<!-- $Id: intro.xml,v 1.13 2007/10/04 22:02:12 gap Exp $ -->
<Chapter Label="ch:intro"><Heading>Introduction and Main Functions</Heading>

    This chapter provides an overview of the background material, and provides
    documentation for the main functions and data structures of the MapClass
    package.


    <Section Label="sec:background"><Heading>Background Material</Heading>
        <Index >Overview</Index>

        Let <M>G</M> be a finite group, let <M>C_1,\ldots, C_r</M> be a
        collection of conjugacy classes in <M>G</M>. Let <M>\mathcal{E} = 
            \mathcal{E}(G,g,(C_1, \dots, C_r))</M> denote the set of all tuples
        <M> \sigma = (\sigma_1,...,\sigma_{2g+r})\in G^{2g+r}</M> (for natural
        numbers <M>g</M> and <M>r</M>) of elements in <M>G</M> satisfying the
        relation
            
            <Display> \prod_{i=1}^g [\sigma_i,
                    \sigma_{g+i}] \prod_{i=1}^r \sigma_{2g+i} = 1
                    </Display>
        and such that <M>\sigma_{2g +k} \in C_k</M>. If the tuple also satisfies
        <M>\langle  \sigma_1, \ldots, \sigma_{2g+r}\rangle = G</M> it is said
        to be <E>generating</E>.<P /> 
        
        One may associate the
        elements of the tuple <M>\sigma</M> with the standard generators of
        the fundamental group of a compact connected surface <M>S</M> (genus
        <M>g</M>, <M>r</M> punctures). The mapping class group of <M>S</M> is
        naturally isomorphic to <M>Out(\pi_1(S))</M> and so gives rise to   
        an action on the fundamental group of <M>S</M> modulo inner automorphisms.
        This action can be transferred to an action on the set <M>\mathcal{E}</M>
        (up to conjugation in <M>G</M>). The <E>mapping class
            orbits</E> are the orbits of <M>\mathcal{E}</M> under this action. <P />

        The package can be used to compute the set <M>\mathcal{E}(G, g,
            (C_1,\ldots,C_r))</M> and the corresponding partition into mapping
        class orbits for  a given  group <M>G</M> and a set of conjugacy classes
        <M>(C_1,...,C_r)</M> (although the programs expect a tuple of class
        representatives). For an example application see Section <Ref
            Sect="sec:application" />. We call the tuple <M>(g;C_1,...,C_r)</M>
        the signature. The package is an extension of the <E>Braid</E> package
        for &GAP;. <P/>

    </Section>
    <Section Label="mainfunction"><Heading>Overview of Main
            Functions</Heading>
        The following are the principal ways for calculating the mapping class
        orbits for a given signature and group. We require our
        groups to be permutation groups, and the tuple in question to have length
        at least two.
        <ManSection>
            <Func Name="AllMCOrbits" Arg="group, genus, tuple"/>
            <Description>
                This function calculates the orbits for the given group, genus
                and tuple. This function is a wrapper for the function <Ref
                    Filt="AllMCOrbitsCore" />, and so can make use of GAP's
                <Keyword>OptionsStack</Keyword>. The options are described in
                more detail in the documentation for 
                <Ref Filt="AllMCOrbitsCore" />. We draw attention to two useful
                options: the <Code>OutputStyle</Code> and <Code>SaveOrbit</Code>
                options. The <Code>SaveOrbit</Code> option takes values of
                either <Keyword>false</Keyword> - in which case the orbit is
                not saved to a file - or
                it accepts a string that is the name of a directory
                in which the routine saves the orbits. See 
                <Ref Filt="AllMCOrbitsCore" /> for more details on the
                saving process. 

                The <Code> OutputStyle </Code> option controls the verbosity of
                the output of the function. It accepts three possible
                values: 
                <List>
                    <Item>
                        <Code>"silent"</Code> -- the routine prints no output
                        except in the case of an Error.
                    </Item>
                    <Item>
                        <Code>"single-line"</Code> -- the routine prints output
                        to a single line. An intermediate output style for those
                        who want some output but do not want to see all
                        diagnostic output.
                    </Item>
                    <Item>
                        <Code>"full"</Code> -- the routine provides full
                        diagnostic output.
                    </Item>
                </List>

