hyperelliptic surfaces and their moduli

Hyperelliptic Surfaces and their Moduli

ANTHONY WEAVER?

Department of Mathematics and Computer Science, BCC, City University of New York,

University Avenue and West 181 St., Bronx, NY 10453, U.S.A. e-mail: [email protected]

(Received: 11 July 2002; in final form: 7 April 2003)

Abstract. The hyperelliptic portion of the moduli space of compact Riemann surfaces of genusg5 2 is decomposed into a lattice of nondisjoint subvarieties corresponding precisely with thelattice of maximal g-hyperelliptic group actions (classified up to topological equivalence).

The resulting stratification of the hyperelliptic moduli space exhibits regularities which dependon the parity of g and can be detected at the level of groups of order 8.

Mathematics Subject Classifications (2000). primary: 32G15; secondary: 30F20, 14J50.

Key words. automorphism group; hyperelliptic Riemann surface; moduli space.

1. Introduction

The space of conformal equivalence classes of compact Riemann surfaces of a given

genus g5 2 has complex dimension 3g� 3 and is called the Riemann moduli space

ðRgÞ. The set of surface classes admitting nontrivial automorphisms forms a sub-

variety Sg, which can be stratified into sets of surface classes admitting a particular

(topological equivalence class of ) finite group action.

The hyperelliptic surfaces constitute a connected subvariety of Sg, of dimension

2g� 1. The stratification of this portion of the moduli space is described in this paper.

Only a small list of groups can act on hyperelliptic surfaces, namely, the central

extensions of Z2 by a finite Mobius group (Zn, D2n, A4, S4 or A5). These extensions

fall into eight infinite families; in addition there are eight other individual groups.

We mostly restrict our attention to the eight infinite families (the central exten-

sions of Z2 by Zn and D2n). The actions of these groups are the most interesting from

the point of view of the moduli space: the group orders grow with the genus and

consequently the actions distinguish unique surface classes (exceptional points)

and/or subvarieties of small dimension in the moduli space in all genera.

Brandt and Stichtenoth [2] determined all the hyperelliptic groups as well as the

covering signatures for their hyperelliptic actions; Bujalance, Gamboa and

Gromadzki [5] determined, for each g, those groups which act as the full automor-

phism group of a hyperelliptic surface of genus g. A major difference between our

approach and that of [2] and [5] is that here we classify not just the groups, but their

?Research supported in part by a grant form the City University of New York PSC-CUNY Research

Award Program.

Geometriae Dedicata 103: 69–87, 2004. 69# 2004 Kluwer Academic Publishers. Printed in the Netherlands.

actions up to topological equivalence. This was the approach taken by Broughton in

his treatment of group actions in genus 2 and 3 [3] (see also [4, 11, 12]).

The outline of the paper is as follows. In Sections 2 and 3 we lay the necessary

groundwork on Fuchsian groups and their Teichmuller spaces, define the lattice

of full action classes, and its relation to the stratification of the moduli space

(Thoerem 2.1). In Section 4, we define hyperelliptic surfaces and actions, and the

reduced group of a hyperelliptic action. In Sections 5 and 6, we use finite group

theory to prove that hyperelliptic actions with reduced group Zn and D2n are unique

up to topological equivalence; this aspect was not treated in earlier papers [2, 5]

on hyperelliptic actions. Finally, in Section 7, we give the stratification of the

hyperelliptic portion of the moduli space by drawing the lattices of maximal hyper-

elliptic actions in arbitrary genera >2. We briefly consider the hyperelliptic

actions by groups of order 8, which partially determine the actions of reduced

polyhedral type (central extensions of Z2 by A4, S4 and A5).

2. Preliminaries

Rg denotes the Riemann (moduli) space of conformal equivalence classes of compact

Riemann surfaces of genus g. An automorphism of X 2 Rg is a conformal self-homeo-

morphism; Aut ðX Þ denotes the full group of automorphisms of X. A group G acts

on X if there is a monomorphism of G into Aut ðX Þ.

2.1. A G-action on X determines (and is determined by) a short exact sequence of

homomorphisms

1 ! L ! G!rG ! 1 ð1Þ

in which G is a Fuchsian group (discrete group of orientation-preserving isometries

of the upper half plane U) with compact quotient space, and L is isomorphic to the

fundamental group of X. L is called a surface group of genus g; X is conformally

equivalent to the quotient U=L. r is called smooth epimorphism, where

smooth means that the kernel ðLÞ is torsion-free. G is called the covering group

of the G-action. The triple ðG; r;GÞ denotes the action determined by the

sequence (1).

[15] and [10] are good general references on the relation between Fuchsian groups

and Riemann surface automorphisms.

G has a signature of the form

s ¼ ðh;m1;m2; . . . ;mrÞ; ð2Þ

which specifies the presentation

generators: a1; b1; . . . ; ah; bh; gl; . . . ; grrelations: orderðgiÞ ¼ mi; ði ¼ 1; . . . ; rÞ; ð3Þ

Yhi¼1

½ai; bi�Yrj¼1

gj ¼ 1;

70 ANTHONY WEAVER

where ½a; b� denotes the commutator a�1b�1ab. The signature is unique up to reor-

dering of the periods m1; . . . ;mr, which are in one-to-one correspondence with con-

jugacy classes of maximal finite cyclic subgroups of G. Thus G is a surface group if

and only if there are no periods; a surface group of genus g has signature ðg;�Þ.

