math0205314.tex
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\def\fge{{$L/k(x)$\ }} \def\bC{{\mathbb C}} \def\t{{\tau}}
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\def\Aut{\mbox {Aut}}

%*************************************************************




\begin{document}


\centerline{\Large The locus of curves with prescribed automorphism group}
\bigskip

\centerline{April 24, 2002}
\bigskip

\centerline{K. Magaard\footnote{Partially supported by NSA grant
MDA904-01-1-0037}, T.Shaska, S. Shpectorov\footnote{Partially supported by
NSA
grant   MDA904-01-1-0023}, and H. V\"olklein\footnote{Partially supported by
NSF
grant DMS-9970357 }}


\bigskip

\begin{center}Wayne State University, University of California at Irvine,\\
Bowling Green State University and University
of Florida
\end{center}
%\vspace{.5cm}
%{\large Math. Subj. Class. 12F12, 20G40}

\vspace{2cm}
\noindent {\bf Abstract:} \ Let $G$ be a finite group, and $g\ge2$.
We study the locus of genus $g$ curves that admit a $G$-action of
given type, and inclusions between such loci. We use this to study the locus
of genus $g$ curves with prescribed automorphism group $G$.
We completely classify these loci for $g=3$ (including equations for the
corresponding curves), and for $g\le10$ we classify
those loci corresponding to  ``large'' $G$.




%***************************************************************
\section{Introduction}


There is a vast literature on automorphism groups of compact
Riemann surfaces, beginning in the 19th century with Schwartz,
Klein, Hurwitz, Wiman and others. However, most of the literature
is quite recent. In the first part of the paper, we survey the
main results.

By covering space theory, a finite group $G$ acts (faithfully) on
a genus $g$ curve if and only if it has a genus $g$ generating
system (see section \ref{fullsection} below). Using this purely
group-theoretic condition, Breuer \cite{Br} classified all groups
that act on a curve of genus $\le 48$. This was a major
computational effort using the computer algebra system GAP
\cite{GAP}. It greatly improved on several papers dealing with
small genus, by various authors (see the references in part I).

Of course, for each group in Breuer's list, all subgroups are also
in the list. This raises the question how to pick out those groups
that occur as the {\bf full automorphism group} of a genus $g$
curve. This question is answered in Part II of the paper. Let
$\M_g$ be the moduli space of genus $g$ curves. We study the locus
$\L$ in $\M_g$ of curves admitting a $G$-action of given
ramification type (resp., signature). All components of $\L$ have
the same dimension which depends only on the signature of the
$G$-action. Restricting the action to a subgroup $H$ of $G$ yields
an inclusion of $\L$ into the corresponding locus $\L'$ for the
action of $H$. If $\dim \L<\dim L'$ then for "most" curves in
$\L'$, the $H$-action does not extend to a $G$-action of the type
defining $\L$. Thus one is led to classify the pairs $(\L,\L')$
with $\dim \L=\dim\L'$. This is done in Lemma \ref{Lemma2} below:
It turns out that such pairs exist only in very restricted cases,
in particular only if $\dim\L\le3$. From that we derive a
necessary and sufficient condition for a group to occur as the
full automorphism group of a genus $g$-curve (Theorem \ref{thm2}).

After Part II was written, we found references to papers of
Singerman \cite{Si} and Ries \cite{Ri} which contain similar
results. Their method is different, of analytic nature, using
Teichm\"uller theory and Fuchsian groups. Our approach is based on
algebraic geometry, using the algebraic structure of Hurwitz
spaces and the moduli spaces $\M_g$; therefore, it can be used to
obtain information on fields of definition. This aspect may be
studied in later work.

In Part III we apply the above criterion to the data compiled by
Breuer (available from his website, see 1.5 below). Our first
application is in the case of genus 3, which is already quite rich
and shows the power of our group-theoretic method. We obtain the
full list of automorphism groups in genus 3 plus equations for the
corresponding curves. This result is scattered over
several long papers by Kuribayashi
and various co-authors \cite{KuKo1},
\cite{KuKo2}, \cite{KK1}, \cite{KK2}, some of which
contain errors (see section 6 below) and some are not available in most
libraries. None of them seems to give a complete account of this
basic result. Here we show how to derive it quickly from the
group-theoretic data.

Further, we obtain the list of ``large'' groups $\Aut(X_g)$ (see 1.3 below
for the definition) up to $g=10$. Plus the dimension and
number of components of the corresponding loci in $\M_g$, see Table 4.

The locus of curves in $\M_g$ with fixed automorphism group
consists of finitely many components; to determine their number
requires tools that will be developed in later work (mapping class
group action on generating systems). In the special case of
components corresponding to curves of orbit genus $0$ (i.e., the
quotient by the automorphism group of the curve has genus $0$),
the mapping class group action is just the braid group action
studied in \cite{FV}, \cite{V}, \cite{Buch};
and those components correspond to
braid orbits on generating systems, which can be computed
with our GAP package BRAID \cite{MSV}. A curve with large automorphism
group always has orbit genus $0$, so we were able to use the BRAID package
to compute the information in Table 4. The details of this computation
will be given in later work (in a more general situation).





\bigskip

\bigskip

\bigskip \noindent
{\bf Notation:} We will use the term ``curve'' to mean ``compact
Riemann surface''; and $X=X_g$ denotes a curve of genus $g\ge2$.
Further, $\Aut(X)$ denotes the group of analytic (equivalently,
algebraic) automorphisms of $X$.


\part{Survey of known results}
%****************************************************************

\subsection{The beginnings in the 19th century}

The group $\Aut(X)$ acts on the finite set of Weierstra\ss\ points of $X$.
This action is faithful unless $X$ is
hyperelliptic, in which case its kernel is the group of order 2
containing the hyperelliptic involution of $X$. Thus in any case,
$\Aut (X)$ is a finite group. This was first proved by Schwartz.

F. Klein \cite{Kl1} (1879) studied the genus 3 projective curve
\begin{equation}
x^3y\ +\ y^3z\ +\ z^3x\ \ =\ \ 0
\label{Klein}
\end{equation}

\noindent which is now known as Klein's quartic.
He found all its automorphisms: They form the simple group of order 168.
Further, he showed that there is a degree 168
covering from this curve to $\P$ branched over three points with
ramification indices 2,3,7.
See \cite{Le} for a detailed history on Klein's quartic
(including an English translation of Klein's original article)
as well as an account of further developments influenced by it.

\par Poincar\'e introduced Fuchsian groups in 1882. They are basic for the
analytic theory of Riemann surfaces and have been used heavily in the study
of
$\Aut(X)$ (see \cite{Br}, \cite{Ri} for example). In this paper we take a
different (algebraic) point of view.


%*****************************************************************

The next milestone was Hurwitz's seminal paper \cite{Hu} in 1893,
where he discovered what is now called the Riemann-Hurwitz formula.
>From this he derived that
$$| \Aut (X_g)| \ \ \leq \ \ 84 \, \, (g-1)$$ now known as the Hurwitz
bound.
This bound is attained by Klein's quartic, for example.
Curves which attain this bound
are called {\bf Hurwitz curves} and their automorphism
groups {\bf Hurwitz groups}. Hurwitz proved that a finite group is a Hurwitz
group
if and only if it has generators $a,b,c$ of orders $2,3,7$, respectively,
with $abc=1$.


\subsection{Hurwitz groups}
%***************************************************

Klein's quartic is the only Hurwitz curve of genus $g\le3$.
Fricke showed that the next Hurwitz group occurs for $g=7$
and has order 504. Its group is $SL(2,8)$, and an equation for it
was computed by Macbeath \cite{Mb2} in 1965.
Klein's quartic and Macbeath's curve are the only
Hurwitz curves whose equations are known. Further Hurwitz curves occur for
$g=13$ and $g=17$ (and for no other values of $g\le19$).

There are a lot of papers by group-theoretists on Hurwitz groups,
surveyed by Conder \cite{Co}. It follows from Hurwitz's presentation
that a Hurwitz group is {\bf perfect}. Thus every quotient is again a
Hurwitz
group, and if such a quotient is minimal then it is a non-abelian simple
group. Several infinite series of simple Hurwitz groups have been found by
Conder, Malle, Kuribayashi, Zalessky, Zimmermann and
others. In 2001, Wilson \cite{Wl} showed the monster is a Hurwitz
group.





\subsection{Large automorphism groups.}
%***********************************************************
For a fixed $g\geq 2$ denote by $N(g)$ the maximum of the
$|\Aut(X_g)|$. Accola \cite{Ac1} and Maclachlan \cite{Mc1} independently
show that $N(g) \geq 8 (g+1)$ and this bound is sharp for
infinitely many $g$'s. If $g$ is divisible by 3 then $N(g) \geq 8
(g+3)$.

The following terminology is rather standard. We say $G\le\Aut(X_g)$ is a
{\bf large automorphism group} in genus $g$ if
$$|G|\ \ >\ \ 4 (g-1)$$
Then the quotient of $X_g$ by $G$ is a curve of genus
$0$, and the number of points of this quotient ramified in $X_g$ is 3 or 4
(see \cite{Br}, Lemma 3.18, or \cite{FK}, pages 258-260).
Singerman \cite{S1} (1974) shows that Riemann surfaces with large
cyclic, Abelian, or Hurwitz groups are symmetric (admit an
involution). Kulkarni \cite{Ku}(1997) classifies Riemann surfaces
admitting large cyclic automorphism groups and works out full
automorphism groups of these surfaces. Matsuno \cite{Mt}(1999)
investigates the Galois covering of the projective line from
compact Riemann surfaces with large automorphism groups.

\subsection{Cyclic groups as automorphism groups.}
%******************************************************
Let $t $ be the order of an automorphism of $X_g$.
Hurwitz \cite{Hu} showed $ t\leq 10 (g-1)$. In 1895, Wiman
improved this bound to be $t\le 2(2 g+1)$ and showed this is best possible.
If $t$ is a prime then $t \leq 2 g +1$. Homma \cite{Ho} (1980)
shows that this bound is achieved if and only if the curve is birationally
equivalent to
$$y^{m-s} (y-1)^s = x^q, \quad for \quad 1 \leq s < m \leq g_x+1$$



\subsection{ The canonical representation of $\Aut(X)$}
%*******************************************************************

Each group $G\le \Aut(X_g)$ acts faithfully on the $g$-dimensional vector
space
of holomorphic differential forms on $X$. The resulting finite subgroups of
$GL(g,\bC)$ satisfy a number of special
conditions, studied by I. Kuribayashi, A. Kuribayashi,
Kimura, Ohmori, Kobayashi, Hayakawa and others.
A. Kuribayashi and others (see [KuKi] and the references in that paper)
have completely listed these finite linear groups
up to genus 5.


Using computational group theory (more precisely, the computer algebra
system
GAP \cite{GAP}), Breuer \cite{Br} extended this up to genus 48:
Computing all groups $G\le\Aut(X_g)$ together with their character
on the space of holomorphic differential forms. He avoids
ad-hoc methods by using the GAP-library of ``small groups''
by Besche and Eick \cite{BE}. His results are collected in a
database which is available from his website at
{\tt www.math.rwth-aachen.de/~Thomas.Breuer/}.
The GAP library
of small groups is available at \\
{\tt http://www-public.tu-bs.de:8080/\~{}beick/so.html}.
In our tables a group $G$ is identified by its Group ID from the
Small Groups Library. A Group ID is a pair $(n,m)$ where $n$ is the
order of $G$ and $m$ the number of $G$ among the groups of order $n$.
\bigskip


%***********************************************
\subsection{Full automorphism groups.}
%************************************************


For each group in Breuer's list, all subgroups are also in the
list. This raises the question how to pick out those groups that occur
as the {\bf full automorphism group} of a genus $g$ curve. Answering
this question requires a moduli space argument: Either analytically,
using Fuchsian groups and Teichm\"uller space, as done by
Singerman \cite{Si} and Ries \cite{Ri}, or algebraically, as
done in Part II of this paper.
Using these general results, one can compute explicitly all groups
$\Aut(X_g)$
for small $g$. We give the complete list for $g=3$ in Part III, and
the list of large groups $\Aut(X_g)$ up to $g=10$. Further results will be
given in later work, together with additional information.

For $g>3$, there are no lists of full automorphism groups in the literature.
The case $g=2$ is easy, but already for $g=3$ there is no complete
account (see the remarks in the introduction).


\part{Subgroups of $\Aut(X_g)$ not occurring as full automorphism groups
in genus $g$}

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%\parindent 11pt

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\def\TG{{\cal T}_r(G)} \def\oor{{\cal O}_r} \def\T{{\cal T}}
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\def\A{{\bf A}} \def\UR{{\cal U}^{(r)}}
\def\Ni{{\mbox{Ni}}} \def\Vtt{(\tilde V)\tilde{}}
\def\Vht{(\hat V)\tilde{}}


\def\d{{\delta }} \def\a{{\alpha }} \def\b{{\beta }} \def\g{{\gamma }}
\def\M{{\cal M}} \def\FF{{\cal F}} \def\s{{\sigma }}
\def\H{{\cal H}} \def\NN{{\cal N}}
\def\HHR{{\cal H}^{(r)}} \def\HHr{{\cal H}_{r}}
\def\U{{\cal U}} \def\UR{{\cal U}^{(r)}} \def\Ur{{\cal U}_{r}}

% \def\B{{\frak B}} \def\F{{\frak F}}
% \def\H{{\frak H}} \def\E{{\frak E}} \def\N{{\frak N}}
% \def\HR{{\frak H}^{(r)}} \def\Hr{{\frak H}_{r}}
\def\Aut{\mbox {Aut}}

\section{Ramification type and signature of a $G$-curve}

Fix an integer $g\ge2$ and a finite group $G$. Let $C_1,...,C_r$
be conjugacy classes $\ne\{1\}$ of $G$. Let $\C=(C_1,...,C_r)$,
viewed as unordered tuple, repetitions are allowed. We allow $r$
to be zero, in which case $\C$ is empty.

We will use the term ``curve" to mean ``compact Riemann surface".
Consider pairs $(X,\mu)$, where $X$ is a curve and $\mu:
G\to\Aut(X)$ is an injective homomorphism. Mostly we will suppress
$\mu$ and just say $X$ is a curve with $G$-action, or a $G$-curve,
for short. Two $G$-curves $X$ and $X'$ are called equivalent if
there is a $G$-equivariant isomorphism $X\to X'$.
%Let $|X,\mu|$ denote the equivalence class of the
%pair $(X,\mu)$.

%\subsection{The ramification type and signature of a $G$-curve}

We say a $G$-curve $X$ is {\bf of ramification type}
$(g,G,\C)$ if the following holds: Firstly,
$g$ is the genus of $X$. Secondly, the points of the quotient
$X/G$ that are ramified in the cover $X\to X/G$ can be labelled as
$p_1,...,p_r$
such that $C_i$ is the conjugacy class in $G$ of distinguished inertia
group generators over $p_i$ (for $i=1,...,r$).
(Distinguished inertia group generator means the generator that acts in the
tangent space as multiplication by $\exp(2\pi\sqrt{-1}/e)$, where $e$ is the
ramification index). For short, we will just say $X$ is of type $(g,G,\C)$.

If $X$ is a $G$-curve of type $(g,G,\C)$ then the genus $g_0$ of
$X/G$ is given by the Riemann-Hurwitz formula $$\frac{2\
(g-1)}{|G|}\ \ \ = \ \ \ 2\ (g_0-1)\ +\ \sum_{i=1}^r\
(1-\frac{1}{c_i}\ ) \leqno{(1)}$$ where $c_i$ is the order of the
elements in $C_i$.