            </Description>
        </ManSection>
        <ManSection>
            <Func Name="AllMCOrbitsCore" Arg="group, genus, tuple,
                partition, constant"/>
            <Description>
                This function calculates the orbits for the given group, genus
                and tuple, with the <M>r</M> branch points partitioned as in 
                <A>partition</A>. It uses the given <A>constant</A> to determine how
                many of the tuples it want to account for before exiting.
                 This function also make use of GAP's
                 <Keyword>OptionsStack</Keyword> if one desires to alter how the
                 algorithm runs. The following options and their defaults are
                 given below.
                <Table Align="|r|c|">
                    <HorLine/>
                    <Row>
                        <Item>Option Name</Item><Item>Default Value</Item>
                    </Row>
                    <HorLine/>
                    <Row><Item><Code>MaximumWeight</Code></Item><Item>40
                    </Item></Row>
                    <Row><Item><Code>MinimumWeight</Code></Item><Item>20
                    </Item></Row>
                    <Row><Item><Code>InitialWeight</Code></Item><Item>20
                    </Item></Row>
                    <Row><Item><Code>BumpUp</Code></Item><Item>7</Item></Row>
                    <Row><Item><Code>KnockDown</Code></Item><Item>7</Item></Row>
                    <Row><Item><Code>InitialNumberOfRandomTuples
                    </Code></Item><Item>20</Item></Row>
                    <Row><Item><Code> SaveOrbit </Code></Item><Item>
                            <Keyword>false</Keyword>
                    </Item></Row>
                    <Row><Item><Code> OutputStyle </Code></Item><Item>
                            <Keyword>"full"</Keyword>
                    </Item></Row>
                    <HorLine/>
                </Table>
                
                When trying to search for orbits it can be the case that
                the routine struggles to find a small orbit because of the low
                probability of randomly choosing a tuple in the orbit. To combat
                this problem the routine does not choose tuples entirely
                randomly but uses a weighted random selection to increase the
                probability of selecting tuples appearing in small tuples.
                To small subgroups of our group we have an associated weight.
                When a subgroup is generated by a tuple in our orbit
                frequently then we reduce its weight. Subgroups which do not
                appear often have their weight increased. The options
                <Code>MaximumWeight</Code>,
                <Code>MinimumWeight</Code>,
                <Code>InitialWeight</Code>,<Code>BumpUp</Code>, and
                <Code>KnockDown</Code>, control this subgroup weighting.
                Each option takes positive integer values. They play the
                following roles in the weighting process:
                <List>
                    <Item><Code>MaximumWeight</Code> : The maximum weight that a
                        subgroup can be.
                    </Item>
                    <Item><Code>MinimumWeight</Code> : The minimum weight that a
                        subgroup can be.
                    </Item>
                    <Item><Code>InitialWeight</Code> : The weight that a new
                        subgroup receives when added to to the list of small
                        subgroups.
                    </Item>
                    <Item><Code>BumpUp</Code> : The amount we increase the
                        weight of a subgroup by when it does not appear
                        frequently.
                    </Item>
                    <Item><Code>KnockDown</Code> : The amount we decrease the
                        weight of a subgroup by when it appears too
                        frequently.
                    </Item>
                </List>
                The default options were chosen experimentally and so it may be
                beneficial to tune these values for a specific case.<P />

                The option <Code>InitialNumberOfRandomTuples</Code>
                decides how many tuples the routine collects before trying to
                see if they are in some pre-existing orbit.<P />
                
                The option <Code>SaveOrbit</Code> which is by default
                <Keyword>false</Keyword> can be set to the name of a directory
                in which you want to save orbits. This option then saves the
                orbits to files in the folder with "<File>_name</File>". So for
                example if you wish to save your orbits into the file
                <File>_example</File> then you would run
                <Code>AllMCOrbits(group, genus, tuple:
                    SaveOrbit:="example");</Code>. The orbits are then saved in
                orbits which are named numerically. Following on from the above
                example then the first orbit will be saved as "_example/0".

                The <Code> OutputStyle </Code> option controls the verbosity of
                the output. It accepts three possible values: 
                <List>
                <Item>
                    <Code>"silent"</Code> - the routine prints no output except in
                    the case of an Error.
                </Item>
                <Item>
                    <Code>"single-line"</Code> - the routine output to a single
                    line. An intermediate output style for those who
                    want some output but do not want to see all
                    diagnostic output.
                </Item>
                <Item>
                    <Code>"full"</Code> - the routine provides full diagnostic output.
                </Item>
                </List>