We adopt the following conventions with regard to the signature (2): (a) if h ¼ 0

we write s ¼ ðm1; . . . ;mrÞ, omitting h; (b) if a period m occurs k 1 times, we

write m½k� instead of m; . . . ;m (k times).

Two short exact sequences

1 ! Li ! Gi !riGi ! 1; i ¼ 1; 2

are called isomorphic if there exist group isomorphisms a:G1 ! G2 and b:G1 ! G2

such that r2a ¼ br1.Actions ðGi; ri;GiÞ; i ¼ 1; 2, are called topologically equivalent if the corresponding

short exact sequences are isomorphic [14]. ½G; r;G� denotes the equivalence class

of ðG; r;GÞ. Occasionally, we refer to the g-action class ½G; r;G�, to indicate that

the actions in the class take place on surfaces of genus g (i.e., that the kernel of ris a surface group of genus g).

There are just finitely many actions (hence, finitely many action classes) in a given

genus g5 2. We make the set of actions in genus g into a partially ordered set by

defining

ðG; r;GÞ5 ðD; t;H Þ

whenever G contains a subgroup ~DD ’ D, such that ½ ~DD; rj ~DD; rð~DDÞ� ¼ ½D; t;H �.

2.2. We call a g-action class ½G; r;G� full if it is topologically equivalent to the action

of the full group of automorphisms of at least one surface of genus g. For g > 2, the

trivial g-action class [{1}] is full because there exist surfaces U=L admitting no non-

trivial automorphisms.

Let Ag denote the (finite) set of full g-action classes. The partial order defined

above descends to equivalence class thus ðAg; 5Þ is a partially ordered set called

the lattice of full g-action classes. Strictly, it is not a lattice since there is no unique

maximal element.

To each full g-action class ½G; r;G� we associate a subset Vgð½G; r;G�Þ of Rg.

X 2 Vgð½G; r;G�Þ if and only if, up to topological equivalence, the action of

Aut ðX Þ restricts to ½G; r;G�. The set

Sg ¼ fVgðAÞ jA 2 Agg; ð4Þ

is partially ordered by subset containment ð�Þ. Thus, it has the structure of a lattice,

except that there is no unique minimal element. There is a unique maximal

element, since Vgð½f1g�Þ ¼ Rg. ðSg;�Þ is called the stratification of the moduli space

in genus g.

THEOREM 2.1. The map A 7!VgðAÞ is a one-to-one, lattice-preserving, order-

reversing correspondence between ðAg;5Þ and ðSg;�Þ.

HYPERELLIPTIC SURFACES AND THEIR MODULI 71

The proof involves considerations from the Teichmuller theory of Fuchsian

groups, and is given in Section 3.

2.3. GENERATING VECTORS

A g-action ðG; r;GÞ with covering group G ¼ GðsÞ, s ¼ ðh;m1; . . . ;mrÞ, exists if and

only if G has a s-generating vector, that is, an ordered set of elements

ða1; b1; . . . ; ah; bh; c1; . . . ; crÞ ð5Þ

which generate G, such that ord ðciÞ ¼ mi, andQh

i¼1½ai; bi�Qr

j¼1 cj ¼ 1. This follows

from the fact that G ¼ rðG Þ, and the fact that r, being smooth, preserves the orders

of elements. Indeed we may take ai ¼ rðaiÞ; bi ¼ rðbiÞ; i ¼ 1; . . . ; h; cj ¼ rðgjÞ;j ¼ 1; . . . ; r. Conversely, if a finite group G has a s-generating vector, there is an

obvious smooth epimorphism from the Fuchsian group G ¼ GðsÞ onto G.

Taking advantage of the latter fact, one can specify an action of G by just giving

a generating vector (in terms of a fixed presentation of G). Of course, different

generating vectors may specify the same action class. Thus we make the following

definition.

Let v1 ¼ ðx1; . . . ; xnÞ and v2 ¼ ðy1; . . . ; ynÞ be two s-generating vectors for a group

G. We define v1 and v2 to be equivalent if there exists o 2 AutðGÞ such that

oðxiÞ ¼ yi; i ¼ 1; . . . ; n.

LEMMA 2.2. Equivalent s-generating vectors determine topologically equivalent

actions.

Proof. Let G ¼ GðsÞ, and let ri:G ! G; i ¼ 1; 2 be the smooth epimorphisms

determined by the s-generating vectors v1 and v2, respectively. Let o 2 Aut ðGÞ effect

an equivalence between v1 and v2. Then r2 ¼ or1. It follows that the short exact

sequences 1 ! kerð riÞ ! G!ri

G ! 1; i ¼ 1; 2 are isomorphic. &

3. Teichmuller Spaces of Fuchsian Groups

The study of spaces of discrete subgroups of a Lie group was initiated by A. Weil and

C. Chabauty and is treated comprehensively by Macbeath and Singerman in the

important paper [16]. In that paper, particular attention is paid to the Lie group

PGLð2;RÞ, and its index 2 subgroup PSLð2;RÞ because of their relevance to the

problem of moduli for Riemann surfaces. Indeed, the latter group is the group of

conformal automorphisms of U, and its discrete subgroups are the Fuchsian groups.

We describe the results of [16] which are relevant to our purpose. Other useful refer-

ences are [7, 8, 11, 18, 19].