%We call $g_0$ the {\bf orbit genus} associated with the ramification type
$(g,G,\C)$.
Note that $g_0$ (the {\bf orbit genus}) depends only on $g$, $|G|$
and the {\bf signature} $\c=(c_1,\ldots,c_r)$ of the $G$-curve
$X$.

\section{Hurwitz spaces and moduli of curves}
\label{section2}

Define $\H=\H(g,G,\C)$ to be the set of equivalence classes of $G$-curves
of type $(g,G,\C)$. By covering space theory (or the theory of Fuchsian
groups),
$\H$ is non-empty if and only if $G$ can be generated by elements
$\a_1,\b_1,...,\a_{g_0},\b_{g_0},\g_1,...,\g_r$ with $\g_i\in C_i$ and
$$\prod_{j}\ [\a_j,\b_j]\ \ \prod_{i}\ \g_i\ \ \ \
= \ \ \ 1 \leqno{(2)}$$
Here $[\a,\b]=\ \a^{-1}\b^{-1}\a\b$.
%\subsection{The Hurwitz space as algebraic variety}

Let $\M_{g}$ be the
moduli space of genus $g$ curves, and $\M_{g_0,r}$ the moduli space
of genus $g_0$ curves with $r$ distinct marked points, where we view the
marked points as unordered (contrary to usual procedure).
Consider the map
$$\Phi:\ \H\ \to \ \M_{g}$$
forgetting the $G$-action, and the map
$$\Psi:\ \H\ \to \ \M_{g_0,r} $$
mapping (the class of) a $G$-curve $X$ to the class of the quotient
curve $X/G$ together with the (unordered) set of branch points
$p_1,...,p_r$.
If $\H\ne\emptyset$ then $\Psi$ is surjective and has finite fibers,
by covering space theory.
Also $\Phi$ has finite fibers, since the
automorphism group of a curve of genus $\ge2$ is finite.


By \cite{Be}, the set $\H$ carries a structure of quasi-projective variety
(over $\bC$) such that the maps $\Phi$ and $\Psi$ are finite
morphisms. If $\H\ne\emptyset$ then
all components of $\H$ map surjectively to $\M_{g_0,r}$
(through a finite map), hence they all have the same dimension

$$\d(g,G,\C):= \ \ \dim\ \M_{g_0,r} \ \ = \ \ 3g_0-3+r$$
%\ \ \cases{3g_0-3+r, & if $g_0\ge2$\cr
%r, & if $g_0=1$\cr
%r-3, & if $g_0=0$}$$
(This is correct because $r\ge1$ if $g_0=1$ and $r\ge3$ if $g_0=0$;
the latter holds because $g\ge2$).
Since also $\Phi$ is a finite map, we get

\begin{Lemma} \label{Lemma1} Let $\M(g,G,\C)$ denote the image of $\Phi$,
i.e., the locus of genus $g$ curves admitting a $G$-action
of type $(g,G,\C)$. If this locus is non-empty then each of its components
has dimension $\d(g,G,\C)$.
\end{Lemma}

Note that $\d(g,G,\C)$ depends only on $g$, $|G|$ and the
signature, so we write $\d(g,G,\c):=\d(g,G,\C)$. The Lemma
continues to hold if we replace $\C$ by $\c$.

%\begin{Corollary} Let $\M(g,G,\c)$ be the locus of genus $g$ curves
admitting
%a $G$-action of given signature $(g,G,\c)$.
%If this locus is non-empty then each of its components
%has dimension $\d(g,G,\c)$.
%\end{Corollary}

\section{Restriction to a subgroup} \label{restr}

Let $H$ be a subgroup of $G$.
Then each $G$-curve can be viewed as an $H$-curve by restriction of action.
Let $X$ be a $G$-curve of type $(g,G,\C)$.
Then the resulting $H$-curve is of type
$(g,H,\D)$, where $\D$ is obtained as follows:
Choose $\g_i\in C_i$ and let $\s_{i,1}, \s_{i,2},...$ be a set of
representatives
for the double cosets $<\g_i>\s H$ in $G$. Let $m_{ij}$ be the smallest
integer
$\ge1$ such that the element $\s_{ij}^{-1}\g_{i}^{m_{ij}}\s_{ij}$ lies in
$H$,
and let $D_{ij}$ be the conjugacy class of this element in $H$.
Then $\D$ is the tuple consisting of all $D_{ij}$
with $D_{ij}\ne\{1\}$. (More precisely, the tuple $\D$ is indexed by the set
of possible pairs $(i,j)$, and its $(i,j)$-entry is $D_{ij}$ ).
The definition of $\D$ does not depend on the choice of the $\g_i$ and
$\s_{ij}$.
Note that the signature of the $H$-curve depends on the type of the
$G$-curve, not only on its signature.

We have $$ \M(g,G,\C) \ \ \subset \ \ \M(g,H,\D)$$
hence their dimensions satisfy
$$\d(g,G,\C)\ \ \ \le\ \ \ \d(g,H,\D)$$
If this is a strict inequality then the complement of the closure of $
\M(g,G,\C)$
in $\M(g,H,\D)$ is open and dense; then in particular, it is not true
that every $H$-curve of type $(g,H,\D)$
is the restriction of a $G$-curve of type $(g,G,\C)$.


\begin{Lemma} \label{Lemma2} Suppose $H$ is a subgroup of $G$ of index
$n>1$.
We let $\bar G$ denote the permutation group induced by $G$ on the coset
space
$G/H$.
Let $X$ be a $G$-curve of type $(g,G,\C)$, where we label the $C_i$ such
that
$c_1\le \ldots \le c_r$. Let $\c=(c_1,\ldots,c_r)$. The $H$-curve obtained
by restriction of action is of type $(g,H,\D)$, where $\D$ is defined above;
let $(g,H,\dd)$ be its signature.
Let $s$ (resp., $r$) be the length of $\D$ (resp., $\C$).
Let $h_0$ (resp., $g_0$) be the genus of $X/H$ (resp., $X/G$).
If $\d:=\d(g,G,\C)=\d(g,H,\D)$ then $g_0=0$, and one of the following holds:
\begin{enumerate}
\item[(I)]\ \ $\d=3$: \ Then $n=2$, $h_0=2$, $s=0$, $r=6$, all $c_i=2$ and
$C_i\not
\subset H$.

\item[(II)]\ \ $\d=2$:\ \
Then $n=2$, $h_0=1$, $s=2$, $r=5$,
$C_i\not\subset H$ and $c_i=2$ for $i=1,...,4$ and\ $\dd=(c_5,c_5)$.
%\subitem \hskip .3in
\item[(III)]\ \ $\d=1$:

\subitem (a) \ $n=2$, $h_0=1$, $s=1$, $r=4$, \ $C_i\not\subset H$
for all $i$, \ $c_1=c_2=c_3=2$ and \ $\dd=(c_4/2)$.
%$\D=(C_4^2)$.

\subitem (b) \ $n=2$, $h_0=0$, $s=4=r$, $c_1=c_2=2$, $C_3,C_4\subset H$,
$\dd=(c_3,c_3,c_4,c_4)$.

\subitem (c) \ $h_0=0$, $s=4=r$, $c_1=c_2=c_3=2$, $C_4\subset H$, and
$H$ is normal in $G$ with $G/H$ a Klein 4-group;
\subitem \hskip .3in and $\dd=(c_4,c_4,c_4,c_4)$.


\medskip
\item[(IV)]\ \ \ $\d=0$: \ Then $h_0=0$, $s=3=r$ and


\subitem (a) \ $n=2$, $h_0=0$, $r=3=s$, $c_1=2$, $c_i>2$ is even and
$C_j\subset
H$
where $\{i,j\}=\{2,3\}$;
\subitem \hskip .3in and $\dd=(c_i/2,c_j,c_j)$.

\subitem (b)\ $n=3$, $\bar G\cong \bZ/3$, one $C_i\subset H$ and the other
two
have $c_j=3$; here $\dd=(c_i,c_i,c_i)$.
\subitem (c) \ $n=3$, $\bar G\cong S_3$, $c_1=2$, $c_2=3$, $c_3>2$ is even,
$\dd=(2,c_3/2,c_3)$.
\subitem (d) \ $n=4$, $\bar G\cong A_4$, $c_1=2$, $c_2=3$, $c_3>3$ is
divisible
by 3,
$\dd=(3,c_3/3,c_3)$.
\subitem (e) \ $n=6$, $\bar G\cong \mbox{PGL}_2(5)$, $\c=(2,4,5)$,
$\dd=(4,4,5)$,
$g\equiv 1$ mod 3.
\subitem (f) \ $n=8$, $\bar G\cong \mbox{PSL}_2(7)$, $\c=(2,3,7)$,
$\dd=(3,3,7)$,
$g\equiv 1$ mod 2.
\subitem (g) \ $n=9$, $\bar G\cong \mbox{PSL}_2(8)$, $\c=(2,3,7)$,
$\dd=(2,7,7)$,
$g\equiv 1$ mod 6.
\subitem (h) \ $n=10$, $\bar G\cong \mbox{PGL}_2(9)$, $\c=(2,3,8)$,
$\dd=(3,8,8)$,
$g\equiv 1$ mod 15.
%\subitem (i) \ $n=10$, $\bar G\cong S_{10}$, $\c=(2,3,7)$, $\d=(2,4,5)$,
$g\equiv 1$
% mod 43200.
\subitem (i) \ There are 6 further cases when $H$ is not maximal in $G$, and
$n=\ 4,6,6,12,12,24$, respectively.
\end{enumerate}
\end{Lemma}

\noindent
The cases in (IV) are taken from \cite{M}, Table 2 (c.f. 5b below).
%\begin{Remark} \label{Remark1}
%(see 5b below and \cite{M}, table 2).
%\end{Remark}


\smallskip\noindent
\Proof We denote the hypothesis $\d(g,G,\C)=\d(g,H,\D)$ by (Hyp).
Let $r_1$ be the number of points of $X/G$ that are ramified in
$X/H$.
Applying Riemann-Hurwitz to the cover $X/H\to X/G$
of degree $n$ yields
$$ 2\ (h_0-1) \ \ \ge \ \ 2n\ (g_0-1) \ +\ r_1 \leqno{(3)}$$
Let $s$ be the number of points of $X/H$ that are ramified in $X$,
and $r_2:=r-r_1$.
The $r_2$ points of $X/G$ that are ramified in $X$, but not in $X/H$,
have a total of $nr_2$ distinct pre-images in $X/H$. The latter points
are ramified in $X$, hence $s\ge nr_2$.

\smallskip\noindent
{\bf Case 1:} \ $g_0>1$ \newline
Then (Hyp) gives
$$ 3g_0\ +\ r_1 \ +\ r_2 \ \ = \ \ 3h_0\ +\ s \ \ \ge \ \ 3h_0\ +\ r_2$$
hence $$ 3g_0\ +\ r_1 \ \ \ge \ \ 3h_0$$
With (3) this gives
$$3g_0\ +\ 2\ (h_0-1) \ -\ 2n\ (g_0-1)\ \ \ge \ \ 3h_0$$
$$ g_0\ (3-2n)\ +\ 2n \ \ \ge \ \ h_0 \ +\ 2 \ \ \ge \ \ g_0 \ +\ 2 $$
%$$ 2g_0\ (1-n)\ +\ 2n \ \ \ge \ \ 2$$
$$g_0\ \ <\ \ \frac{n}{n-1}\ \ \le \ \ 2$$
a contradiction.

\smallskip\noindent
{\bf Case 2:} \ $g_0=1$ \newline
If $r=0$ then by (1) we have $g=1$, contrary to assumption. Thus $r>0$.
Now (3) gives $ 2\ (h_0-1) \ge r_1$.
If $h_0>1$ then (Hyp) gives $r_1\ge\ 3(h_0-1)$ (as in Case 1), a
contradiction.
If $h_0=1$ then $r_1=0$, hence (Hyp) gives $r_2=s\ge nr_2$.
Contradiction, since $r_2=r>0$.

\medskip
We have now proved $g_0=0$. Thus if $h_0>0$ then $r_1\ge3$.
Further, we can
choose $\g_i\in C_i$ with $\g_1\cdots\g_r=1$ and $G=<\g_1,...,\g_r>$ (by
(2)).
Then $c_i$ is the order of $\g_i$.
Let $p_1,...,p_r$ (resp., $q_1,...,q_{s}$) be the points of $X/G$
(resp., $X/H$) that are ramified in $X$. We choose the labelling such that
$p_i$ corresponds to $C_i$ and
$$c_1\ \ \le \ \ldots \ \le \ \ c_r\leqno{(4)}$$
%Let $d_j$ be the ramification index of $q_j$ over $X/G$. If $q_j$ lies over
$p_i$
%then $d_j$ divides $c_i$, and $d_j<c_i$.
The orbits of $\g_i$ on $G/H$ correspond to the points of $X/H$ over $p_i$
(where the length of the orbit equals the ramification index of the
corresponding
point over $p_i$).
Let $\mbox{Ind}(C_i)$ (=$\mbox{Ind}(\g_i)$ ) be
$n$ minus the number of those orbits. Those orbits that correspond to
points $\not\in\{q_1,...,q_{s}\}$
have length $c_i\ge2$. Thus the total number
of orbits of all $\g_i$ is $\le s+(nr-s)/2$. Hence Riemann-Hurwitz for
$X/H\to
X/G$
yields
$$2\ (h_0-1+n) \ \ \ =\ \ \ \sum_{i=1}^{r}\ \mbox{Ind}(C_i) \ \ \ \ge\ \ \
nr\ - \ \frac{nr+s}{2} \ \ \ =\ \ \ \frac{nr-s}{2}\leqno{(5)}$$



\medskip\noindent
{\bf Case 3:} \ $h_0\ge2$ \newline
Here (Hyp) gives $r=3h_0+s$. Plugging this into (5) gives
$$2\ (h_0-1+n) \ \ \ \ge\ \ \ \frac{n\ (3h_0+s)\ - \ s}{2}\ \ \ =\ \ \
\frac{3}{2}h_0 n\ +\ \frac{s(n-1)}{2}\ \ \ \ge\ \ \ \frac{3}{2}h_0
n\leqno{(6)}$$
This yields $n\le 2(h_0-1)/(\frac{3}{2}h_0-2)\le2$ since $h_0\ge2$.
Plugging $n=2$ into (6) gives $h_0\le2$. Thus $h_0=2$ and the inequalities
in
(6) are equalities, which implies $s=0$. This is
case (I) in the Lemma.


\medskip\noindent
{\bf Case 4:} \ $h_0=1$ \newline
If $s=0$ then $g=1$, contrary to assumption. Thus $s>0$, and so (Hyp) gives
$r-3=s$. Thus $r\ge4$. Plugging $s=r-3$ into (5) gives
$$2n \ \ \ \ge\ \ \ \frac{r(n-1)\ +\ 3}{2}\leqno{(7)}$$
which implies $r\le\ 4+\frac{1}{n-1}$.
If $n=2$ then we are in case (II) resp. (III).a of the Lemma. Assume now
$n>2$.
Then
$r=4$, $s=1$.
If all $c_i=2$ then (7) would have to hold with equality, which is not
possible.
Thus $c_4\ge3$ (by (4)).

Let $p_{i_0}$ be the image in $X/G$ of $q_1$. If $i_0<4$ then
Riemann-Hurwitz yields $2n\ \ge \ \ \frac{3n-1}{2}+\frac{2}{3}n$,
hence $n=3$ and $c_1=c_2=c_3=2$, a contradiction (since $n$ is divisible by
those
$c_i$ with $i\ne i_0$). Thus $i_0=4$.
If $n=3$ then $c_1=c_2=c_3=3$ contradicting Riemann-Hurwitz. Thus $n>3$.
Riemann-Hurwitz yields $2n\ \ge \ 3\ \frac{n}{2}+ \frac{2}{3}(n-1)$, hence
$n=4$,
$c_1=c_2=c_3=2$ and $c_4=3$.
Thus $\g_1$, $\g_2$,$\g_3$ act as
double transpositions on the 4-set $G/H$, and $\g_4$ acts as 3-cycle.
However, three double transpositions in $S_4$ cannot have a 3-cycle as their
product (since the double transpositions together with the identity form a
normal subgroup). Contradiction.



