purely-non-free.

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GENERALIZEDQUASI-DIHEDRALGROUPASAUTOMORPHISMGROUPOFRIEMANNSURFACESRUB´ENA.HIDALGO,YERIKAMAR´INMONTILLAANDSA´ULQUISPEAbstract.Inthispaper,wediscusscertaintypesofconformal/anticonformalactionsofthegeneralizedquasi-dihedralgroupGnoforder8n,forn2,onclosedRiemannsur-faces,pseudo-realRiemannsurfacesandco...

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GENERALIZED QUASI-DIHEDRAL GROUP AS AUTOMORPHISM GROUP OF RIEMANN SURFACES RUB EN A. HIDALGO YERIKA MAR IN MONTILLA AND SA UL QUISPE

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