            </Description>
        </ManSection>
        <ManSection>
            <Func Name="GeneratingMCOrbits" Arg="group, genus, tuple" />
            <Description> 
                This calculates the orbits for the given arguments.
                 Unlike the <Ref Filt="AllMCOrbits" /> function,
                <Code>GeneratingMCOrbits</Code> calculates only those orbits
                whose tuples generate the whole of our original group.
            </Description>
        </ManSection>
        <ManSection>
            <Func Name="GeneratingMCOrbitsCore" Arg="group, genus, tuple,
                partition, constant" />
            <Description> 
                This calculates the orbits for the given arguments.  Unlike the
                <Ref Filt="AllMCOrbits" /> function,
                <Code>GeneratingMCOrbitsCore</Code> calculates only those orbits
                whose tuples generate the whole of our original group. As with
                <Ref Filt="AllMCOrbitsCore" />, the argument <A>partition</A>
                must be a partition of the conjugacy classes represented in list
                form. We also have access to the full value of the options stack
                as in <Ref Filt="AllMCOrbitsCore" />.
            </Description>
        </ManSection>
        <ManSection>
            <Func Name="MappingClassOrbit" Arg="group, genus, principaltuple,
                partition, tuple"
                />
            <Returns> an orbit record for an orbit containing tuple or returns
                <K>fail</K>
            </Returns> 
            <Description> 
                Calculates the orbit of the <A>tuple</A> with respect
                to the given <A>group</A>, <A>principaltuple</A> and
                <A>genus</A>.
            </Description>
        </ManSection>
        <ManSection>
            <Func Name="PrepareMinTree" Arg="principaltuple,
                group, ourR, genus" />
            <Returns> a record with two keys <Code>MinimizationTree</Code> and
                <Code>MinimumSet</Code>. If <Code>record</Code> is the returned
                record then <Code>record.MinimizationTree</Code> is the list
                encoding the tree used to help minimize tuples. The list <Code>
                record.MinimumSet</Code> is a list of minimal elements which
                is also used to speed up tuple minimization. </Returns>
        </ManSection>
        <ManSection>
            <Func Name="MinimizeTuple" Arg="tuple, minimizationTree, minimumSet,
                numberOfGenerators"/>
            <Returns> the minimal tuple in the same orbit of
                <A>tuple</A>.</Returns>
            <Description> Take the minimisation data provided by <Ref
                    Filt="PrepareMinTree" /> and minimizes the given
                <A>tuple</A>.
            </Description>
        </ManSection>
        <ManSection>
          <Func Name="EasyMinimizeTuple" Arg="group, genus, tuple" />
            <Returns> the minimal tuple in the same orbit as
                <A>tuple</A>.</Returns>
        </ManSection>
        <ManSection>
            <Func Name="IsInOrbit" Arg="orbit, tuple" />
            <Returns><Keyword>True</Keyword> if the <A>tuple</A> lies in the
                <A>orbit</A>.</Returns>
        </ManSection>
        <ManSection>
            <Func Name="FindTupleInOrbit" Arg="orbit, tuple" />
            <Returns> the index of <A>tuple</A> in <Code>orbit.TupleTable</Code> if in the
                <A>orbit</A>. If the tuple is not in <A>orbit</A> returns
                <Keyword>fail</Keyword>.
            </Returns>
        </ManSection>
        <ManSection>
            <Func Name="IsEqualOrbit" Arg="orbit1, orbit2" />
            <Returns><Keyword>true</Keyword> if the two orbits are equal else
                returns <Keyword>fail</Keyword>.
            </Returns>
        </ManSection>
        <ManSection>
            <Func Name="SelectTuple" Arg="orbit, index" />
            <Returns>  the tuple 
                <Code>orbit.TupleTable[index].tuple</Code>.
            </Returns>
        </ManSection>
        <ManSection>
            <Func Name="NumberGeneratingTuples" Arg="group, partition, tuple,
              genus" />
            <Returns> the total number of possible generating tuples for the
                group and tuple. 
            </Returns>
        </ManSection>
        <ManSection>
            <Func Name="TotalNumberTuples" Arg="group, tuple, genus"
              />
            <Returns> the total number of tuples we have to account for.
            </Returns>
        </ManSection>
        <ManSection>
            <Func Name="CalculateTuplePartition" Arg="group, tuple" />
            <Returns> A partition of <Math>{1,\ldots, r}</Math> where
              <Math>r</Math> is the length of the tuple.
            </Returns>
            <Description>
              The function returns a partition of <Math>{1,\ldots, r}</Math>
              such that <M>i</M> and <M>j</M> lie in the same block if and only