\medskip\noindent
{\bf Case 5:} \ $h_0=0$ \newline
If $r\le2$ or $s\le2$ then $g=0$, contradiction.
Thus we have $r\ge3$ and $s\ge3$. Hence (Hyp) gives
$r=s$. Plugging this into (5) gives $r\le4$.

\medskip\noindent
{\bf Case 5.a:} \ $r=4=s$ \newline
Here (5) holds with equality, hence the $q_j$'s are unramified over $X/G$,
and all other points of $X/H$ over some $p_i$ have ramification index
$2=c_i$.
If all $c_i=2$ then (1) gives $g=1$,
contradiction. Thus some $p_i$, say $p_{i_0}$, is unramified in $X/H$.
If two $p_i$'s are unramified in $X/H$ then $s\ge2n$,
hence $n\le s/2=2$; this is case III.b in the Lemma.

Now suppose
$p_i$ is unramified in $X/H$ only for $i=i_0$. Then $c_i=2$ for $i\ne i_0$,
and $n$ is even (since not each $p_i$, $i\ne i_0$, can have a $q_j$ over
it);
also, $n\le s=4$ and $n\ne2$ (because $X/H\to X/G$ has 3 branch points).
Hence $n=4$ and the $\g_i$ with $i\ne i_0$ act as double
transpositions on the 4-set $G/H$ (and $\g_{i_0}$ acts trivially).
This is case III.c.
%Since $G=<\g_1,\g_2,\g_3, \g_4>$ it follows that $G$ acts imprimitively on
%$G/H$, contradicting the hypothesis that $H$ is maximal in $G$

\medskip\noindent
{\bf Case 5.b:} \ $r=3=s$. \newline
This is the last case to be considered.
Now $X/H$ and $X/G$ have genus zero, and the cover
$X/H\to X/G$ (resp., $X\to X/H$) is ramified at 3 points of $X/G$
(resp., of $X/H$). This gives the following condition on the action of
$\g_1,\g_2,\g_3$ on the $n$-set $G/H$: Apart from exactly three orbits of
$\g_1,\g_2,
\g_3$,
all orbit lengths of $\g_i$ equal $c_i$; and if one of the three exceptional
orbits belongs to $\g_i$ then its length is $<c_i$. From (1) we also get
$\frac{1}{c_1}+\frac{1}{c_2}+\frac{1}{c_3}<1$. Finally, we have the genus
zero
condition
$2(n-1)=\sum_{i=1}^3\ \mbox{Ind}(\g_i)$. Triples of permutations with these
properties (and with product 1, and generating a transitive group)
have been classified by Malle \cite{M}: His Table 2 gives the cycle types of
the three permutations plus the group they generate (plus the $c_i$).
This is part (IV) of the Lemma. (Actually, Malle's
situation is slightly more general than ours, since he doesn't require $c_i$
to
be the
order of $\g_i$; thus two of his cases, where $n=10$ and $n=18$, don't occur
here). \qed

%Since we assume
%$H$ is maximal in $G$, only those triples from Malle's list show up which
generate
%a primitive group.

\section{The full automorphism group} \label{fullsection}

Now we change perspective and fix the finite group $H$.
An $H$-curve $X$ is called {\bf generic} if the following holds:
The curve $X/H$, together with its points $q_1,...,q_s$
that are ramified in $X$,
defines a generic point of $\M_{h,s}$. Here $h$ is the genus of $X/H$,
and \lq generic\rq \ means generic over $\overline\bQ$
(the algebraic closure of $\bQ$).

Let $\Aut(X)$ denote the automorphism group of the curve $X$ (without
regard of $H$-action).

\begin{Theorem} \label{thm1}
Suppose $X$ is a generic $H$-curve of genus $\ge2$. Then $[\Aut(X): H]\le
24$.
Let $h$ be the genus of $X/H$, and $s$ the number
of points of $X/H$ ramified in $X$.
Then $\Aut(X)=H$ unless $(h,s)$ equals $(2,0)$, $(1,2)$, $(1,1)$, $(0,4)$ or
$(0,3)$.
%Then there is an integer $n\le24$ such that
%$[\Aut(Y): H]\le n$ for all curves $Y$ corresponding to a dense open subset
%of the locus $\M(g,H,\D)$. This is true with $n=1$ unless one of the
following
holds:
If $h>0$ we have $[\Aut(X): H]\le 2$, and in the case
$(0,4)$ we have $H$ normal in $\Aut(X)$ and $[\Aut(X): H]\le 4$.
\end{Theorem}

\noindent \Proof Let $G=\Aut(X)$.
Let $X$ be of type $(g,H,\D)$ when viewed as $H$-curve,
and of type $(g,G,\C)$ when viewed as $G$-curve.

Note that all maps
and spaces in section \ref{section2} are defined over $\overline\bQ$.
Hence the condition that $X$ is a generic $H$-curve means that
its corresponding point in $\M_g$ is a generic point
of a component of the locus $\M(g,H,\D)$. This point lies in the sublocus
$\M(g,G,\C)$, hence the dimensions are the same: \ $\d(g,G,\C)=\d(g,H,\D)$.
Now the claim follows from Lemma \ref{Lemma2}. \qed

%By Lemma \ref{Lemma2}, $X$ is not the restriction of a $G$-curve with the
%same $\d$-invariant. As remarked before Lemma \ref{Lemma2}, it follows that
%whenever an $H$-curve of type $(g,H,\D)$ is the restriction of a $G$-curve
%of some type $(g,G,\C)$ with $|G|>|H|$ then
%the complement of $ \M(g,G,\C)$ in $\M(g,H,\D)$ is open and dense.
%Since there is only finitely many ramification types of genus $g$ with
%$ \M(g,G,\C)$ non-empty ( see \cite{Br}), the claim follows.

%Let $\A(g)$ be the set of isomorphism classes of the groups $\Aut(Y)$,
%$Y$ a genus $g$ curve. We say $\Hg$ to mean the isomorphism class of $H$
%is in $\A(g)$.

\begin{Lemma} \label{Lemma3}
\ (i) Let $\G$ be a group with a presentation on generators $\s_1,...\s_6$
and relations $$\s_1^2\ =...=\ \s_6^2\ =\ \ 1\ \ =\ \s_1\cdots\s_6$$
Let $\Pi$ be the
kernel of the map $\G\to\{\pm1\}$, $\s_i\mapsto-1$. Then $\Pi$ is generated
by
$$\begin{array}{cc}
\a_1\ =\ \s_2\s_1\ \ \ \ \ \ & \a_2\ =\ \s_5\s_4 \\
\b_1\ =\ \s_2\s_3\ \ \ \ \ \ & \b_2\ =\ \s_5\s_6
\end{array}
$$
which gives a presentation for $\Pi$ with the single relation
$$ [\a_1,\b_1] \ [\a_2,\b_2] \ \ = \ \ 1$$
We have $\G=\ \Pi\xs\langle \s_2\rangle$ where $\s_2$ acts on $\Pi$ via
$$\begin{array}{cc}
\a_1 \mapsto \ \a_1^{-1}\ \ \ \ \ \ &
\a_2\mapsto \ \ (\a_2^{-1})^{\a_2\b_2\a_1^{-1}\b_1^{-1}} \\
\b_1\mapsto \ \b_1^{-1}\ \ \ \ \ \ &
\b_2\mapsto \ \ (\b_2^{-1})^{\a_2\b_2\a_1^{-1}\b_1^{-1}}
\end{array} \leqno{(9)}$$
(ii)\ Let $\G=\ \langle \s_1,...\s_5\rangle$ subject to the relations
$$\s_1^2\ =...=\ \s_4^2\ =\ \ 1\ \ =\ \s_1\cdots\s_5$$ Let $\Pi$ be the
kernel of the map $\G\to\{\pm1\}$, $\s_i\mapsto-1$ for $i=1,...,4$
and $\s_5\mapsto1$. Then $\Pi$ is generated by
$$\begin{array}{cc}
\a\ =\ \s_2\s_1,\ \ \ \ \ & \g_1\ =\ \s_5^{\s_4} \\
\b\ =\ \s_2\s_3,\ \ \ \ \ & \g_2\ =\ \s_5
\end{array}
$$
which gives a presentation for $\Pi$ with the single relation
$$ [\a,\b] \ \g_1\g_2 \ \ = \ \ 1$$
Now $\s_2$ acts on $\Pi$ via
$$\begin{array}{cc}
\a \mapsto \ \a^{-1}\ \ \ \ \ \ &
\g_1\mapsto \ \ \g_2^{\a^{-1}\b^{-1}} \\
\b\mapsto \ \b^{-1}\ \ \ \ \ \ &
\g_2\mapsto \ \ \g_1^{\b^{-1}\a^{-1}}
\end{array} \leqno{(10)}$$

(iii)\ Let $\G=\ \langle \s_1,...\s_4\rangle$ subject to the relations
$$\s_1^2\ =\ \s_2^2\ =\ \s_3^2\ =\ \ 1\ \ =\ \s_1\cdots\s_4$$
%$$\s_1^2\ =\ \s_2^2\ =\ \ 1\ \ =\ \s_1\cdots\s_4$$
Let $\Pi$ be the
kernel of the map $\G\to\{\pm1\}$, $\s_i\mapsto-1$ (resp.,
%(resp.,$\s_1,\s_2\mapsto-1$ and $\s_3,\s_4\mapsto1$).
%$\G\to\{\pm1\}^2$, $\s_1\mapsto(1,-1)$, $\s_2\mapsto(-1,1)$,
$\s_3\mapsto(-1,-1)$,
%$\s_4\mapsto1$).
the normal subgroup of $\G$ generated by the conjugates of $\s_4$).
Then $\Pi$ is generated by
$$ \begin{array}{ccccc}
\begin{array}{cc}
\a\ =\ \s_2\s_1\ \ \ \ \ & \g\ =\ \s_4^2 \\
\b\ =\ \s_2\s_3\ \ \ \ \ &
\end{array} & \ \ \ \ & \mbox{resp.,} &\ \ \ \ &
%\begin{array}{cc}
%\g_1\ =\ \s_4\ \ \ \ \ & \g_3\ =\ \s_4^{\s_1} \\
%\g_2\ =\ \s_3^{\s_1},\ \ \ \ \ & \g_4\ =\ \s_3
%\end{array}
\begin{array}{cc}
\g_1\ =\ \s_4\ \ \ \ \ & \g_3\ =\ \s_4^{\s_2} \\
\g_2\ =\ \s_4^{\s_1},\ \ \ \ \ & \g_4\ =\ \s_4^{\s_3}
\end{array}
\end{array}
$$
which gives a presentation for $\Pi$ with the single relation
$$ [\a,\b] \ \g \ = \ 1 \ \ \ \ \ \mbox{resp.,} \ \ \ \ \
\g_1\cdots \g_4\ = \ 1$$
Now $\s_2$ acts on $\Pi$ by inverting $\a$ and $\b$ (resp.,
$\s_2$ switches $\g_1,\g_3$ and switches $\g_2,\g_4$, and
$\s_1$ switches $\g_1,\g_2$ and maps $\g_3\mapsto\g_4^{\g_1}$,
$\g_4\mapsto\g_3^{\g_2^{-1}}$, and
$\s_3$ switches $\g_1,\g_4$ and maps $\g_2\mapsto\g_3^{\g_4}$,
$\g_3\mapsto\g_2^{\g_1^{-1}}$).
\end{Lemma}

\noindent \Proof (i) All straightforward computation. One computes that the
relation
among the $\a_j$, $\b_j$ implies the relations among the $\s_i$, hence
the former yields a presentation for $\Pi$ since $\G=\ \Pi\xs\langle
\s_2\rangle$.
The proof of (ii) and (iii) is similar. \qed

\medskip

We say that
$\a_1,\b_1,...,\a_{h},\b_{h},\g_1,...,\g_s$ form a
{\bf genus $g$ generating system } of $H$ if these elements generate $H$
and satisfy the basic relation (2), where $g$ is given by
Riemann-Hurwitz $$\frac{2\ (g-1)}{|H|}\ \ \ = \ \ \ 2\ (h-1)\ +\
\sum_{i=1}^s\ (1-\frac{1}{\mbox{ord} (\g_i)}\ ) \leqno{(8)}$$ Call
a genus $g$ curve $X$ {\bf exceptional} if $X/\Aut(X)$ has genus
$0$ and exactly three of its points are ramified in $X$.

\begin{Theorem} \label{thm2} Fix $g\ge2$. A finite group $H$ is isomorphic
to
the full
automorphism group of a non-exceptional genus $g$ curve
if and only if it has a genus $g$ generating system
$\a_1,\b_1,...,\a_{h},\b_{h},\g_1,...,\g_s$ with $(h,s)\ne (0,3)$
satisfying:
\begin{enumerate}
\item[(a)]\ \ If $(h,s)=(2,0)$ (resp., $(1,2)$ )
then there is no automorphism
of $H$ acting on the generators via (9) (resp., (10) ).
\item[(b)]\ \ If $(h,s)=(1,1)$ (resp., $(0,4)$ )
then there is no automorphism
of $H$ inverting $\a_1$ and $\b_1$ (resp., acting on the generators
like $\s_1$, $\s_2$ or $\s_3$ in (iii) of the Lemma).
\end{enumerate}
\end{Theorem}


For the proof we need the following set-up: Let $Y$ be a curve of genus $h$,
and let $q_1,...,q_s$ be distinct points of $Y$ such that the pair $(h,s)$
is as in (a) or (b). Then there is a Galois cover $Y\to\P$ such that
the group $A=\Aut(Y/\P)$ permutes $q_1,...,q_s$; further,
$|A|=2$ unless $(h,s)=(0,4)$
in which case $A$ is a Klein 4 group whose non-trivial elements act
as double transpositions on $\{q_1,...,q_4\}$. If $(Y,q_1,...,q_s)$
corresponds to a generic point of $\M_{h,s}$ then $A$ is the full
stabilizer of $\{q_1,...,q_s\}$ in $\Aut(Y)$.

Let $\{p_1,...,p_r\}$
be the subset of $\P$ consisting of the points ramified in $Y$ plus
the images of $q_1,...,q_s$. Let $t$ be the identity function on $\P$,
so that $\bC(t)$ is the function field of $\P$. View $\bC(t)$ as a
subfield of $\bC(Y)$ naturally. Let $\tilde\M$ be a maximal
(algebraic) extension of $\bC(t)$ unramified outside
the places $t=p_1,...,p_r$,
and containing $\bC(Y)$.
Its Galois group $\tilde\G$ is a profinite group generated by
elements $\tilde\s_1,...,\tilde\s_r$ subject to the single relation
$\tilde\s_1\cdots\tilde\s_r=1$; here $\tilde\s_i$ can be chosen to
be a distinguished inertia group generator over the place
$t=p_i$. Let $\Omega$ be
the smallest (closed) normal subgroup of $\tilde\G$
containing those $\tilde\s_i^2$
for which there is no $q_j$ over $p_i$. Then $\G:=\tilde\G/\Omega$
is the Galois group of $\M:=\tilde\M^\Omega$ over $\bC(t)$, and
$\bC(t)\subset \bC(Y)\subset \M$. Let $\Pi=G(\M/\bC(Y))$, and
let $\s_1,...,\s_r$ be the images of the $\tilde\s_i$ in $\G$.
Then $\G$ (resp., $\Pi$) is the profinite
completion of the group denoted by the same symbol in the Lemma;
the cases $(h,s)=\ (2,0)$, $(1,2)$, $(1,1)$ and $(0,4)$
correspond to cases (i), (ii) and the two cases in (iii), respectively.