Let G be a Fuchsian group with signature (2) and presentation (3). Let

L ’ PSLð2;RÞ. The set WðG Þ of monomorphisms G ,!L with discrete image and

72 ANTHONY WEAVER

compact quotient is topologized as a subset LN;N ¼ 2hþ r, by associating to

f 2 WðG Þ the point ðfða1Þ;fðb1Þ; . . . ;fðahÞ;fðbhÞ;fðglÞ; . . . ;fðgrÞÞ 2 LN.

L acts on WðG Þ by conjugation: for l 2 L;f 2 WðG Þ, the map

g 7! l�1 � fðgÞ � l

ðg 2 G Þ defines another element of WðG Þ. The quotient space WðG Þ=L is called the

Teichmuller space of G and denoted TðG Þ. TðG Þ is homeomorphic to a cell of

complex dimension

dðTðG ÞÞ ¼ 3h� 3þ r: ð6Þ

It admits a metric (the Teichmuller metric) which induces the same topology as

the quotient topology induced from WðG Þ [16, Theorem 8.6].

Let AðG Þ be the group automorphisms of G and IðG Þ the subgroup of inner auto-

morphisms. The modular group ModðG Þ ¼ AðG Þ=IðG Þ acts properly discontinuously

on TðG Þ as a group of isometries [16, Theorem 9.12, 9.14]. The action is defined by

�aa: ½t� 7! ½t � a�1�

where �aa 2 MðG Þ denotes the equivalence class of a 2 AðG Þ, and ½t� 2 TðG Þ denotes

the equivalence class of t 2 WðG Þ. The quotient space TðG Þ=MðG Þ, denoted RðG Þ,

is the moduli space of surfaces uniformized by G. TðG Þ is a branched covering of

RðG Þ and the locus of branching is the set of points in TðG Þ whose stabilizer is a

nontrivial (finite) subgroup G4ModðG Þ. These points project to surface classes

in RðG Þ admitting a G-action.

In the special case where G is a surface group with signature ðg;�Þ;ModðG Þ is iso-

morphic to the mapping class groupModg;TðG Þ is homeomorphic to the Teichmuller

space T g of marked surfaces of genus g (having complex dimension 3g� 3, by (6)),

and RðG Þ is the full moduli space Rg [19, 20].

3.1. Let G0 and G be Fuchsian groups with compact quotient space, and j: G0 ! G a

monomorphism. j induces a map

�jj:TðG Þ ! TðG0Þ ð7Þ

defined by ½t� 7! ½t � j �; t 2 WðG Þ, which turns out to be an isometric imbedding

[16, Corollary 8.9]. The following lemma is a simple consequence of the definition

of �jj (see, e.g. [18]); the accompanying corollary leads to a proof of Theorem 2.1.

LEMMA 3.1. Let G0 and G be Fuchsian groups with compact quotient space. Let

ji:G0 ,!G; i ¼ 1; 2 be monomorphisms. If there exists an automorphism a:G ! G such

that aj1 ¼ j2, then the images �j1j1ðTðG ÞÞ; �j2j2ðTðG ÞÞ coincide in TðG0Þ.

COROLLARY 3.2. Let ðG; r;GÞ be a g-action, with L ¼ kerð rÞ;L a surface group of

genus g. The inclusion i:L ,!G induces an imbedding �ii:TðG Þ ! T g whose image

depends only on the class ½G; r;G�.


Proof. If ðG0; r0;G0Þ 2 ½G; r;G�, the short exact sequences determined by r and r0

are isomorphic. Thus the monomorphisms kerð rÞ ,!G and kerð r0Þ ,!G0 are related

as in the Lemma. &

Let Tð½G; r;G�Þ � T g denote the unique subvariety associated with the g-action

class ½G; r;G� as in the Corollary. Tð½G; r;G�Þ projects (via the action of Modgon T g) to a subvariety Vð½G; r;G�Þ � Rg whose dimension is equal to the dimension

of TðG Þ, given at (6). It follows from Teichmuller’s extremal mapping theorem that

Vð½G; r;G�Þ is connected (see [6, Theorem 7] for a short proof).

We now show that Vð½G; r;G�Þ ¼ Vgð½G; r;G�Þ, where the latter set has the defini-tion given in Section 2.2. Clearly Vð½G; r;G�Þ � Vgð½G; r;G�Þ. For the opposite

inclusion, let X 2 Vgð½G; r;G�Þ. Then X ¼ U=L, where L is a surface group with

signature ðg;�Þ, and there is a short exact sequence

1 ! L!kG ! G ! 1;

in which k is a monomorphism. This sequence is isomorphic to the sequence deter-

mined by r, hence, by Lemma 3.1, the image �kkðTðGÞÞ coincides with Tð½G; r;G�Þ andtherefore X 2 Vð½G; r;G�Þ.

Proof of Theorem 2:1: We can now prove that

A 7!VgðAÞ; A 2 Ag ð8Þ

is a one-to-one, lattice-preserving, order-reversing correspondence between ðAg; 5Þ

and ðSg;�Þ. Let ½G; r;G�; ½D; t;H � 2 Ag, with ½G; r;G�5 ½D; t;H �. Suppose X 2

Vgð½G; r;G�Þ. Since G5 �DD ’ D and the sequence determined by t is isomorphic to

the sequence determined by rj �DD, it is also true that X 2 Vgð½D; t;H �Þ. Thus

Vgð½G; r;G�Þ � Vgð½D; t;H �Þ. Hence the correspondence (8) is lattice-preserving

and order-reversing.