              if the elements <Code>tuple[i]</Code> and <Code>tuple[j]</Code>
              are member of the same conjugacy class. The program currently
              requires that the elements of the tuple be ordered such that if
              <Code>tuple[i]</Code> and <Code>tuple[j]</Code> are in the same
              conjugacy class with <M>i \le j</M> then so is<Code>tuple[k]</Code>
              for all <M>i \le k \le j</M>.
            </Description>
        </ManSection>
        <ManSection>
            <Func Name="RestoreOrbitFromFile" Arg="n, name [, path]" />
            <Returns> the n-th orbit record from the project with project
                <Code>"name"</Code>
            </Returns>
            <Description> By default the function searches the current working
                directory for the saved project folder and searches inside this
                for the n-th orbit. If no such orbit exists it returns
                <Keyword>fail</Keyword>. If an optional argument <A>path</A> is
                provided then it searches this path for a folder with the name
                specified (note that path expects a <Code>Directory</Code>
                object). If an orbit exists then it is returned as a record as
                outlined in the data structure section.
            </Description>
        </ManSection>
        <ManSection>
            <Func Name="SaveOrbitToFile" Arg=" orbit, n, name" />
            <Description>
                Saves the orbit to filename "<A>n</A>" in the directory
                <Code>'_name'</Code>. The directory must already exist.
            </Description>
        </ManSection>
    </Section>
    <Section Label="datastructures"><Heading>Key Data Structures</Heading>
        Many of the above functions require or return key data structures
        which we aim to document.
        <Subsection Label="orbitrecord"><Heading>The Orbit Record</Heading>
            Many of the functions return or expect an orbit "object". This
            object is in fact record with the following values:
            <List>
                <Item>
                    <Code>PrincipalFiniteGroup </Code>- the finite group
                </Item>
                <Item>
                    <Code>OurG </Code>- genus
                </Item>
                <Item>
                    <Code>OurR </Code>- length of our primary tuple
                </Item>
                <Item>
                    <Code>OurN </Code>- number of points on which our group
                    acts
                </Item>
                <Item>
                    <Code>NumberOfGenerators </Code>- 2 <Code>OurG</Code>+
                    <Code>OurR </Code>
                </Item>
                <Item>
                    <Code>OurFreeGroup </Code>- a free group on
                    <Code>NumberOfGenerators </Code>letters
                </Item>
                <Item>
                    <Code>AbsGens </Code>- generators for
                    <Code>OurFreeGroup</Code>
                </Item> 
                <Item>
                    <Code>OurAlpha
                    </Code> - generators of
                    <Code>OurFreeGroup</Code>corresponding to the
                    <M>\alpha_i</M> type loops in the fundamental group ( the
                    first <M>g</M> elements of <Code>AbsGens</Code>)
                </Item>
                <Item>
                    <Code>OurBeta </Code>- elements of <Code>OurFreeGroup
                    </Code>corresponding to <M>\beta</M> type loops 
                </Item>
                <Item>
                    <Code>OurGamma </Code>- generators of <Code>OurFreeGroup
                </Code>corresponding to the loops around branch points
                </Item>
                <Item>
                    <Code>TupleTable </Code>- a table containing all the
                    tuples in the orbit
                </Item>
                <Item><Code>HashLength</Code></Item>
                <Item><Code>Hash</Code></Item>
                <Item><Code>PrimeCode</Code></Item>
                <Item><Code>OurAction</Code></Item>
                <Item><Code>ActionOnOrbit</Code></Item>
                <Item><Code>MinimizationTree</Code>- minimization structure</Item>
                <Item><Code>MinimumSet</Code>- minimizaton structure</Item>
            </List>
        </Subsection>
        <Subsection Label="tupletable">
            <Heading>The Tuple Table</Heading>
            The tuple table is a list. Each element of the list is a record with
            the names, tuple and next. If <Code>orbit</Code> is an orbit object
            then <Code>orbit.TupleTable[n].tuple</Code> returns the tuple at
            index <M>n</M> of the tuple table.
        </Subsection>
    </Section>
    <Section><Heading>A Sample Session</Heading>
        We demonstate how one might use the package.
    <Log><![CDATA[
gap> group:=AlternatingGroup(5);
Alt( [ 1 .. 5 ] )
gap> tuple:=[ (1,2)(3,4), (1,2)(3,4), (1,2)(3,4) ];
[ (1,2)(3,4), (1,2)(3,4), (1,2)(3,4) ]
gap> orbits:=AllMCOrbits(group,1,tuple);;