\medskip\noindent
\Proof (of the Theorem)
\ First we prove that the conditon is necessary. So assume $H=\Aut(X)$
for a non-exceptional
genus $g$ curve $X$. Let $h$ be the genus of $Y:=X/H$, and
$q_1,...,q_s$ the points of $Y$ ramified in $X$. It is well-known that
then $H$ has a genus $g$ generating system with parameters $h,s$.
So we may assume that $(h,s)$ is as in (a) or (b). Then we
are in the set-up discussed before this proof, with
$\bC(t)\subset \bC(Y)\subset \bC(X)\subset \M$.
The generators of $\Pi$ from the Lemma yield generators of $H$
via the natural surjection $\Pi=G(\M/\bC(Y))\to G(\bC(X)/\bC(Y))=H$.
It remains to show that these generators satisfy the condition
in (a) and (b).

Let $f$ be the automorphism of $\Pi$ induced by conjugation action
of $\s_2$ unless $(h,s)=(0,4)$ when $f$ could be induced by any of
$\s_1,\s_2,\s_3$. If $H$ has an automorphism $\bar f$ making the
following diagram commutative
$$\matrix{\Pi & \mathop{\longrightarrow}\limits^{f} & \Pi\cr
\Big\downarrow & & \Big\downarrow\cr
H & \mathop{\longrightarrow}\limits^{\bar f} & H\cr}$$
then the kernel $\Gamma$ of $\Pi\to H$ is $f$-invariant.
Then the normalizer $\Delta$ of $\Gamma$ in $\G$ is strictly bigger than
$\Pi$; thus $\Delta/\Gamma$ is strictly bigger than $\Pi/\Gamma\cong H$.
But $\Delta/\Gamma$ acts faithfully on $X$ (since $\Gamma=G(\M/\bC(X))$.
This contradicts the assumption
$H=\Aut(X)$. Thus there is no such commutative diagram, which implies
(a) and (b).

Assume now $H$ has a genus $g$ generating system as in the Theorem.
It is well-known that then there exists a generic $H$-curve $X$ of genus $g$
with parameters $h$, $s$.
If $(h,s)$ is not as in (a) or (b) then $H=\Aut(X)$
by Theorem \ref{thm1} and we are done.
Now assume $(h,s)$ is as in (a) or (b).
Choose data $(Y,q_1,...,q_s)$ as in the discussion before this proof,
corresponding to a generic point of $\M_{h,s}$. Mapping the generators
of $\Pi$ from the Lemma to the given generators of $H$ defines a
surjection $\Pi\to H$. Let $K$ be the fixed field (in $\M$)
of the kernel of this
surjection. Then $K=\bC(X)$ for a generic $H$-curve $X$ of genus $g$.
Thus $H$ is normal in $G:=\Aut(X)$ by Theorem \ref{thm1}.
Hence $G/H$ embeds into $A$, and so $\bC(t)\subset K^G\subset \bC(Y)$.
Thus the map $\Pi\to H$ extends to a map
from a group between $\Pi$ and $\G$ onto $G$.
Thus if $G\ne H$ then there is an element of $G$ that acts on the
given generators of $H$ in the same way that $\s_2$ (resp.,
$\s_1$, $\s_2$ or $\s_3$ in case $(h,s)=(0,4)$ )
acts on the generators of $\Pi$ (see Lemma).
But this is excluded by the condition in (a) resp. (b).
Hence $G=H$ and we are done.\qed








\begin{Remark} The proof shows more: Given $g$, $H$ and a signature $\e$
(resp., a ramification type $(g,H,\E)$ ), the existence of a generating
system
of $H$ as in the Theorem with $\mbox{ord}(\g_i)=e_i$
(resp., $\g_i\in E_i$) is equivalent to the existence of a non-exceptional
genus $g$ curve $X$ with $\Aut(X)\cong H$ such that the resulting $H$-curve
is of given signature (resp., type).
\end{Remark}

\part{Classification of automorphism groups}


Recall that a group $G$ acts faithfully on a genus $g$ curve if
and only if it has a genus $g$ generating system (see (8) above).
For $g$ up to 48, all such groups and the signatures of all their
genus $g$ generating systems have been listed by Breuer \cite{Br}.
More precisely, for each $g\le48$, he produced a list containing
all {\bf signature-group pairs} in genus $g$, i.e., pairs
consisting of a group $G$ together with the signature of a genus
$g$ generating system of $G$.

If $G$ acts on $X_g$ then so does each subgroup of $G$. This shows
that Breuer's lists have to be long, and contain some redundancy.
Part II of our paper was written in order to eliminate those
signature-group pairs that do not yield the full automorphism
group of a curve. This can also be done by the results of
\cite{Si} and \cite{Ri} (as we learned later). It turns out that
the larger $g$ gets, the larger is the ratio of entries in
Breuer's lists that do occur as full automorphism group in genus
$g$. That can already be seen from the fact (see Lemma
\ref{Lemma2}) that if a signature-group pair does not yield the
full automorphism group of a curve, then its $\delta$-invariant
(dimension of corresponding locus in $\M_g$) is at most 3.


For small $g$, a relatively large portion of those groups do not
occur as full automorphism group in genus $g$. Among those that do
occur, we distinguish those that occur for a particularly simple
class of curves: Call a group homocyclic if it is a direct product
of isomorphic cyclic groups. A cyclic (resp., homocyclic) cover
$X\to\P$ is defined to be a Galois cover with cyclic (resp.,
homocyclic) Galois group $C$. We call it a {\bf normal cyclic
(resp., homocyclic) cover of} $\P$ if $C$ is normal in
$G=\Aut(X)$. Then $\bar G:=G/C$ embeds as a finite subgroup of
$PSL(2,\bC)$ and it is easy to write down an equation for $X$
(e.g., in the cyclic case, $X$ has an equation of the form
$y^c=f(x)$, where $c=|C|$). To illustrate this, we work out the
equations in the genus 3 case. The hyperelliptic case (cyclic with
$|C|=2$) is the simplest case and can be done as in the genus 2
case (see \cite{ShV}); see also \cite{Bra} for the cyclic case (in
arbitrary characteristic).


\section{Genus 3}

For $g=3$, there are 49 signature-group pairs. Applying Theorem
\ref{thm2} and Lemma \ref{Lemma2}
we obtain that 23 of them correspond to $G$-curves
having $G$ as full automorphism group. For each of these 23
signature-group pairs, we find that the corresponding curves form
an irreducible locus in $\M_3$, and we compute the equation of a
general curve in that locus. There is only two groups (namely
$C_2$ and $V_4$) that occur with more than one signature, and as
soon as $|G|>2$, then the orbit genus $g_0$ (genus of $X_3/G$) is always
$0$.

Each row in Table 1,2 or 3 corresponds to an irreducible family of
curves $X_3$ with automorphism group $G$ (given in the first
column). We let $K$ denote the function field of $X_3$ over the
algebraically closed field $k$ of characteristic $0$ (and identify
$G$ with $\Aut(K/k)$). We display an equation for $K$. For almost
all values (Zariski-open set) of the parameters in these
equations, the equation describes a genus 3 curve with $G$ as full
automorphism group. For the exceptional values, the automorphism
group may be larger, or the curve may degenerate. (We do not
specify these exceptional values). Most of these equations agree
with those found in \cite{KuKo1}, \cite{KuKo2}, but they missed
two cases: The hyperelliptic case with $\bar G=D_{12}$, and the
3-dimensional hyperelliptic case with $\bar G=V_4$. Further, they
display an equation for the hyperelliptic case with $\bar G=S_4$
which actually belongs to the case $\bar G=D_{12}$. Some of the
loci they describe with more parameters than the dimension
requires. The cases with $g_0=1$ and group of order 3 or 4 should
be omitted in their list, because the full automorphism group of
these curves is larger (as shown by our Theorem \ref{thm2} above).
They note this for two of these cases in the Errata at the end of
their paper, but again missed one case.

Now we show how to obtain these equations quickly from the
group-theoretic data. Starting point is Breuer's data in genus 3
(see above). We used Part II to eliminate those signature-group pairs
that do not
yield the full automorphism group of a genus 3 curve. Now we discuss
the remaining signature-group pairs.

\subsection{Homocyclic non-hyperelliptic case}

We show how to derive the information in Table 2.

\smallskip\noindent
$G=C_4^2\xs S_3$: \ The fixed field of $C_4^2$ is of
genus $0$, call it $k(u)$. The extension $K/k(u)$ is ramified at 3
places of $k(u)$, say $u=0,1,\infty$, and the corresponding
inertia groups are cyclic of order 4. The fixed fields of these 3
inertia groups extend $k(u)$ with 2 ramified places each, hence
they are of genus $0$; call them $k(x)$, $k(y)$ and $k(z)$. We may
assume $x^4=u-1$, $y^4=u$. Thus $K=k(x,y)$ with $y^4=u=x^4+1$. The
equation $y^4=x(x^2-1)$ given by \cite{KuKo1} is also correct.
(Indeed, the four roots of $x^4+1$ have $j$-invariant 1728, same
as $\infty$ plus the 3 roots of $x(x^2-1)$).

\medskip\noindent
$G=(48,33)$: Here $G$ is a central extension of $C_4$ by quotient
$A_4$. Let $H\cong C_3$ be a Sylow 3-subgroup of $G$. The fixed
field of $H$ is a $k(x)$, and $K/k(x)$ is ramified at 5 places of
$k(x)$. Let $h_1,...,h_5$ be the corresponding distinguished
inertia group generators in $H$. Their product is 1, hence we may
assume $h_1=...=h_4=h_5^{-1}$. Take the place with generator $h_5$
to be $x=\infty$, and the other 4 ramified places to be the roots
of $f(x)$. Then $K=k(x,y)$ with $y^3=f(x)$. Further, the $C_4$
acts on $k(x)$ permuting the 4 roots of $f(x)$. We can choose $x$
such that $C_4$ acts on $k(x)$ by the maps $x\mapsto \zeta x$,
$\zeta^4=1$. Then $f(x)=x^4-b$. We can get each $b\ne0$ by further
normalization of $x$. This agrees with \cite{KuKo1}.

\medskip\noindent
$G=(16,13)$: Here $G$ is a central extension of $H\cong C_4$ by
quotient $V_4$. The fixed field of $H$ is a $k(x)$, and $K/k(x)$
is ramified at 4 places $p_1,...,p_4$ of $k(x)$. Let $h_1,...,h_4$
be the corresponding inertia elements in $H$. The Klein 4 group
$G/H$ permutes $p_1,...,p_4$. If this action were not transitive,
then $p_1,...,p_4$ could not be 4 general points in $\P$,
contradicting the fact that the covers under consideration form a
1-dimensional family. Thus $G/H$ permutes $p_1,...,p_4$
transitively, hence permutes $h_1,...,h_4$ transitively (by
conjugation). Thus $h_1=...=h_4$ since $H$ is central in $G$.
(Alternatively, this can be shown by the algorithm before Lemma
\ref{Lemma2}). Hence $K=k(x,y)$ with $f$ of degree 4. The generic
curve of this type has automorphism group of order 16, because
there is always a Klein 4 group in Aut$(k(x))$ permuting the 4
roots of $f$.

\medskip\noindent
$G=C_9,C_6$ or $C_3$: Let $H$ be the subgroup of $G$ of order 3.
The fixed field of $H$ is a $k(x)$, and $K/k(x)$
is ramified at 5 places $p_1,...,p_5$ of $k(x)$. Let $h_1,...,h_5$
be the corresponding inertia elements. They have product 1, hence
we may assume $h_1=...=h_4=h_5^{-1}$. The group $G/H$ permutes
$p_1,...,p_5$, fixing $p_5$. Take $p_1,...,p_5$ to be the places
$x=0,1, s,t,\infty$, respectively. Then $K=k(x,y)$ with
$y^3=x(x-1)(x-s)(x-t)$. Let $\alpha$ be the automorphism of $k(x)$
induced by a generator of $G/H$. If it has order 3 then we may
assume that $x=0$ is the other fixed point of $\alpha$ (besides
$x=\infty$); then $\alpha(x)=\zeta_3x$ and the equation becomes
$y^3=x(x^3-1)$. Now assume $\alpha$ has order 2. We may assume it
switches $0$ and $1$. Then $\alpha(t)=1-t$, again as claimed.

\medskip\noindent
$G=S_4, D_8$ or $V_4$: Let $H$ a normal Klein 4-group in $G$.
The fixed field of $H$ is a $k(w)$, and $K/k(w)$
is ramified at 6 places $p_1,...,p_6$ of $k(w)$. Let $h_1,...,h_6$
be the corresponding inertia elements. If they don't comprise all
involutions in $H$ then there is $h\in H$ that occurs 4 times
among the $h_i$; then $K^h$ extends $k(w)$ with 2 branch points,
hence $K^h$ has genus $0$ and so $K$ is hyperelliptic. We excluded
that case. Thus we may assume $h_1=h_2$, $h_3=h_4$, $h_5=h_6$ and
$\{h_1,h_3,h_5\}= H\setminus\{1\}$.

Let $u,v$ be independent transcendentals over $k$ with $w=u/v$.
The places of $k(w)$ correspond to homogeneous degree 1
polynomials in $u,v$, modulo scalar multiples. Let $P,Q,R$ be
homogeneous degree 2 polynomials in $k[u,v]$ such that the degree
1 factors of $P$ (resp., $Q$, resp., $R$) correspond to $p_1,p_2$
(resp., $p_3,p_4$, resp., $p_5, p_6$). The quotients $P/Q$, $R/Q$,
$P/R$ are naturally elements of $k(w)$. Then $K^{h_3}=k(w,x)$ with
$x^2=P/Q$, and $K^{h_1}= k(w,y)$ with $y^2=R/Q$, and
$K^{h_2}=k(w,z)$ with $z^2=P/R$. Thus $K=k(w,x,y)$. We claim $K
=k(x,y)$. Assume wrong. Then $[K:k(x,y)]=2$ since $w$ satisfies
the equation $P(w,1)-x^2Q(w,1)=0$ over $k(x,y)$. Let $g_0$ be the
generator of $G(K/k(x,y))$. Then $g_0$ is not in $H$, since none
of the $h_i$ fixes both $x$ and $y$. But each $h_i$ maps $x$
(resp., $y$) to $\pm x$ (resp., $\pm y$), hence $H$ leaves
$k(x,y)$ invariant and thus centralizes $g_0$. Hence $<H,g_0>\cong
C_2^3$, contradiction (since $G$ contains no such subgroup). This
proves $K=k(x,y)$.
%(The latter follows by using the two
%equations $P-x^2Q=0$ and $R-y^2Q=0$ for $w$ over $k(x,y)$).
%Similarly, $K=k(x,z)=k(y,z)$.

The 6 homogeneous polynomials $P^2, Q^2, R^2, PQ, PR, QR$ of
degree 4 are linearly dependent in $k[u,v]$. This gives a
non-trivial relation $$aPR\ +\ bPQ\ +\ cQR\ +\ d P^2\ +\ eQ^2\ +\
fR^2\ = \ 0$$ If $f=0$ then dividing by $Q^2$ gives an equation
for $y$ over $k(x)$ of $y$-degree $\le2$ and so $K$ is
hyperelliptic or of genus $0$ -- contradiction. Thus $f$ (and
similarly $d,e$) are non-zero. Replacing $P$, $Q$, $R$ by suitable
scalar multiples we get a relation $$ P^2\ +\ Q^2\ +\ R^2\ +\ aPR\
+\ b PQ\ +\ c QR \ = \ 0$$ with $a,b,c\in k$. Dividing by $Q^2$
gives $$x^4\ +\ 1\ +\ y^4\ +\ ax^2y^2\ +\ bx^2\ +\ cy^2 \ = \ 0$$
This settles the case $G=V_4$.