If the correspondence (8) is not one-to-one, there exist full g-action classes A 6¼ B

for which VgðAÞ ¼ VgðBÞ ¼ V. Then for any X 2 V, the action of Aut ðX Þ must

restrict to both A and B. Thus A and B cannot be full action classes, unless

A ¼ B, contrary to assumption. This completes the proof of Theorem 2.1. &

3.2. Let Gi, i ¼ 1; 2 be Fuchsian groups such that G1 4G2 with finite index

½G1:G2� > 1. ðG1;G2Þ is called a Greenberg–Singerman pair if for every monomorph-

ism j:G1 ! G2, the induced imbedding �jj:TðG2Þ ! TðG1Þ is actually a surjection.

There are just finitely many Greenberg–Singerman pairs [16, Theorem 9.15], [8,

21]; they occur if and only if dðTðG2ÞÞ ¼ dðTðG1ÞÞ. The implication for g-action

classes is as follows. Let ðG1;G2Þ be a Greenberg–Singerman pair, and suppose Gi

covers a g-action class At, i=1, 2. If, in addition, A1 4A2 then VgðA1Þ ¼ VgðA2Þ,

and A1 is not a full g-action class since there is no surface of genus g admitting

A1 but not A2. (But see Remark 7.1, below.)

74 ANTHONY WEAVER

4. Hyperelliptic Surfaces

X ¼ U=L 2 Rg is hyperelliptic if and only if the normalizer of L in PSL2ðRÞ contains

a unique (hence normal) subgroup Lg, with signature ð2½2gþ2�Þ such that the index

½Lg:L� ¼ 2 [18]. As usual L denotes a surface group of genus g. It follows that X

admits a unique Z2 action, covered by Lg, having 2gþ 2 fixed points, such that

the quotient is the Riemann sphere. The generator of this group is called the hyper-

elliptic involution, and it will be denoted t throughout the paper. There is just one

possible smooth epimorphism r from Lg onto Z2, hence the action of hti ’ Z2

is unique up to topological equivalence; we denote the class ½Lg; r;Z2� by [t] for

brevity.

The subvariety Vgð½t�Þ � Rg consisting of hyperelliptic surfaces has complex

dimension 2g� 1 by (6). This is strictly less than the dimension of Rgð3g� 3Þ unless

g ¼ 2, where equality holds. It follows that all surfaces of genus 2 are hyperelliptic;

and 2 is the only genus where this is the case.

An action ðG; r;GÞ is called hyperelliptic if it restricts to ½t�. This means that G con-

tains a unique normal subgroup isomorphic to Lg; hyperellipticity depends only on

the class ½G; r;G�. The same is true of the finite quotient group G=Lg, which is called

the reduced group of ½G; r;G�. Since the reduced group acts on the sphere U=Lg, it is a

finite Mobius group, hence it is either cyclic, dihedral, or isomorphic to A4, S4 or A5.

It follows that G is a central extension of Z2 by one of these groups. This restricts the

possibilities for G to the groups in Table I, together with eight other groups of order

4120 which are central extensions of Z2 by the polyhedral groups A4, S4 and A5.

By contrast, every finite group can be represented as the full automorphism group

of a (not necessarily hyperelliptic) surface [9].

In Table I we use TDN, SDN, DDN to indicate, respectively, ‘twisted dihedral’,

‘semidihedral’, and ‘doubled dihedral’ groups of order N. GQN denotes a generalized

quaternion group of order N. We note the following isomorphisms among the

groups in Table I:

Table I. Central Extensions of Z2 ¼ hti by Zn and D2n.

Name Presentation t ¼

Z2 � Zn hx; t j xn ¼ t2 ¼ ½x; t� ¼ 1i

Z2n hx j x2n ¼ 1i xn

Z2 �D2n hx; y; t j xn ¼ y2 ¼ t2 ¼ ½x; t� ¼ ½ y; t� ¼ 1; yxy�1 ¼ x�1i

D4n hx; y jx2n ¼ y2 ¼ 1; yxy�1 ¼ x�1i xn

TD4n hx; y; t j xn ¼ y2 ¼ t2 ¼ ½x; t� ¼ ½ y; t� ¼ 1; yxy�1 ¼ tx�1i

SD4n hx; y jx2n ¼ y2 ¼ 1; yxy�1 ¼ xn�1i xn

GQ4n hx; y jx2n ¼ 1; y2 ¼ xn; yxy�1 ¼ x�1i xn

DD4n hx; y jxn ¼ y4 ¼ 1; yxy�1 ¼ x�1i y2


Z2 �D2n ’ D4n if n odd; ð9Þ

DD4n ’ GQ4n if n odd; ð10Þ

TD8 ’ D8;

DD8 ’ SD8 ’ Z2 � Z4: ð11Þ

We also point out that TD4n and SD4n exist if and only if n is even.

The possible covering signatures for the groups in Table I were determined in [2],

where it was shown that a hyperelliptic action corresponding to each signature exists.

This information is incorporated into our Table II (we omit the information

for groups of reduced polyhedral type). We have added a representative generating

vector for each action. In the next two sections we prove that these generating

vectors are unique up to equivalence and hence that Table II may be considered a

table of action classes.

Table II determines g-hyperelliptic actions as follows: give g5 2, an action from

the table exists if and only if the corresponding value of q ¼ qðn; gÞ is a nonnegative

integer and n is not specifically excluded in column one. q denotes the number of

occurrences of t in the generating vector. The excluded values of n involve groups

of order48, for which there are redundancies due to low-order isomorphisms

between the groups.