Suppose now $G=D_8$. Then $G/H$ induces an involutory automorphism
$\alpha$ of $k(w)$ switching (without loss) the sets $\{p_1,p_2\}$
and $\{p_5,p_6\}$.
%$k(w)$ switching (without loss) the sets $\{p_1,p_2\}$ and
%$\{p_5,p_6\}$.
%Then $K/K^G$ is ramified at 5 places of
%$K^G$, and $G/H$ induces an involutory automorphism $\alpha$ of
%$G/H$ induces an involutory automorphism $\alpha$ of
%$k(w)$ switching (without loss) the sets $\{p_1,p_2\}$ and
%$\{p_5,p_6\}$. If $\alpha$ fixes both of $p_3$, $p_4$ then $K/K^G$
%would be ramified at only 4 places of $K^G$, hence $\alpha$
%switches $p_3, p_4$.
We may assume $\alpha(w)=1/w$. Extend it to $k(u,v)$ by
$\alpha(u)=v$, $\alpha(v)=u$. Then $\alpha$ fixes $Q$ and switches
$P,R$ up to scalar multiples. Since $\alpha^2=\mbox{id}$, we have
$\alpha(Q)=\pm Q$. Applying $\alpha$ to the relation between
$P,Q,R$ yields that $\alpha(P)=\pm R$. Replacing $R$ by $-R$ if
necessary we may assume $\alpha(P)=R$ (resp., $\alpha(P)=-R$) if
$\alpha(Q)=Q$ (resp., $\alpha(Q)=-Q$). Then $b=c$.
% since $PR$, $PQ$ and $QR$ are linearly independent.
Finally, suppose $G=S_4$.
Then there is another automorphism like $\alpha$, fixing $R$ and
switching $P,Q$ up to scalar multiples. Replacing $Q$ by $-Q$ if
necessary yields $a=b=c$.

\subsection{Hyperelliptic case}
Here we derive the information in Table 3.

In the hyperelliptic case, $G$ has a central subgroup $C$ of
order 2 whose fixed field in $K$ is a $k(x)$. There are 8 places
of $k(x)$ that ramify in $K$, say $x=a_1,...,a_8$. Then $\bar
G=G/C$ embeds into $PGL(2,k)$ as a subgroup permuting
$a_1,...,a_8\in\P$. Let $\sigma_m$ (resp., $\tau$) be the element
of $PGL(2,k)$ mapping $x$ to $\zeta_m x$ (resp., $1/x$), where
$\zeta_m$ is a primitive $m$-th root of 1 in $k$. Then each
subgroup of $PGL(2,k)$ isomorphic to $C_m$ resp., $D_{2m}$ is
conjugate to $<\sigma_m>$ resp., $<\sigma_m, \tau>$. We identify
$C_m$ (resp., $D_m$) with $<\sigma_m>$ (resp., $<\sigma_m,\tau>$).
In particular, $D_2=V_4=<\sigma_2,\tau>$.

\smallskip\noindent
$\bar G=S_4$: Let $i\in k$ with $i^2=-1$. We may assume $\bar G$
contains $D_2$ and $\rho:x\mapsto(x+i)/(x-i)$. ($\rho$ is of order
3, permuting the 6 fixed points of the involutions in $V_4$). The
two fixed points of $\rho$ are the roots of $X^2-(i+1)X-i=0$. The
group $<V_4,\rho>\cong A_4$ has exactly one orbit of length 8 on
$\P$, containing those fixed points of $\rho$. Hence the orbit
consists of the roots of $(X^2-(i+1)X-i)
(X^2+(i+1)X-i)(-iX^2-(i+1)X+1)(-iX^2+(i+1)X+1) \ = \ X^8+14X^4+1$.

\medskip\noindent
$\bar G=D_{16}, D_{12},C_7$: In the dihedral case, $\bar G$
leaves exactly one set of length $8$ in $\P$ invariant, consisting
of the roots of $X^8-1$, resp., the roots of $X^6-1$ plus $0$ and
$\infty$. In the third case, we may assume $a_1=1$ by a coordinate
change; then $a_1,...,a_8$ comprise the roots of $X^7-1$ plus
either $0$ or $\infty$. (The latter two choices are conjugate
under the map $\tau$). The case $\bar G=D_{12}$ is missing in
\cite{KuKo1} and the corresponding equation $y^2=x(x^6-1)$ is
erroneously associated with the case $\bar G=S_4$.

\medskip\noindent
$\bar G=D_8,S_3$: Here $a_1,...,a_8$ comprise a regular orbit of
$D_8$ resp., $0$ and $\infty$ plus a regular orbit. Thus in the
first case, $a_1,...,a_8$ are the roots of $(X^4-t)(X^4-1/t)=
X^8-(t+1/t)X^4+1$. The other case is similar.

\medskip\noindent
$\bar G=V_4$: Here $a_1,...,a_8$ either comprise two regular
orbits of $\bar G$, or a regular orbit plus two of three orbits of
length 2. The orbits of length 2 are all conjugate under $S_4$, so
we may take $\{0,\infty\}$ and $\{\pm1\}$ as the two orbits of
length 2. Now the claim follows as in the previous case.

\medskip\noindent
$\bar G=C_2$: Here $a_1,...,a_8$ either comprise 4 regular orbits,
or $0$ and $\infty$ plus 3 regular orbits. We may assume $a_1=1$
by a coordinate change.


\subsection{Remaining cases, collected in Table 1}

Here we take $X_3$ in its canonical embedding in the projective
plane associated with the dual of the 3-dimensional (faithful)
$G$-module $\Omega$ of holomorphic differential forms on $X_3$.
Let $x,y,z$ be coordinates of this module $\Omega^*$. Let
$f(x,y,z)=0$ be the equation of $X_3$ in these coordinates. Then
$f$ spans a 1-dimensional fixed space for the representation of
$G$ on the space of quartic homogeneous polynomials in $x,y,z$.

To identify the representation of $G$ on $\Omega$, we use that the
space of fixed points of $G$ on $\Omega$ has dimension equal to
the genus $g_0$ of $X_3/G$ (see \cite{Br}); this fixed point
space can be identified with $\Omega(X_3/G)$).


\medskip\noindent
$G=L_3(2)$: This is the well-known Hurwitz group of order 168. The
$(2,3,7)$ triple is weakly-rigid in $G$, so there is exactly one
associated curve found already by Klein.

\medskip\noindent
$G=S_3, g_0=0$: The group $S_3$ has only one faithful
3-dimensional representation with trivial fixed space: It is the
natural permutation representation tensored with the 1-dimensional
sign-representation. Thus $G$ acts in the projective plane
associated with $\Omega^*$ just by permuting the homogeneous
coordinates $x,y,z$ (for suitable choice of such coordinates).
Thus the quartic polynomial $f(x,y,z)$ defining $X_3$ is either
symmetric in $x,y,z$, or has the property that it is invariant
under a cyclic permutation of $x,y,z$, but changes sign under a
transposition. The only possibility (up to scalar multiples) for
the latter case is $f= x^3y+y^3z+z^3x-y^3x-z^3y-x^3z$, but this
polynomial is not irreducible (product of linear factors). Hence
$f$ is symmetric, thus is of the form
$f=a(x^4+y^4+z^4)+b(x^2y^2+x^2z^2+y^2z^2)+c(x^2yz+y^2xz+z^2xy)+
d(x^3y+x^3z+y^3x+y^3z+z^3x+z^3y)$. The centralizer of $S_3$ in
$GL_3(\bC)$ consists of the matrices $tI+sJ$, where $I$ (resp.,
$J$) is the identity matrix (resp., the all $1$'s matrix). By a
coordinate change with such a matrix we can make $d=0$, but not
uniquely: Generically, there is always 4 triples $(a,b,c)$ (up to
scalar multiples) that give the same curve (up to isomorphism).

\medskip\noindent
$G=C_2, g_0=1$:\ We may assume that the involution in $G$ acts as
$x\mapsto-x, y\mapsto y, z\mapsto z$. Thus $f$ either contains
only even powers of $x$ (i.e., $f$ is fixed by $G$) or $f$
contains only odd powers of $x$ (i.e., $f$ is mapped to $-f$ by
the involution in $G$). In the latter case, $f$ is reducible
(product of $x$ and a genus 1 equation). Thus $f$ contains only
even powers of $x$, hence $$f\ \ =\ \ a_0x^4\ +\ x^2\
(a_1y^2+a_2yz+a_3z^2)\ + \
a_4y^4+a_5y^3z+a_6y^2z^2+a_7yz^3+a_8z^4$$ By a linear change of
the variables $y,z$ we can make $a_1=1$ and $a_2=0$. (If
$a_1=a_2=a_3=0$ then $X_3$ has an automorphism of order 4, namely
multiplying $x$ by $\zeta_4$, so we don't consider this case
here). Further we can replace $x$ and $z$ by scalar multiples to
get the normalizations from Table 1. (If either $a_0=0$ or
$a_7=a_8=0$ then the curve is hyperelliptic).



\begin{table}[h] \label{genus 3.nothomo}

\caption{$\Aut(X_3)$ for $X_3$ not a normal homocyclic cover
of $\P$} \vskip 0.5cm


\begin{center}
\begin{tabular}{||c|c|c|c|c||}

\hline \hline &&&&\\ $G$ & signature & orbit& proj. equation &
Group
\\ &&genus &&ID\\
\hline \hline &&&&\\ $L_3(2)$ & $(2,3,7)$ & 0
&$x^3y+y^3z+z^3x=0$ &(168,42) \\



\hline &&&&\\
$S_3$ & $(2,2,2,2,3)$ &0 &
$a(x^4+y^4+z^4)+b(x^2y^2+x^2z^2+y^2z^2)+$&(6,1)\\

&&&$c(x^2yz+y^2xz+z^2xy)=0$&\\

\hline &&&&\\ $C_2$ & $(2,2,2,2)$ &1&$ x^4+x^2(y^2+az^2) +
by^4+cy^3z+dy^2z^2+eyz^3+gz^4=0$&(2,1)\\ &&& either $e=1$ or $g=1$
&\\ \hline \hline
\end{tabular}
\end{center}
\end{table}


\begin{table}[h] \label{genus 3.homo}

\caption{$\Aut(X_3)$ for normal homocyclic covers of $\P$, not
hyperelliptic}
\vskip 0.5cm

\begin{center}
\begin{tabular}{||c|c|c|c|c|c||}
\hline \hline &&&&&\\ $G$ & $C$ &$ G/C$ & signature & equation &
Group
\\ &&&&&ID\\ \hline

%\hline $L_3(2)$ & $(2,3,7)$ & 0 &1&1& large&0&(168,42) \\

\hline &&&&&\\
$V_4$ & $V_4$ & $\{1\}$ & $(2^6)$ &$x^4+y^4+ax^2y^2+bx^2+cy^2+1=0$&(4,2) \\

$D_8$ & $V_4$ & $C_2$ & $(2^5)$ & take\ $b=c$& (8,3)\\

$S_4$ &$V_4$& $S_3$ & $(2,2,2,3)$ & take\ $a=b=c$ &(24,12) \\

$C_4^2\xs S_3$ &$V_4$ & $S_4$ & $(2,3,8)$ &
%$y^4=x^4+1$ \
take\ $a=b=c=0$ \ or\ $y^4=x(x^2-1)$& (96,64)\\

\hline

$16$ & $C_4$ & $V_4$& $(2,2,2,4)$ &$y^4=x(x-1)(x-t)$&(16,13)
\\

$48$ &$C_4$& $A_4$& $(2,3,12)$ &$y^4=x^3-1$
&(48,33)\\

\hline
%\hline $S_3$ & $(2^4,3)$ &2 &1& ?& &0&(6,1)\\

$C_3$ &$C_3$& $\{1\}$ &$(3^5)$ &$y^3=x(x-1)(x-s)(x-t)$&(3,1)\\

$C_6$ & $C_3$ & $C_2$&$(2,3,3,6)$ & take\ $s=1-t$& (6,2)\\

$C_9$ &$C_3$ &$C_3$ & $(3,9,9)$
&$y^3=x(x^3-1)$&(9,1)\\

\hline \hline


\end{tabular}
\end{center}
\end{table}


\begin{table}[h] \label{genus 3.hyperelliptic}

\caption{$\Aut(X_3)$ for hyperelliptic $X_3$}
\vskip 0.5cm

\begin{center}
\begin{tabular}{||c|c|c|c|c|c||}
\hline \hline &&&&&\\ $G$ & $ G/C$ & signature & dim. of&equation
& Group
\\ &&&locus&$y^2=$&ID

\\ \hline \hline &&&&&\\
$48$ & $S_4$ &$(2,4,6)$ &0 & $x^8+14x^2+1$ &$(48,48)$
\\

$32$ &$D_{16}$ &$(2,4,8)$ &0& $x^8-1$& $(32,9)$\\

$24$ &$D_{12}$ & $(2,4,12)$ &0&$x(x^6-1)$ & $(24,5)$\\

$C_{14}$ &$C_7$& $(2,7,14)$ &0&$x^7-1$ &$(14,2)$ \\

$16$ &$D_8$ &$(2,2,2,4)$&1&$x^8+ax^4+1$ &$(16,11)$ \\

$D_{12}$ &$S_3$& $(2,2,2,6)$ &1&$x(x^6+ax^3+1)$ &$(12,4)$\\

$C_2\times C_4$ &$V_4$ &$(2,2,4,4)$ & 1&$x(x^2-1)(x^4+ax^2+1)$&
$(8,2)$
\\

$C_2^3$ &$V_4$& $(2^5)$ &2&$(x^4+ax^2+1)(x^4+bx^2+1)$ &$(8,5)$\\

$C_4$ &$C_2$& $(2,2,2,4,4)$ &2&$x(x^2-1)(x^4+ax^2+b)$& $(4,1)$\\

$V_4$ &$C_2$& $(2^6)$ & 3&$(x^2-1)(x^6+ax^4+bx^2+c)$ &$(4,2)$\\

$C_2$ &$\{1\}$& $(2^8)$ & 5&$x(x-1)(x^5+ax^4+bx^3+cx^2+dx+e)$&
$(2,1)$
\\ \hline\hline


\end{tabular}
\end{center}
\end{table}

\section{Curves of genus $\le10$ with large automorphism group}

In the genus 3 case, we were able to write out explicit equations for
the curves with any given automorphism group. This yields an explicit
description of the corresponding loci in $\M_3$. For higher genus,
we cannot expect to obtain explicit equations. Still,
computational group theory allows us to determine the dimensions and
number of components of these loci. This uses the
general set-up of section 3.

The number
of signature-group pairs grows quickly with the genus.
E.g., in genus 10
there are already 174 signature-group pairs with $g_0 = 0$,
and most of them yield the
full automorphism group of a genus 10 curve. So it would not be feasible
to display all automorphism groups up to genus 10.
Therefore, we only display
the {\bf large} groups Aut$(X_g)$ (in the sense of section 1.3),
see Table 4. Surprisingly, their number remains relatively small.
They comprise the most interesting groups in each genus, and we avoid
listing the many group-signature pairs with group of order 2, 3 etc.

\subsection{The general set-up}

We return to the set-up of section 3. Let $\c=(c_1,...,c_r)$ be the
signature of a genus $g$ generating system of $G$. Let $\H(g,G,\c)$ be
Hurwitz space parameterizing equivalence classes of $G$-curves of genus $g$
and signature $\c$; i.e., $\H(g,G,\c)$ is the (disjoint) union of the
spaces $\H(g,G,\C)$ with $\C$ of signature $\c$ (see section 3). The map
$$\Phi:\ \ \H(g,G,\c) \ \ \to \ \ \M_{g}$$
forgetting the $G$-action is a finite morphism.
Let $\M(g,G,\c)$ be its
image (the locus of genus $g$ curves admitting a $G$-action
of signature $\c$).
All components of $\M(g,G,\c)$ have dimension $\delta=\delta(g,G,\c)$
(by Lemma 3.1). In particular, if $G$ is large then $g_0=0$ and $r=3,4$,
hence $\delta$ is $0$ or $1$.

Define $\H^*(g,G,\c)$ (resp., $\H^*(g,G,\C)$ with $\C$ a tuple
of conjugacy classes, see section 3) to be the union of all components
$C$ of $\H(g,G,\c)$ (resp., $\H(g,G,\C)$ ) with the following property:
There is at least one point on $C$ such that the associated $G$-curve
has $G$ as full automorphism group. Let
$\M^*(g,G,\c)$ (resp., $\M^*(g,G,\C)$ ) be the $\Phi$-image of
the corresponding space $\H^*(...)$.
Then $\Phi$ is generically injective on $\M^*(g,G,\c)$, and so the
spaces $\M^*(g,G,\c)$ and $\H^*(g,G,\c)$ have the same number of components.
In the case $g_0=0$, the spaces $\H(g,G,\c)$ coincide with the Hurwitz
spaces studied in \cite{FV}, \cite{V}, \cite{Buch}. Thus
the components of $\H(g,G,\c)$ correspond to the braid
orbits of (genus $0$) generating systems of $G$ of signature $\c$, taken
modulo $\Aut(G)$. These braid
orbits can be computed with the BRAID package \cite{MSV}.


\subsection{The table of large automorphism groups up to genus $10$}

In Table 4 we list all group-signature pairs $(G,\c)$ of genus
$g$, where $4 \le g \le 10$, with the
following property: There exists a $G$-curve $X_g$ of genus
$g$ and signature $\c$ such that $G$ is the full automorphism group
of $X_g$ and $G$ is large; i.e, $|G|> 4(g-1)$. More precisely,
the rows of Table 4 correspond to the components of the loci
$\M^*(g,G,\c)$ associated with these parameters. It turns out that
these loci are mostly irreducible, with only 6 exceptions listed
in Table 5. In these exceptional cases, they have two components,
and these components are all of the form $M^*(g,G,\C)$, with $\C$ a tuple
of conjugacy classes (see 7.1). Thus the spaces $M^*(g,G,\C)$ are
always irreducible in the situation of Table 4, and they correspond
bijectively to the rows of Table 4. In particular, duplicate rows
occur iff the corresponding locus $\M^*(g,G,\c)$ is reducible. The
group $G$ is identified via its ID from the Small
Group Library. In the last column of Table 4, we also indicate the inclusion
relations between components of dimension
0 and 1. They can be computed by the algorithm given before Lemma
\ref{Lemma2}.
%\newpage




\begin{table}[] \label{genus 4-10}

      \caption{Components of the Hurwitz loci $\M^*(g,G,\C)$ for large $G$}

      \vskip 0.5cm
      \begin{center}
      \renewcommand{\arraystretch}{1.24}
      \begin{tabular}{||c|c|c|c||c|c|c|c||}
      \hline
      \hline
      $\#$ & Group ID & signature & contains & $\#$ & Group ID & signature &
contains \\
      \hline
      \hline
      \multicolumn{8}{c}{Genus 4, $\delta = 0$} \\
      \hline
      \hline
      {\bf 1} & (120,34) & $(2, 4, 5)$ & &%\\
      {\bf 2} & (72,42) & $(2, 3, 12)$ & \\
      {\bf 3}& (72,40) & $(2, 4, 6)$ & &%\\
      {\bf 4} & (40,8) & $(2, 4, 10)$ & \\
      {\bf 5} & (36,12) & $(2, 6, 6)$ & &%\\
      {\bf 6} & (32,19) & $(2, 4, 16)$ & \\
      {\bf 7} & (24,3) & $(3, 4, 6)$ & &%\\
      {\bf 8} & (18,2) & $(2, 9, 18)$ & \\
      {\bf 9} & (15,1) & $(3, 5, 15)$ &&&&&\\
      \hline
      \hline
      \multicolumn{8}{c}{Genus 4, $\delta = 1$} \\
      \hline
      \hline
      {\bf 10} & (36,10) & $(2, 2, 2, 3)$ & 3 &%\\
      {\bf 11} & (24,12) & $(2, 2, 2, 4)$ & 1, 2  \\
      {\bf 12} & (20,4) & $(2, 2, 2, 5)$ & 4 &%\\
      {\bf 13} & (18,3) & $(2, 2, 3, 3)$ & 2, 5  \\
      {\bf 14} & (16,7) & $(2, 2, 2, 8)$ & 6 &&&& \\
      \hline
      \hline
      \multicolumn{8}{c}{Genus 5, $\delta = 0$} \\
      \hline
      \hline
      {\bf 1} & (192,181) & $(2, 3, 8)$ & &%\\
      {\bf 2} & (160,234) & $(2, 4, 5)$ & \\
      {\bf 3} & (120,35) & $(2, 3, 10)$ & &%\\
      {\bf 4} & (96,195) & $(2, 4, 6)$ & \\
      {\bf 5} & (64,32) & $(2, 4, 8)$ & &%\\
      {\bf 6} & (48,14) & $(2, 4, 12)$ & \\
      {\bf 7} & (48,30) & $(3, 4, 4)$ & &%\\
      {\bf 8} & (40,5) & $(2, 4, 20)$ & \\
      {\bf 9} & (30,2) & $(2, 6, 15)$ & &%\\
      {\bf 10} & (22,2) & $(2, 11, 22)$ & \\
      \hline
      \hline
      \multicolumn{8}{c}{Genus 5, $\delta = 1$} \\
      \hline
      \hline
      {\bf 11} & (48,48) & $(2, 2, 2, 3)$ & 1, 4 &%\\
      {\bf 12} & (32,43) & $(2, 2, 2, 4)$ & \\
      {\bf 13} & (32,28) & $(2, 2, 2, 4)$ & 1 &%\\
      {\bf 14} & (32,27) & $(2, 2, 2, 4)$ & 2, 4, 5 \\
      {\bf 15} & (24,14) & $(2, 2, 2, 6)$ & 6 &%\\
      {\bf 16} & (24,8) & $(2, 2, 2, 6)$ & 4 \\
      {\bf 17} & (24,13) & $(2, 2, 3, 3)$ & 3, 7 &%\\
      {\bf 18} & (20,4) & $(2, 2, 2, 10)$ & 3, 8 \\

      \hline
      \hline
      \multicolumn{8}{c}{Genus 6, $\delta = 0$} \\
      \hline
      \hline
      {\bf 1} & (150,5) & $(2, 3, 10)$ & &%\\
      {\bf 2} & (120,34) & $(2, 4, 6)$ & \\
      {\bf 3} & (72,15) & $(2, 4, 9)$ & &%\\
      {\bf 4} & (56,7) & $(2, 4, 14)$ & \\
      {\bf 5} & (48,6) & $(2, 4, 24)$ & &%\\
      {\bf 6} & (48,29) & $(2, 6, 8)$ & \\
      {\bf 7} & (48,15) & $(2, 6, 8)$ & &%\\
      {\bf 8} & (39,1) & $(3, 3, 13)$ & \\
      {\bf 9} & (30,1) & $(2, 10, 15)$ & &%\\
      {\bf 10} & (26,2) & $(2, 13, 26)$ & \\
      {\bf 11} & (21,2) & $(3, 7, 21)$ &&&&&\\
      \hline
      \hline
      \multicolumn{8}{c}{Genus 6, $\delta = 1$} \\
      \hline
      \hline
      {\bf 12} & (60,5) & $(2, 2, 2, 3)$ & 2 &%\\
      {\bf 13} & (28,3) & $(2, 2, 2, 7)$ & 4 \\
      {\bf 14} & (24,12) & $(2, 2, 3, 4)$ & 2 &%\\
      {\bf 15} & (24,8) & $(2, 2, 3, 4)$ & 3 \\
      {\bf 16} & (24,6) & $(2, 2, 2, 12)$ & 5 &%\\
      {\bf 17} & (24,6) & $(2, 2, 3, 4)$ & 7 \\

      \hline
      \hline
      \end{tabular}
      \end{center}
      \end{table}


      \addtocounter{table}{-1}
      %
      \begin{table}

      \caption{(Cont.)}

      \vskip 0.5cm
      \begin{center}
      \renewcommand{\arraystretch}{1.24}
      \begin{tabular}{||c|c|c|c||c|c|c|c||}
      \hline
      \hline
      $\#$ & Group ID & signature & contains & $\#$ & Group ID & signature &
contains \\

\hline
      \hline

      \multicolumn{8}{c}{Genus 7, $\delta = 0$} \\
      \hline
      \hline
      {\bf 1} & (504,156) & $(2, 3, 7)$ & &%\\
      {\bf 2} & (144,127) & $(2, 3, 12)$ & \\
      {\bf 3} & (64,41) & $(2, 4, 16)$ & &%\\
      {\bf 4} & (64,38) & $(2, 4, 16)$ & \\
      {\bf 5} & (56,4) & $(2, 4, 28)$ & &%\\
      {\bf 6} & (54,6) & $(2, 6, 9)$ & \\
      {\bf 7} & (54,6) & $(2, 6, 9)$ & &%\\
      {\bf 8} & (54,3) & $(2, 6, 9)$ & \\
      {\bf 9} & (48,32) & $(3, 4, 6)$ & &%\\
      {\bf 10} & (42,4) & $(2, 6, 21)$ & \\
      {\bf 11} & (32,11) & $(4, 4, 8)$ & &%\\
      {\bf 12} & (32,10) & $(4, 4, 8)$ & \\
      {\bf 13} & (30,4) & $(2, 15, 30)$ &&&&& \\
      \hline
      \hline
      \multicolumn{8}{c}{Genus 7, $\delta = 1$} \\
      \hline
      \hline
      {\bf 14} & (48,48) & $(2, 2, 2, 4)$ & &%\\
      {\bf 15} & (48,41) & $(2, 2, 2, 4)$ & 2 \\
      {\bf 16} & (48,38) & $(2, 2, 2, 4)$ & &%\\
      {\bf 17} & (36,10) & $(2, 2, 2, 6)$ & \\
      {\bf 18} & (32,43) & $(2, 2, 2, 8)$ & &%\\
      {\bf 19} & (32,42) & $(2, 2, 2, 8)$ & 3 \\
      {\bf 20} & (32,39) & $(2, 2, 2, 8)$ & 4 &%\\
      {\bf 21} & (28,3) & $(2, 2, 2, 14)$ & 5 \\
      \hline
      \hline
      \multicolumn{8}{c}{Genus 8, $\delta = 0$} \\
      \hline
      \hline
      {\bf 1} & (336,208) & $(2, 3, 8)$ & &%\\
      {\bf 2} & (336,208) & $(2, 3, 8)$ & \\
      {\bf 3} & (84,7) & $(2, 6, 6)$ & &%\\
      {\bf 4} & (84,7) & $(2, 6, 6)$ & \\
      {\bf 5} & (72,8) & $(2, 4, 18)$ & &%\\
      {\bf 6} & (64,53) & $(2, 4, 32)$ & \\
      {\bf 7} & (60,8) & $(2, 6, 10)$ & &%\\
      {\bf 8} & (48,25) & $(2, 6, 24)$ & \\
      {\bf 9} & (48,17) & $(2, 8, 12)$ & &%\\
      {\bf 10} & (48,28) & $(3, 4, 8)$ & \\
      {\bf 11} & (40,10) & $(2, 10, 20)$ & &%\\
      {\bf 12} & (34,2) & $(2, 17, 34)$ & \\
      \hline
      \hline
      \multicolumn{8}{c}{Genus 8, $\delta = 1$} \\
      \hline
      \hline
      {\bf 13} & (42,1) & $(2, 2, 3, 3)$ & 1, 2, 3, 4 &%\\
      {\bf 14} & (36,4) & $(2, 2, 2, 9)$ & 5 \\
      {\bf 15} & (32,18) & $(2, 2, 2, 16)$ & 6 &%\\
      {\bf 16} & (30,3) & $(2, 2, 3, 5)$ & 7 \\

      \hline
      \hline
      \end{tabular}
      \end{center}
      \end{table}


      \addtocounter{table}{-1}
      %
      \begin{table}

      \caption{(Cont.)}

      \vskip 0.5cm
      \begin{center}
      \renewcommand{\arraystretch}{1.24}
      \begin{tabular}{||c|c|c|c||c|c|c|c||}
      \hline
      \hline
      $\#$ & Group ID & signature & contains & $\#$ & Group ID & signature &
contains \\
      \hline
      \hline
      \multicolumn{8}{c}{Genus 9, $\delta = 0$} \\
      \hline
      \hline
      {\bf 1} & (320,1582) & $(2, 4, 5)$ & &%\\
      {\bf 2} & (192,194) & $(2, 3, 12)$ & \\
      {\bf 3} & (192,990) & $(2, 4, 6)$ & &%\\
      {\bf 4} & (192,955) & $(2, 4, 6)$ & \\
      {\bf 5} & (128,138) & $(2, 4, 8)$ & &%\\
      {\bf 6} & (128,136) & $(2, 4, 8)$ & \\
      {\bf 7} & (128,134) & $(2, 4, 8)$ & &%\\
      {\bf 8} & (128,75) & $(2, 4, 8)$ & \\
      {\bf 9} & (120,35) & $(2, 5, 6)$ & &%\\
      {\bf 10} & (120,34) & $(2, 5, 6)$ & \\
      {\bf 11} & (96,187) & $(2, 4, 12)$ & &%\\
      {\bf 12} & (96,186) & $(2, 4, 12)$ & \\
      {\bf 13} & (96,13) & $(2, 4, 12)$ & &%\\
      {\bf 14} & (80,14) & $(2, 4, 20)$ & \\
      {\bf 15} & (72,5) & $(2, 4, 36)$ & &%\\
      {\bf 16} & (64,6) & $(2, 8, 8)$ & \\
      {\bf 17} & (57,1) & $(3, 3, 19)$ & &%\\
      {\bf 18} & (48,5) & $(2, 8, 24)$ & \\
      {\bf 19} & (48,4) & $(2, 8, 24)$ & &%\\
      {\bf 20} & (48,30) & $(4, 4, 6)$ & \\
      {\bf 21} & (42,3) & $(2, 14, 21)$ & &%\\
      {\bf 22} & (40,12) & $(4, 4, 10)$ & \\
      {\bf 23} & (38,2) & $(2, 19, 38)$ & &&&&\\
      \hline
      \hline
      \multicolumn{8}{c}{Genus 9, $\delta = 1$} \\
      \hline
      \hline
      {\bf 24} & (96,193) & $(2, 2, 2, 3)$ & 3 &%\\
      {\bf 25} & (96,227) & $(2, 2, 2, 3)$ & 4 \\
      {\bf 26} & (64,190) & $(2, 2, 2, 4)$ & &%\\
      {\bf 27} & (64,177) & $(2, 2, 2, 4)$ & 5 \\
      {\bf 28} & (64,140) & $(2, 2, 2, 4)$ & &%\\
      {\bf 29} & (64,138) & $(2, 2, 2, 4)$ & 4 \\
      {\bf 30} & (64,135) & $(2, 2, 2, 4)$ & 1, 3, 6 &%\\
      {\bf 31} & (64,134) & $(2, 2, 2, 4)$ & 7 \\
      {\bf 32} & (64,128) & $(2, 2, 2, 4)$ & &%\\
      {\bf 33} & (64,73) & $(2, 2, 2, 4)$ & 2, 8 \\
      {\bf 34} & (48,43) & $(2, 2, 2, 6)$ & 13 &%\\
      {\bf 35} & (48,38) & $(2, 2, 2, 6)$ & \\
      {\bf 36} & (48,15) & $(2, 2, 2, 6)$ & 3 &%\\
      {\bf 37} & (48,48) & $(2, 2, 2, 6)$ & 4, 12 \\
      {\bf 38} & (48,48) & $(2, 2, 2, 6)$ & 11 &%\\
      {\bf 39} & (40,13) & $(2, 2, 2, 10)$ & 14 \\
      {\bf 40} & (40,8) & $(2, 2, 2, 10)$ & &%\\
      {\bf 41} & (36,4) & $(2, 2, 2, 18)$ & 15 \\
      \hline
      \hline
      \end{tabular}
      \end{center}
      \end{table}