Table II. g-hyperelliptic action classes with reduced group Zn and D2n.

Action type Group Signature q Generating vector

A1n Z2 � Zn ð2½q�; n; nÞ 2gþ2

n ðt½q�;x�1;xÞ q even

ðt½q�; tx�1; xÞ q odd

A2n Z2n ð2½q�; 2n; 2nÞ 2g

n ððxnÞ½q�;x�1;xÞ q evenðn > 1Þ ððxnÞ½q�;xn�1; xÞ q odd

Cn Z2n ð2½q�; n; 2nÞ 2gþ1n ððxnÞ½q�;xn�1; xÞ q odd

(n odd, >1Þ

J1n Z2 �D2n ð2½q�; 2; 2; nÞ gþ1n ðt½q�; yx; y;xÞ q even

ðn > 1Þ ðt½q�; yx; ty;xÞ q odd

J2n D4n ð2½q�; 2; 2; 2nÞ gn ððxnÞ½q�; yx; y;xÞ q even

ðn > 1Þ ððxnÞ½q�; y; yxn�1;xÞ q odd

K1n TD4n ð2½q�; 2; 4; nÞ gþ1

n � 12 ððt½q�; y; yx�1;xÞ q even

ðn even, > 2Þ ðt½q�; y; tyx�1;xÞ q odd

K2n SD4n ð2½q�; 2; 4; 2nÞ g

n �12 ððxnÞ½q�; y; yx�1; xÞ q even

ðn even, > 2Þ ððxnÞ½q�; y; yxn�1;xÞ q odd

L1n DD4n ð2½q�; 4; 4; nÞ gþ1

n � 1 ððy2Þ½q�; yx; y�1; xÞ q evenðn > 1Þ ððy2Þ½q�; yx; y;xÞ q odd

L2n GQ4n ð2½q�; 4; 4; 2nÞ g

n � 1 ððxnÞ½q�; y; yxn�1;xÞ q evenðn > 1Þ ððxnÞ½q�; y; yx�1; xÞ q odd

76 ANTHONY WEAVER

5. Actions with Reduced Group Zn

From Table II, these actions are of type A1n, A

2n and Cn. We deal with type A1

n and

omit the (similar) arguments for the other two types.

Let g5 2 be given and let n be a divisor of 2g + 2. An A1n-generating vector for

Z2 � Zn ¼ ht; xjt2 ¼ xn ¼ ½t; x� ¼ ei has one of the forms

ðt½q�; x�j; x jÞ or ðt½q�; tx�j; tx jÞ ðif q is evenÞ; ð12Þ

ðt½q�; tx�j; x jÞ or ðt½q�; x�j; tx jÞ ðif q is oddÞ: ð13Þ

where j is a positive integer coprime with n such that 14 j < n. (t½q� means that t is

repeated q times in the generating vector.) The definition of generating vector

(Section 2.3) makes the parity of q relevant: the product of the elements must be

the identity, hence the number of occurrences of the central involution t must be

even. There is an automorphism of Z2 � Zn which fixes t while interchanging x

and tx, and it effects an equivalence between the left and right vectors in both

(12) and (13), for fixed j. The automorphism which fixes t and sends x 7!x j (applied

finitely many times) effects an equivalence between any generating vector of the

form (12) or (13) and one in which j ¼ 1. Thus, given n and g, if an A1n-generating

vector exists it is equivalent to ðt½q�; x�1; xÞ if q is even or ðt½q�; tx�1; xÞ if q is odd.

These are the representative generating vectors given in Table II. It follows from

Lemma 2.2 that the g-hyperelliptic action class ½A1n� is unique.

6. Actions with Reduced Group D2n

Let G4n denote one of the six central extensions of Z2 by D2n, with order 4n. A J in-,

K1n- or L

in-generating vector for a hyperelliptic G4n-action is of the form

ðt ½q�; c1; c2; c3Þ ð14Þ

where c1c2c3 ¼ 1 (if q is even) or =t (if q is odd). c3 ¼ x j or tx j, for some j coprime

with n (or 2n, as appropriate). Suppose j 6¼ 1. The map which fixes t; y while mapping

x 7! x j is an automorphism of all six groups, which may be applied finitely many

times, yielding an equivalent generating vector of the form (14) in which c3 ¼ x or

tx . If c3 ¼ tx, and n is even, the map which fixes t; y while mapping x 7! tx is an

automorphism which yields an equivalent generating vector of the form (14) in which

c3 ¼ x. If n is odd. G4n ’ D4n or ’ GQ4n by (9), (10), whence we may assume x has

order 2n, and the same map determines and automorphism in this case as well. Thus

we may assume in all cases that (14) has the form

ðt½q�; c1; c2; xÞ: ð15Þ

It follows that the problem of finding a generating vector (up to equivalence) for a

hyperelliptic G4n-action reduces to the problem of finding ordered pairs ðc1; c2Þ of

elements of G4n (having appropriate orders), whose product is x�1 (if q is even) or

tx�1 (if q is odd). The order of an element in the pair ðc1; c2Þ is either 2 or 4.


THEOREM 6.1. Let ci, i ¼ 1; 2 be elements of G4n such that ð15Þ is a generating

vector for a hyperelliptic G4n-action. For each i separately, either order ðciÞ ¼ 4, and

c2i ¼ t; or order ðciÞ ¼ 2, and ci 6¼ t.