      \addtocounter{table}{-1}
      %
      \begin{table}

      \caption{(Cont.)}

      \vskip 0.5cm
      \begin{center}
      \renewcommand{\arraystretch}{1.24}
      \begin{tabular}{||c|c|c|c||c|c|c|c||}
      \hline
      \hline
      $\#$ & Group ID & signature & contains & $\#$ & Group ID & signature &
contains \\
      \hline
      \hline
      \multicolumn{8}{c}{Genus 10, $\delta = 0$} \\
      \hline
      \hline
      {\bf 1} & (432,734) & $(2, 3, 8)$ & &%\\
      {\bf 2} & (432,734) & $(2, 3, 8)$ & \\
      {\bf 3} & (360,118) & $(2, 4, 5)$ & &%\\
      {\bf 4} & (324,160) & $(2, 3, 9)$ & \\
      {\bf 5} & (216,92) & $(2, 3, 12)$ & &%\\
      {\bf 6} & (216,158) & $(2, 4, 6)$ & \\
      {\bf 7} & (216,87) & $(2, 4, 6)$ & &%\\
      {\bf 8} & (216,153) & $(3, 3, 4)$ & \\
      {\bf 9} & (180,19) & $(2, 3, 15)$ & &%\\
      {\bf 10} & (168,42) & $(2, 4, 7)$ & \\
      {\bf 11} & (162,14) & $(2, 3, 18)$ & &%\\
      {\bf 12} & (144,122) & $(2, 3, 24)$ & \\
      {\bf 13} & (108,25) & $(2, 6, 6)$ & &%\\
      {\bf 14} & (108,15) & $(2, 4, 12)$ & \\
      {\bf 15} & (88,7) & $(2, 4, 22)$ & &%\\
      {\bf 16} & (80,6) & $(2, 4, 40)$ & \\
      {\bf 17} & (72,28) & $(2, 6, 12)$ & &%\\
      {\bf 18} & (72,23) & $(2, 6, 12)$ & \\
      {\bf 19} & (72,42) & $(3, 4, 6)$ & &%\\
      {\bf 20} & (63,3) & $(3, 3, 21)$ & \\
      {\bf 21} & (60,10) & $(2, 6, 30)$ & &%\\
      {\bf 22} & (42,6) & $(2, 21, 42)$ & \\
      {\bf 23} & (42,2) & $(3, 6, 14)$ & &%\\
      {\bf 24} & (42,2) & $(3, 6, 14)$ & \\
      \hline
      \hline
      \multicolumn{8}{c}{Genus 10, $\delta = 1$} \\
      \hline
      \hline
      {\bf 25} & (108,40) & $(2, 2, 2, 3)$ & 4, 6 &%\\
      {\bf 26} & (108,17) & $(2, 2, 2, 3)$ & 1, 2, 7 \\
      {\bf 27} & (72,43) & $(2, 2, 2, 4)$ & 5 &%\\
      {\bf 28} & (72,40) & $(2, 2, 2, 4)$ & 1, 2, 6 \\
      {\bf 29} & (72,15) & $(2, 2, 2, 4)$ & &%\\
      {\bf 30} & (60,5) & $(2, 2, 2, 5)$ & 3, 9 \\
      {\bf 31} & (54,8) & $(2, 2, 2, 6)$ & 11, 14 &%\\
      {\bf 32} & (54,5) & $(2, 2, 3, 3)$ & 5, 8, 13 \\
      {\bf 33} & (48,29) & $(2, 2, 2, 8)$ & 1, 2, 12 &%\\
      {\bf 34} & (44,3) & $(2, 2, 2, 11)$ & 15 \\
      {\bf 35} & (40,6) & $(2, 2, 2, 20)$ & 16 &&&&\\
      \hline
      \hline
      \end{tabular}
      \end{center}
      \end{table}


  \begin{table}[] \label{multicomponents}

      \caption{Reducible Hurwitz loci $\M^*(g,G,\c)$}
      \vskip 0.5cm
      \begin{center}
      \renewcommand{\arraystretch}{1.24}
      \begin{tabular}{||c|c|c|c||}
      \hline
      \hline
       genus & Group ID & signature & components of locus \\
      \hline
      \hline

       7 & (54,6) & (2,6,9) & 6,7 \\
       \hline
       8 & (336,208) & (2,3,8) & 1,2 \\
       \hline
       8 & (84,7) & (2,6,6) & 3,4 \\

       \hline
       9 & (48,48) & (2,2,2,6) & 37,38 \\

       \hline
       10 & (432,734) & (2,3,8) & 1,2 \\
       \hline

       10 & (42,2) & (3,6,14) & 23,24 \\
       \hline
       \hline
      \end{tabular}
      \end{center}
      \end{table}

\newpage



\begin{thebibliography}{99}
%***********************************************************************
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      \bibitem[Ac2]{Ac2} {\sc R. Accola}, Two theorems on Riemann surfaces
      with noncyclic automorphism groups, {\it Proc. Amer. Math. Soc.}
      {\bf 25} (1970), 598--602.

      \bibitem[Ba]{Ba} {\sc W. Baily}, On the automorphism group of a
      generic curve of genus $>2$, {\it J. Math. Kyoto Univ.} {\bf 1}
      (1961/1962), 101--108; correction, 325.


      \bibitem[Be]{Be} {\sc Jos\'e Bertin}, Compactification des sch\'emas
      de Hurwitz, {\it C.R. Acad. Sci. Paris I}, vol. 322 (1996),
      1063--1066.

      \bibitem[BE]{BE} {\sc H. Besche and B. Eick}, The groups of order at
      most 1000 except 512 and 768, {\it J. Symbolic Comput.} {\bf 27}
(1999),
      no. 4, 405--413.

      \bibitem[BS]{BS} {\sc R. Brandt and H. Stichtenoth},
      Die Automorphismengruppen hyperelliptischer Kurven, {\it Man.
      Math} {\bf 55} (1986), 83--92.

      \bibitem[Bra]{Bra} {\sc R. Brandt}, \"Uber die Automorphismengruppen
      von algebraischen Funktionenk\"orpern, PhD thesis.
      Universit\"at-Gesamthochschule Essen, 1988.

      \bibitem[Br]{Br} {\sc Th. Breuer}, Characters and automorphism groups
      of compact Riemann surfaces, {\it London Math. Soc. Lect. Notes}
      {\bf 280}, Cambridge Univ. Press 2000.

      \bibitem[BGG]{Bu} {\sc E. Bujulance, J. Gamboa, and G. Gromadzki}, The
      full automorphism groups of hyperelliptic Riemann surfaces, {\em
      Manuscripta Math.} {\bf 79} (1993), no. 3-4, 267--282.

      \bibitem[Co]{Co} {\sc M. Conder}, Hurwitz groups: a brief survey.
      {\it Bull. Amer. Math. Soc. (N.S.)} {\bf 23} (1990), no. 2,
      359--370.

      \bibitem[Du]{Du} {\sc A. Duma}, Automorphisms of compact Riemann
      surfaces. {\it Confer. Sem. Mat. Univ. Bari}, No. 207 (1985).

      \bibitem[DT]{DT} {\sc A. Duma, C. Tanasi}, Groups of automorphisms of
      a family of compact Riemann surfaces of genus four. {\it Ricerche
      Mat.} {\bf 34} (1985), no. 2, 377--390.

      \bibitem[FK]{FK} {\sc M. Farkas and I. Kra}, {\sl Riemann Surfaces},
      Springer-Verlag, 1992.

      \bibitem[Fr]{Fr} {\sc R. Fricke}, {\it Lehrbuch der Algebra}, Vieweg,
      Braunschweig, 1926.

      \bibitem[FV]{FV} {\sc M.~Fried and H. V\"olklein}, The inverse Galois
      problem and rational points on moduli spaces, {\it Math. Annalen}
      {\bf 290} (1991), 771--800.

      \bibitem[GAP] {GAP} The GAP~Group,
      {\it GAP -- Groups, Algorithms, and Programming}, Version 4.2;
      2000.\\
      {\tt (http://www.gap-system.org)}

      \bibitem[Ha]{Ha} {\sc W. Harvey}, Cyclic groups of automorphisms of a
      compact Riemann surface, {\it Quart. J. Math. Oxford Ser. (2)} {\bf
      17} (1966), 86--97.

      \bibitem[Ho]{Ho} {\sc M. Homma}, Automorphisms of prime order of
      curves. {\it Manuscripta Math.} {\bf 33} (1980/81), no. 1, 99--109.

      \bibitem[Hu]{Hu} {\sc A. Hurwitz}, \"Uber algebraische Gebilde mit
      eindeutigen Transformationen in sich, {\it Math. Ann.} {\bf 41}
      (1893), 403--442.

      \bibitem[Ki]{Ki} {\sc W. Kiley}, Automorphism groups of compact
      Riemann surfaces. {\it Trans. Amer. Math. Soc.} {\bf 150} (1970),
      557--563.

      \bibitem[Kl]{Kl1} {\sc F. Klein}, \"Uber die Transformationen
      siebenter Ordnung der elliptischen Functionen, {\it Math. Annalen}
      {\bf 14}, (1879), 428--471.

      %\bibitem[Kl2] {Kl2} {\sc F. Klein},
      %On the Order-Seven Transformation of Elliptic Functions,
      %(translated by Silvio Levy), 287-331
      %Math. Sci. Res. Inst. Publ., 35, Cambridge Univ. Press, Cambridge,
      %1999.

      \bibitem[Ku]{Ku} {\sc R. S. Kulkarni}, Riemann surfaces admitting
      large automorphism groups. Extremal Riemann surfaces (San Francisco,
      CA, 1995), 63--79, {\em Contemp. Math.} {\bf 201}, Amer. Math.
      Soc., Providence, RI, 1997.

      \bibitem[KH]{KH} {\sc A. Kuribayashi and K. Hayakawa}, On the Hurwitz
      group. {\it Bull. Fac. Sci. Engrg. Chuo Univ. Ser. I Math.}
      {\bf 37} (1994), 1--23 (1995).

      \bibitem[KuKi]{KuKi1} {\sc A. Kuribayashi and H. Kimura}, Automorphism
      groups of compact Riemann surfaces of genus five, {\it J. Algebra}
      {\bf 134} (1990), 80--103.

      \bibitem[KuKo1]{KuKo1} {\sc A. Kuribayashi, K. Komiya}, On
      Weierstra\ss points and automorphisms of curves of genus three. In:
      Algebraic geometry (Proc. Summer Meeting, Univ. Copenhagen,
      Copenhagen, 1978), pp. 253--299, {\em Lecture Notes in Math.} {\bf
      732}, Springer, Berlin, 1979.

      \bibitem[KuKo2]{KuKo2} {\sc A. Kuribayashi, K. Komiya}, On the
      structure of the automorphism group of a compact Riemann surface of
      genus $3$. {\it Bull. Fac. Sci. Engrg. Chuo Univ.} {\bf 23}
      (1980), 1--34.

      \bibitem[KK1]{KK1} {\sc A. Kuribayashi and I. Kuribayashi}, On
      subgroups of automorphism groups of hyperelliptic Riemann surfaces of
      genus three. {\it Bull. Fac. Sci. Engrg. Chuo Univ.} {\bf 24}
      (1981), 21--37.

      \bibitem[KK2]{KK2} {\sc A. Kuribayashi and I. Kuribayashi}, On
      subgroups of automorphism groups of hyperelliptic Riemann surfaces of
      genus three. II. {\it Bull. Fac. Sci. Engrg. Chuo Univ.} {\bf 25}
      (1982), 5--43.

      \bibitem[KK3]{KK3} {\sc A. Kuribayashi and I. Kuribayashi},
      Automorphism groups of compact Riemann surfaces of genus three and
      four, {\it J. Pure Appl. Algebra} {\bf 65} (1990), 277--292.

      \bibitem[KKO]{KKO} {\sc A. Kuribayashi, A. Kobayashi, and S. Ohmori},
      Automorphism groups of compact Riemann surfaces. Travaux de la
      Conf\'erence Internationale d'Analyse Complexe et du 7e S\'eminaire
      Roumano-Finlandais (1993). {\it Rev. Roumaine Math. Pures Appl.}
      {\bf 40} (1995), no. 1, 65--75.

      \bibitem[LN]{LN} {\sc J. Lehner and M. Newman}, On Riemann surfaces
      with maximal automorphism groups, {\it Glasgow Math. J.} {\bf 8}
      (1967), 102--112.

      \bibitem[Le]{Le} {\sc S. Levy}, The eightfold way. The beauty of
      Klein's quartic curve. Ed. Silvio Levy. {\it Mathematical Sciences
      Research Institute Publications} {\bf 35}, Cambridge University Press,
      Cambridge, 1999.

      %\bibitem[Mb1] {Mb1}
      %{\sc A. M. Macbeath}, Hurwitz groups and surfaces. The eightfold
      %way, 103--113, Math. Sci. Res. Inst. Publ., 35, Cambridge Univ.
      %Press, Cambridge, 1999.

      \bibitem[Mb]{Mb2} {\sc A. M. Macbeath}, On a curve of genus $7$. {\it
      Proc. London Math. Soc. (3)} {\bf 15} (1965), 527--542.

      \bibitem[Mb2]{Mb3} {\sc A. M. Macbeath}, On a theorem of Hurwitz.
      {\it Proc. Glasgow Math. Assoc.} {\bf 5} (1961), 90--96.

      \bibitem[Mc1]{Mc1} {\sc C. Maclachlan}, Abelian groups of
      automorphisms of compact Riemann surfaces, {\it Proc. London Math.
      Soc. (3)} {\bf 15} (1965), 699--712.

      \bibitem[Mc2]{Mc2} {\sc C. Maclachlan}, A bound for the number of
      automorphisms of a compact Riemann surface, {\it J. London Math. Soc.
      (2)} {\bf 44} (1969), 265--272.

      \bibitem[MSV]{MSV} {\sc K. Magaard, S. Sphectorov and H. V\"olklein},
      Computing braid group action, in preparation.

      \bibitem[Ma1]{M} {\sc G.~Malle}, Genus zero translates of three point
      ramified Galois extensions, {\sl Manuscripta Math.} {\bf 71} (1991),
      97--111.

      \bibitem[Ma2]{Ma2} {\sc G.~Malle}, Hurwitz groups and $G\sb 2(q)$,
      {\it Canad. Math. Bull.} {\bf 33} (1990), no. 3, 349--357.

      \bibitem[Mt]{Mt} {\sc T. Matsuno}, Compact Riemann surfaces with large
      automorphism groups, {\it J. Math. Soc. Japan} {\bf 51} (1999), no.
      2, 309--329.

      \bibitem[Ri]{Ri} {\sc J. Ries}, Subvarieties of moduli space
      determined by finite groups acting on surfaces, {\it Trans. Amer.
      Math. Soc.} {\bf 335} (1993), no. 1, 385--406.

      \bibitem[Si1]{S2} {\sc D. Singerman}, Subgroups of Fuchsian groups and
      finite permutation groups, {\it Bull. London Math. Soc.} {\bf 2}
      (1970), 319--323.

      \bibitem[Si2]{Si} {\sc D. Singerman}, Finitely maximal Fuchsian
      groups, {\it J. London Math. Soc.} {\bf 6} (1972), 29--38.

      \bibitem[Si3]{S1} {\sc D. Singerman}, Symmetries of Riemann surfaces
      with large automorphism group, {\it Math. Ann.} {\bf 210} (1974),
      17--32.

      \bibitem[ShV]{ShV} {\sc T. ~Shaska and H. ~V\"olklein}, Elliptic
      Subfields and automorphisms of genus fields, to appear in: {\it
      Proceeding of the Conference on Algebra and Algebraic Geometry with
      Applications: The celebration of the seventieth birthday of Professor
      S.S. Abhyankar}, Springer-Verlag, 2001.

      \bibitem[Ta]{Ta} {\sc N.Takigawa}, Compact Riemann surfaces with an
      automorphism group of order $p$, {\it Math. Rep. Toyama Univ.} {\bf
      4} (1981), 115--127.

      \bibitem[V]{Buch} {\sc H.~V\"olklein}, {\it Groups as Galois Groups --
      an Introduction}, Cambr. Studies in Adv. Math. {\bf 53}, Cambridge
      Univ. Press 1996.

      \bibitem[V1]{V} {\sc H. V\"olklein}, Moduli spaces for covers of
      the Riemann sphere, {\it Israel J. Math.} {\bf 85} (1994), 407--430.

      \bibitem[W1]{W1} {\sc A. Wiman}, \"Uber die hyperelliptischen Curven
      und diejenigen vom Geschlechte $p=3$, welche eindeutige
      Transformationen in sich zulassen, {\it Bihang Kongl. Svenska
      Vetenskaps-Akademiens Handlingar} (1895), no. 21 (1), 1--23.

      \bibitem[W2]{W2} {\sc A. Wiman}, \"Uber die hyperelliptischen Curven
      vom den Geschlechte $p=4,5$, und 6, welche eindeutige Transformationen
      in sich besitzen, {\it Bihang Kongl. Svenska Vetenskaps-Akademiens
      Handlingar} (1895), no. 21 (3), 1--41.

      \bibitem[Wl]{Wl} {\sc R. Wilson}, The Monster is a Hurwitz group.
      {\it J. Group Theory} {\bf 4} (2001), no. 4, 367--374.

\end{thebibliography}
\end{document}




















math0205314.extracted_bib
\begin{thebibliography}{99}
%***********************************************************************
      \bibitem[Ac1]{Ac1} {\sc R. Accola}, On the number of automorphisms of
      a closed Riemann surface, {\it Trans. Amer. Math. Soc.} {\bf 131}
      (1968), 398--408.

      \bibitem[Ac2]{Ac2} {\sc R. Accola}, Two theorems on Riemann surfaces
      with noncyclic automorphism groups, {\it Proc. Amer. Math. Soc.}
      {\bf 25} (1970), 598--602.

      \bibitem[Ba]{Ba} {\sc W. Baily}, On the automorphism group of a
      generic curve of genus $>2$, {\it J. Math. Kyoto Univ.} {\bf 1}
      (1961/1962), 101--108; correction, 325.


      \bibitem[Be]{Be} {\sc Jos\'e Bertin}, Compactification des sch\'emas
      de Hurwitz, {\it C.R. Acad. Sci. Paris I}, vol. 322 (1996),
      1063--1066.

      \bibitem[BE]{BE} {\sc H. Besche and B. Eick}, The groups of order at
      most 1000 except 512 and 768, {\it J. Symbolic Comput.} {\bf 27}
(1999),
      no. 4, 405--413.

      \bibitem[BS]{BS} {\sc R. Brandt and H. Stichtenoth},
      Die Automorphismengruppen hyperelliptischer Kurven, {\it Man.
      Math} {\bf 55} (1986), 83--92.

      \bibitem[Bra]{Bra} {\sc R. Brandt}, \"Uber die Automorphismengruppen
      von algebraischen Funktionenk\"orpern, PhD thesis.
      Universit\"at-Gesamthochschule Essen, 1988.

      \bibitem[Br]{Br} {\sc Th. Breuer}, Characters and automorphism groups
      of compact Riemann surfaces, {\it London Math. Soc. Lect. Notes}
      {\bf 280}, Cambridge Univ. Press 2000.

      \bibitem[BGG]{Bu} {\sc E. Bujulance, J. Gamboa, and G. Gromadzki}, The
      full automorphism groups of hyperelliptic Riemann surfaces, {\em
      Manuscripta Math.} {\bf 79} (1993), no. 3-4, 267--282.

      \bibitem[Co]{Co} {\sc M. Conder}, Hurwitz groups: a brief survey.
      {\it Bull. Amer. Math. Soc. (N.S.)} {\bf 23} (1990), no. 2,
      359--370.

      \bibitem[Du]{Du} {\sc A. Duma}, Automorphisms of compact Riemann
      surfaces. {\it Confer. Sem. Mat. Univ. Bari}, No. 207 (1985).

      \bibitem[DT]{DT} {\sc A. Duma, C. Tanasi}, Groups of automorphisms of
      a family of compact Riemann surfaces of genus four. {\it Ricerche
      Mat.} {\bf 34} (1985), no. 2, 377--390.

      \bibitem[FK]{FK} {\sc M. Farkas and I. Kra}, {\sl Riemann Surfaces},
      Springer-Verlag, 1992.

      \bibitem[Fr]{Fr} {\sc R. Fricke}, {\it Lehrbuch der Algebra}, Vieweg,
      Braunschweig, 1926.

      \bibitem[FV]{FV} {\sc M.~Fried and H. V\"olklein}, The inverse Galois
      problem and rational points on moduli spaces, {\it Math. Annalen}
      {\bf 290} (1991), 771--800.

      \bibitem[GAP] {GAP} The GAP~Group,
      {\it GAP -- Groups, Algorithms, and Programming}, Version 4.2;
      2000.\\
      {\tt (http://www.gap-system.org)}

      \bibitem[Ha]{Ha} {\sc W. Harvey}, Cyclic groups of automorphisms of a
      compact Riemann surface, {\it Quart. J. Math. Oxford Ser. (2)} {\bf
      17} (1966), 86--97.

      \bibitem[Ho]{Ho} {\sc M. Homma}, Automorphisms of prime order of
      curves. {\it Manuscripta Math.} {\bf 33} (1980/81), no. 1, 99--109.

      \bibitem[Hu]{Hu} {\sc A. Hurwitz}, \"Uber algebraische Gebilde mit
      eindeutigen Transformationen in sich, {\it Math. Ann.} {\bf 41}
      (1893), 403--442.

      \bibitem[Ki]{Ki} {\sc W. Kiley}, Automorphism groups of compact
      Riemann surfaces. {\it Trans. Amer. Math. Soc.} {\bf 150} (1970),
      557--563.

      \bibitem[Kl]{Kl1} {\sc F. Klein}, \"Uber die Transformationen
      siebenter Ordnung der elliptischen Functionen, {\it Math. Annalen}
      {\bf 14}, (1879), 428--471.

      %\bibitem[Kl2] {Kl2} {\sc F. Klein},
      %On the Order-Seven Transformation of Elliptic Functions,
      %(translated by Silvio Levy), 287-331
      %Math. Sci. Res. Inst. Publ., 35, Cambridge Univ. Press, Cambridge,
      %1999.

      \bibitem[Ku]{Ku} {\sc R. S. Kulkarni}, Riemann surfaces admitting
      large automorphism groups. Extremal Riemann surfaces (San Francisco,
      CA, 1995), 63--79, {\em Contemp. Math.} {\bf 201}, Amer. Math.
      Soc., Providence, RI, 1997.

      \bibitem[KH]{KH} {\sc A. Kuribayashi and K. Hayakawa}, On the Hurwitz
      group. {\it Bull. Fac. Sci. Engrg. Chuo Univ. Ser. I Math.}
      {\bf 37} (1994), 1--23 (1995).

      \bibitem[KuKi]{KuKi1} {\sc A. Kuribayashi and H. Kimura}, Automorphism
      groups of compact Riemann surfaces of genus five, {\it J. Algebra}
      {\bf 134} (1990), 80--103.

      \bibitem[KuKo1]{KuKo1} {\sc A. Kuribayashi, K. Komiya}, On
      Weierstra\ss points and automorphisms of curves of genus three. In:
      Algebraic geometry (Proc. Summer Meeting, Univ. Copenhagen,
      Copenhagen, 1978), pp. 253--299, {\em Lecture Notes in Math.} {\bf
      732}, Springer, Berlin, 1979.

      \bibitem[KuKo2]{KuKo2} {\sc A. Kuribayashi, K. Komiya}, On the
      structure of the automorphism group of a compact Riemann surface of
      genus $3$. {\it Bull. Fac. Sci. Engrg. Chuo Univ.} {\bf 23}
      (1980), 1--34.

      \bibitem[KK1]{KK1} {\sc A. Kuribayashi and I. Kuribayashi}, On
      subgroups of automorphism groups of hyperelliptic Riemann surfaces of
      genus three. {\it Bull. Fac. Sci. Engrg. Chuo Univ.} {\bf 24}
      (1981), 21--37.

      \bibitem[KK2]{KK2} {\sc A. Kuribayashi and I. Kuribayashi}, On
      subgroups of automorphism groups of hyperelliptic Riemann surfaces of
      genus three. II. {\it Bull. Fac. Sci. Engrg. Chuo Univ.} {\bf 25}
      (1982), 5--43.

      \bibitem[KK3]{KK3} {\sc A. Kuribayashi and I. Kuribayashi},
      Automorphism groups of compact Riemann surfaces of genus three and
      four, {\it J. Pure Appl. Algebra} {\bf 65} (1990), 277--292.

      \bibitem[KKO]{KKO} {\sc A. Kuribayashi, A. Kobayashi, and S. Ohmori},
      Automorphism groups of compact Riemann surfaces. Travaux de la
      Conf\'erence Internationale d'Analyse Complexe et du 7e S\'eminaire
      Roumano-Finlandais (1993). {\it Rev. Roumaine Math. Pures Appl.}
      {\bf 40} (1995), no. 1, 65--75.

      \bibitem[LN]{LN} {\sc J. Lehner and M. Newman}, On Riemann surfaces
      with maximal automorphism groups, {\it Glasgow Math. J.} {\bf 8}
      (1967), 102--112.

      \bibitem[Le]{Le} {\sc S. Levy}, The eightfold way. The beauty of
      Klein's quartic curve. Ed. Silvio Levy. {\it Mathematical Sciences
      Research Institute Publications} {\bf 35}, Cambridge University Press,
      Cambridge, 1999.

      %\bibitem[Mb1] {Mb1}
      %{\sc A. M. Macbeath}, Hurwitz groups and surfaces. The eightfold
      %way, 103--113, Math. Sci. Res. Inst. Publ., 35, Cambridge Univ.
      %Press, Cambridge, 1999.

      \bibitem[Mb]{Mb2} {\sc A. M. Macbeath}, On a curve of genus $7$. {\it
      Proc. London Math. Soc. (3)} {\bf 15} (1965), 527--542.

      \bibitem[Mb2]{Mb3} {\sc A. M. Macbeath}, On a theorem of Hurwitz.
      {\it Proc. Glasgow Math. Assoc.} {\bf 5} (1961), 90--96.

      \bibitem[Mc1]{Mc1} {\sc C. Maclachlan}, Abelian groups of
      automorphisms of compact Riemann surfaces, {\it Proc. London Math.
      Soc. (3)} {\bf 15} (1965), 699--712.

      \bibitem[Mc2]{Mc2} {\sc C. Maclachlan}, A bound for the number of
      automorphisms of a compact Riemann surface, {\it J. London Math. Soc.
      (2)} {\bf 44} (1969), 265--272.

      \bibitem[MSV]{MSV} {\sc K. Magaard, S. Sphectorov and H. V\"olklein},
      Computing braid group action, in preparation.

      \bibitem[Ma1]{M} {\sc G.~Malle}, Genus zero translates of three point
      ramified Galois extensions, {\sl Manuscripta Math.} {\bf 71} (1991),
      97--111.

      \bibitem[Ma2]{Ma2} {\sc G.~Malle}, Hurwitz groups and $G\sb 2(q)$,
      {\it Canad. Math. Bull.} {\bf 33} (1990), no. 3, 349--357.

      \bibitem[Mt]{Mt} {\sc T. Matsuno}, Compact Riemann surfaces with large
      automorphism groups, {\it J. Math. Soc. Japan} {\bf 51} (1999), no.
      2, 309--329.

      \bibitem[Ri]{Ri} {\sc J. Ries}, Subvarieties of moduli space
      determined by finite groups acting on surfaces, {\it Trans. Amer.
      Math. Soc.} {\bf 335} (1993), no. 1, 385--406.

      \bibitem[Si1]{S2} {\sc D. Singerman}, Subgroups of Fuchsian groups and
      finite permutation groups, {\it Bull. London Math. Soc.} {\bf 2}
      (1970), 319--323.

      \bibitem[Si2]{Si} {\sc D. Singerman}, Finitely maximal Fuchsian
      groups, {\it J. London Math. Soc.} {\bf 6} (1972), 29--38.

      \bibitem[Si3]{S1} {\sc D. Singerman}, Symmetries of Riemann surfaces
      with large automorphism group, {\it Math. Ann.} {\bf 210} (1974),
      17--32.

      \bibitem[ShV]{ShV} {\sc T. ~Shaska and H. ~V\"olklein}, Elliptic
      Subfields and automorphisms of genus fields, to appear in: {\it
      Proceeding of the Conference on Algebra and Algebraic Geometry with
      Applications: The celebration of the seventieth birthday of Professor
      S.S. Abhyankar}, Springer-Verlag, 2001.

      \bibitem[Ta]{Ta} {\sc N.Takigawa}, Compact Riemann surfaces with an
      automorphism group of order $p$, {\it Math. Rep. Toyama Univ.} {\bf
      4} (1981), 115--127.

      \bibitem[V]{Buch} {\sc H.~V\"olklein}, {\it Groups as Galois Groups --
      an Introduction}, Cambr. Studies in Adv. Math. {\bf 53}, Cambridge
      Univ. Press 1996.

      \bibitem[V1]{V} {\sc H. V\"olklein}, Moduli spaces for covers of
      the Riemann sphere, {\it Israel J. Math.} {\bf 85} (1994), 407--430.

      \bibitem[W1]{W1} {\sc A. Wiman}, \"Uber die hyperelliptischen Curven
      und diejenigen vom Geschlechte $p=3$, welche eindeutige
      Transformationen in sich zulassen, {\it Bihang Kongl. Svenska
      Vetenskaps-Akademiens Handlingar} (1895), no. 21 (1), 1--23.

      \bibitem[W2]{W2} {\sc A. Wiman}, \"Uber die hyperelliptischen Curven
      vom den Geschlechte $p=4,5$, und 6, welche eindeutige Transformationen
      in sich besitzen, {\it Bihang Kongl. Svenska Vetenskaps-Akademiens
      Handlingar} (1895), no. 21 (3), 1--41.

      \bibitem[Wl]{Wl} {\sc R. Wilson}, The Monster is a Hurwitz group.
      {\it J. Group Theory} {\bf 4} (2001), no. 4, 367--374.

\end{thebibliography}