The proof uses the fact that a hyperelliptic G4n-action restricts to a unique hyper-

elliptic action by an index 2 subgroup H ’ Z2n or ’ Z2 � Zn. We give the proof in

Section 6.3, but first we extract the essential features of this situation.

6.1. Consider actions ðG; r;GÞ5 ðD; t;H Þ, where G5H and the index ½G:H � ¼ 2.

Suppose X ¼ U=L 2 Vgð½G; r;G�Þ. We may assume G5D (with index ½G:D� ¼ 2),

and that both G and D contain L as a normal subgroup. Let Y;Z 2 Rg denote

the surfaces U=D and U=G, respectively. The action of G ’ G=L on X factors

through an action of Z2 ’ G=D ’ G=H on Y. In other words, the diagram below

commutes:

Let BH � Y denote the branch set of theH-action, i.e., the finite set over which the

H-orbit map is branched. If the covering group D has signature ðh;m1; . . . ;mrÞ, then

Y is a surface of genus h and BH consists of r points labelled with branch indices

m1; . . . ;mr, respectively. Over the ith point, there are jHj=mi points in X fixed by

cyclic subgroup of H having order mi. It is convenient to assign the ‘branch index’ 1

to a point in Y nBH.

Assume that h ¼ 0 so that Y is a sphere (this implies that Z is also a sphere). The

Z2 action on Y is equivalent to a rotation by 180� with two fixed points. Let

p: Y ! Z be the Z2 orbit map. For each z 2 Z, let Oz ¼ p�1ðzÞ � Y. Oz consists

of either 1 or 2 , and is either contained in the branch set BH or disjoint from it.

If jOzj ¼ 2, the two points must have the same branch index with respect to the

H-action; thus there is a well-defined branch index rðOzÞ5 1 associated to each

orbit Oz.

Figure 1.

78 ANTHONY WEAVER

We now prove two lemmas pertaining to the commutative diagram in Figure 1.

Recall that Y and Z are assumed to be spheres. Let BG � Z denote the branch set

of the G-action.

LEMMA 6.2. BG ¼ fz 2 Z j sðzÞ > 1g, where

sðzÞ ¼2rðOzÞ if jOzj ¼ 1;

rjOzj if jOzj ¼ 2:

�ð16Þ

sðzÞ is the branch index ðwith respect to the G-actionÞ of z.

Proof. Let Oz ¼ fyg, i.e., jOzj ¼ 1. Then there are jHj=rðfygÞ ¼ jGj=2rðfygÞ points

in X lying over y and hence over z. Thus z has branch index 2rðfygÞ with respect to

the G-action. On the other hand, if Oz ¼ fy1; y2g, i.e., jOzj ¼ 2, then there are

jHj=rðOzÞ points in X lying over both y1 and y2 and hence 2jHj=rðOzÞ ¼ jGj=rðOzÞ

points lying over over z. Thus z has branch index rðOzÞ with respect to the

G-action. &

LEMMA 6.3. p�1ðBGÞ ¼ BH [W, where W \ BH ¼ ; and jWj4 2.

Proof. Commutativity of the diagram in Figure 1 implies BH � p�1ðBGÞ. Suppose

y 2 p�1ðBGÞ. If y 62 BH, y has branch index 1 with respect to the H-action. Let

z ¼ pð yÞ. By the previous lemma, fyg ¼ Oz, since otherwise sðzÞ ¼ 1, z 62 BG and

y 62 p�1ðBGÞ, contrary to assumption. Thus y is one of the two fixed points of the Z2-

action, which implies that W ¼ p�1ðBGÞ n BH has cardinality at most 2. &

6.2. In the set-up of the previous section, let G ¼ G4n and H ¼ Z2n or Z2 � Zn. By

abuse of notation, we also let G and H denote the corresponding g-hyperelliptic

actions; we know that H belongs to one of the action classes ½A1n�; ½A

2n� or ½Cn�,

and we take G to be an action of type J in;K

in; or L

in from Table II.

To further simplify notation, we identify branch sets with their associated branch-

ing indices. Thus if H 2 ½A1n�, we say that BH ¼ f2½q�; n; ng, meaning that BH � Y con-

sists of q points with branch index 2, and two points with branch index n. Similarly,

a point in W ¼ p�1ðBGÞnBH is denoted by its branch index, 1.

We first show that H 62 ½Cn�. Suppose the contrary. In this case BH ¼

f2½q�; n; 2ng � Y. n and 2n must be fixed points (i.e., singleton orbits) of the Z2-action

on Y, since points in the same Z2-orbit must have the same branching index with

respect to the H-action. Let z ¼ pðf2ngÞ. By (16), sðzÞ ¼ 4n, which implies there is

a single point in X lying over z, fixed by every element of G4n. This is a contradiction,

since G4n is not cyclic.

Hence H 2 ½Ain�; i ¼ 1 or i ¼ 2. For fixed i, there are at most three ways to parti-

tion p�1ðBGÞ ¼ f2½q�; ni; nig [W into Z2-orbits, such that points in the same orbit

have the same branching indices with respect to the H-action:


p�1ðBGÞ ¼

f2; 2g½q=2� [ fni; nig [ f1g [ f1g; or

f2; 2g½ðq�1Þ=2� [ f2g [ fni; nig [ f1g; or

f2; 2g½ðq�2Þ=2� [ f2g [ f2g [ fni; nig;

8><>:

where f2; 2g½q=2� means there are q=2 occurrences of an orbit of the form f2; 2g in the

partition. W is the set of orbits of the form f1g in all cases; in the last case W ¼ ;.

Using (16) we recover BG:

BG ¼

f2g½q=2� [ fnig [ f2g� [ f2g�; or

f2g½ðq�1Þ=2� [ f4gy [ fnig [ f2g�; or

f2g½ðq�2Þ=2� [ f4gy [ f4gy [ fnig:

8><>: ð17Þ

The symbol � is used to distinguish points f2g 2 Z for which f2g ¼ pðf1gÞ. f4gy is

used to distinguish f4g from fnig in case ni ¼ 4. The three signature types

J in;K

in;L

in are recovered from (17) by dropping the set theoretical symbols (and

reordering if necessary).

6.3. We can now prove Theorem 6.1. Over a point f2g� 2 Z there are jHj points in X

permuted freely by H4G but fixed by G=H ’ Z2, i.e. by a nonhyperelliptic element

of order 2 in G. Over a point z ¼ f4gy 2 Z there are jGj=4 points in X fixed by a cyclic

group C ¼ hc j c4 ¼ 1i4G. Since Oz ¼ fyg ¼ f2g 2 Y, these same jGj=4 points

lie over y and hence are fixed by t. Thus C must contain t as its unique element of

order 2, i.e., c2 ¼ t.

From the preceding paragraph it is clear that in a generating vector of the form

(15), the elements c1 and c2 must be as described in Theorem 6.1.

6.4. We now return to the problem of determining all generating vectors for G ¼ G4n

of the form (15). Let 2� denote the subset of G consisting of elements of order 2 other

than t, and 4y the subset of elements of order 4 whose square is t. We shall give one

example of the computations involved.

Let G ’ TD4n. Recall that this group exists if and only if n is even. Using the

multiplication rule

yx j � yxk ¼xk�j if j is even,

txk�j if j is odd,

�

one can determine that

2� ¼ fxn=2; txn=2g [ fyx2i; tyx2i j i ¼ 0; . . . ; ðn� 2Þ=2g

and

4y ¼ fyx2iþ1; tyx2iþ1 j i ¼ 0; . . . ; ðn� 2Þ=2g:

We construct all possible ð2½q�; 2�; 4y; 2nÞ-generating vectors of the form (15). The

possible pairs ðc1; c2Þ 2 2� � 4y such that c1c2 ¼ x�1 if q is even and c1c2 ¼ tx�1

if q is odd are:

80 ANTHONY WEAVER

c1 ¼ tEyx2i; c2 ¼ tEyx2iþ1; ðE; iÞ 2 f0; 1g � f0; . . . ; ðn� 2Þ=2g ðq evenÞ;

c1 ¼ tEyx2i; c2 ¼ tEþ1yx2iþ1; ðE; iÞ 2 f0; 1g � f0; . . . ; ðn� 2Þ=2g ðq oddÞ:

Assume that q is odd. A possible generating vector is then

ðt½q�; tEyx2i; tEþ1yx2i�1; xÞ: ð18Þ

We show that all such vectors are equivalent to the one determined by the pair

ðE; iÞ ¼ ð0; 0Þ. Suppose ðE; iÞ 6¼ ð0; 0Þ. If E ¼ 1, apply the automorphism

a: y 7! ty; x 7!x; t 7! t;

making E ¼ 2, or equivalently, E ¼ 0. If i 6¼ 0, apply the automorphism

b: y 7! yx2; x 7!x; t 7! t

finitely many times until 2i ¼ n, or equivalently, i ¼ 0. Thus all vectors (18)

are equivalent to ðt½q�; y; tyx�1; xÞ. It is clear that the vector generates TD4n. The

argument in the case where p is even is similar, and shows that all vectors in this case

are equivalent to ðt½q�; y; yx�1; xÞ. It follows that if a g-hyperelliptic action by TD4n

exists, it is unique up to topological equivalence.

7. The Lattice of Full Hyperelliptic Action Classes

Henceforth the word ‘action’ means ‘action class’. In addition, we now refer to a

hyperelliptic action by its group alone. This could lead to confusion in just one

case: Z2 � Z4 has two distinct actions in odd genera with covering signature

ð2½ðg�1Þ=2�; 4; 4; 2Þ. One action has reduced group Z4 ð½A14�Þ and the other has reduced

group D4 ð½L14�Þ (recall isomorphism (11)). To avoid this confusion, we retain the

notation DD8 for Z2 � Z4 in the latter case.

A g-hyperelliptic action is maximal if (i) it is a full action; and (ii) there is no larger

group of the same type (i.e., from the same row of Table II) having a full g-hyper-

elliptic action.

THEOREM 7.1. Let g5 2 be given. The maximal actions with reduced cyclic or

dihedral group are:

TD8gþ8 SD8g; Z4gþ2; Z2 �D2gþ2; D4g;

and, in addition,

Z2 � Zðgþ1Þ=p; DD4ðgþ1Þ=p; GQ4g=q; Z2g=r;

where


p ¼ the smallest nontrivial proper divisor of gþ 1

q ¼ the smallest nontrivial proper divisor of g

r ¼ the smallest divisor > 2 of g:

Proof. The largest group order attainable in a row of Table II occurs when n is

largest subject to q being a nonnegative integer. This accounts for the maximality of

TD8gþ8; SD8g; Z4gþ2; Z2 �D2gþ2 and D4g. In the remaining cases, the largest

admissible n produces an action which is not full. This occurs because the signature is

the first member of a Greenberg–Singerman pair (Section 3.2), and the second sig-

nature of the pair covers a larger g-hyperelliptic action. These cases are summarized

in Figures 2 and 3, which show portions of the subgroup lattices of SD8g and

TD8gþ8, respectively, together with the signatures of the corresponding

g-hyperelliptic actions. Downward lines signify action inclusion; double lines link

Greenberg–Singerman pairs and signify automatic extensions from nonmaximal to

maximal actions. (If g ¼ 2, Figure 2 is not complete; there are further extensions

involving actions of reduced polyhedral type.) The numbers p, q and r, if they exist,

produce the largest admissible values of n such that the corresponding signature is

not the first member of a Greenberg–Singerman pair. &

Remark 7:1: It is not always the case that a g-action covered by the first member

of a Greenberg–Singerman pair extends automatically to a larger g-action. Consider

the Greenberg–Singerman pair ðG1;G2Þ where G1 has signature s1 ¼ ð2½2�; 4½2�Þ and G2

has signature s2 ¼ ð2½3�; 4Þ. G1 covers DD8 in genus 3. The only 3-hyperelliptic action

by a group of order 16 covered by G2 is Z2 �D8, but this group, having no element

of order 4 whose square is t, cannot contain DD8. In terms of Fuchsian groups: if

Figure 2.

82 ANTHONY WEAVER

L denotes a surface group of genus 3, there is no series L / G1 / G2 in which L / G2.

(/ denotes normal subgroup.)

7.1. Using Table II and elementary subgroup inclusions, one can draw the lattice of

full hyperelliptic actions for arbitrary g > 2, thus obtaining the stratification of the

hyperelliptic moduli space Vgð½t�Þ. For simplicity we include only maximal actions,

and as usual we omit actions of reduced polyhedral type. Once the parity of g is spe-

cified, either p or q ¼ 2 in Theorem 7.1. In Figures 4 and 5 we give the lattices for

even g > 2 and odd g, respectively. (The genus 2 lattice can be found in [13].) We

have drawn the lattices so that if actions G1 and G2 occur at the same horizontal

level, the subvarieties VgðG1Þ and VgðG2Þ have the same complex dimension, cal-

culated from the signatures of the covering groups (using (6)), and indicated by the

numbers running down the right-hand sides of the lattices. In tracing a path

downward through one of the lattices, we are following a chain of inclusions among

hyperelliptic actions; in tracing the same path upward we are following the chain

of inclusions of corresponding subvarieties of Vgð½t�Þ � Rg.

Subvarieties of dimension 0 (which, being connected, are just single points in the

moduli space) are of particular interest; we call them exceptional points. It is evident

from Figures 4 and 5 that there are always at least three distinct exceptional points in

Vgð½t�Þ, namely,

VgðSD8gÞ; VgðTD8gþ8Þ; and VgðZ4gþ2Þ: ð19Þ

VgðZ4gþ2Þ and VgðSD8gÞ were discovered by Wiman ([23]), who showed that 4gþ 2

is the maximum possible order of a cyclic group of automorphisms of a compact

Riemann surface of genus g5 2. Accola and Maclachlan ([1, 17]) discovered

VgðTD8gþ8Þ independently, and showed that there are are infinitely many genera g

Figure 3.


Figure 5. g odd.

Figure 4. g even, g > 2.

84 ANTHONY WEAVER

in which TD8gþ8 is the largest group of automorphisms that any surface of genus

g can admit.

7.2. GROUPS OF ORDER 8; ACTIONS OF REDUCED POLYHEDRAL TYPE

In Figures 6 and 7 we give the lattices of full hyperelliptic actions by groups of order

2, 4 and 8 in even and odd genera, respectively. This information is readily obtained

from Table II, and depends entirely on the parity of g.

Let P denote one of the polyhedral groups A4;S4 or A5. Since A4 is a subgroup of

both A5 and S4, any central extension of Z2 by P must contain one of the central

extensions of Z2 by A4, of which there are two:

(1) The trivial extension, Z2 � A4, with 2-Sylow subgroup Z2 �D4;

(2) The binary extension, SL2ðZ3Þ, with 2-Sylow subgroup GQ8.

The binary extension of P4 SOð3Þ is its preimage Pbin under the 2 : 1 Lie group

covering SUð2Þ ! SOð3Þ [24]. From Table II, GQ8 has a g-hyperelliptic action if

and only if g is even and Z2 �D4 has a g-hyperelliptic action if and if g is odd. It

follows that if Z2 � P has a g-hyperelliptic action then g is odd; and if Pbin has a

g-hyperelliptic action then g is even. When P ¼ S4 there are, besides Z2 � P and

Pbin, two additional nontrivial extensions.

Necessary and sufficient conditions for the existence of any action of reduced

polyhedral type can be given in terms of the congruence class of g modulo 30 [5].

In particular, Z2 � A4 (resp. SL2ðZ3ÞÞ acts if and only if g is odd (resp. even). In

Figure 6. g even, g > 2.


Figures 6 and 7, the symbols �; �� denote possible actions of reduced type S4 and

A5. The dimensions of the corresponding subvarieties are not given since they

depend on the congruence class of g modulo 30. It can be shown that if g > 30 there

are no exceptional points in Vgð½t�Þ whose full automorphism group is of reduced

polyhedral type [22].

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