Total Number of Tuples: 189120


Collecting 20 random tuples... done

Cleaning done; 20 random tuples remaining


Orbit 1:
																	  
Length=3072
Generating Tuple  =[ (1,2,4,5,3), (1,4,5,2,3), (1,2)(4,5),
(1,4)(2,3), (2,5)(3,4) ]
Generated subgroup size=60
Centralizer size=1
4800 tuples remaining
Cleaning current orbit...
Cleaning a list of 20 tuples
Random Tuples Remaining: 0
Cleaning done; 0 random tuples remaining


Collecting 20 random tuples... done

Cleaning orbit 1
Cleaning a list of 20 tuples
Random Tuples Remaining: 0

Cleaning done; 0 random tuples remaining


Collecting 40 random tuples... done

Cleaning orbit 1
Cleaning a list of 40 tuples
Random Tuples Remaining: 3

Cleaning done; 3 random tuples remaining


Orbit 2:
																	  
Length=32
Generating Tuple  =[ (1,4)(2,3), (1,2)(3,4), (1,4)(2,3), (1,2)(3,4),
(1,3)(2,4) ]
Generated subgroup size=4
Centralizer size=4
4320 tuples remaining
Cleaning current orbit...
Cleaning a list of 3 tuples
Random Tuples Remaining: 2
Cleaning done; 2 random tuples remaining


Orbit 3:
																	  
Length=72
Generating Tuple  =[ (1,5,2), (1,3,2), (1,2)(3,5), (1,3)(2,5),
(1,3)(2,5) ]
Generated subgroup size=12
Centralizer size=1
0 tuples remaining
Cleaning current orbit...
Cleaning a list of 2 tuples
Random Tuples Remaining: 0
Cleaning done; 0 random tuples remaining

gap> # Check we have as many orbits as it says...
gap> Length(orbits);
3
gap> # Inspect the first orbit..
gap> orb1:=orbits[1];;
gap> # How long is orb1?
gap> Length(orb1.TupleTable);
3072
gap> # Select element 42 ...
gap> SelectTuple(orb1, 42);
[ (1,3,4), (1,5,3,2,4), (1,5)(2,4), (1,2)(3,5), (2,3)(4,5) ]
gap> # Save the orbit to a file...
gap> SaveOrbitToFile(orb1, 1, "test");
gap> #If the folder doesn't exist we get an error..
gap> SaveOrbitToFile(orb1, 1, "foo");
AppendTo: cannot open '_foo/1' for output at
CallFuncList( APPEND_TO, arg );
gap> #
gap> # Now we try find generating orbits 
gap> group:=SymmetricGroup(5);
Sym( [ 1 .. 5 ] )
gap> # And we will save them using the `SaveOrbit` option
gap> GeneratingMCOrbits(group,1,tuple : SaveOrbit:="example");;

Total Number of Tuples: 607680



Collecting 20 generating tuples .. done

Cleaning done; 20 random tuples remaining


Orbit 1:                    
Length=5064
Generating Tuple  =[ (1,3,2,5), (2,4,3), (1,4)(3,5), (1,3)(2,5),
(1,4)(3,5) ]
Generated subgroup size=120
Saving orbit to file _example/0
Centralizer size=1
0 tuples remaining
Cleaning current orbit...
Cleaning a list of 20 tuples
Random Tuples Remaining: 0
Cleaning done; 0 random tuples remaining

gap> generatingorbits:=last;;
gap> # How many generating orbits are there?
gap> Length(generatingorbits);
1
gap> # Is this orbit equal to the other one we found earlier
gap> IsEqualOrbit(orb1, generatingorbits[1]);
fail
gap> # We can reload the orbits...
gap> orb2:=RestoreOrbitFromFile(0, "example");;
gap> Length(orb2.TupleTable);
5064
    ]]></Log>
    </Section>
    <Section Label="sec:application"><Heading>An Application</Heading>
        This section describes an application of the package. <P/>
        
        Let <M>X</M> be a compact Riemann surface of genus <M>g</M> and <M>f: X
            \rightarrow \mathbb{P}^1\mathbb{C}</M> be a meromorphic function of
        degree <M>n</M>. Let <M>B</M> be the set of branch points for <M>f</M>
        and fix a basepoint <M>b_0 \in \mathbb{P}^1\mathbb{C}- B</M>.
        The fundamental group <M>\pi_1(\mathbb{P}^1 \mathbb{C} - B, b_0) </M>
        acts transitively on the fibre <M>f^{-1}(b_0)</M> and this corresponds
        to a representation
        <Display>
             f^* :\pi_1(\mathbb{P}^1 \mathbb{C} - B, b_0) \rightarrow S_n
        </Display>
        The image of <M>f^*</M> is called  the <E>monodromy group</E> of <M>(X,
            f)</M>. The fundamental group of the punctured Riemann sphere is
        generated by the loops that wind around the points in <M>B</M>. Label the
        branch points <M>b_1,...,b_r</M> and let <M>\tau_i </M>
        be the image under <M>f^*</M> of the loop, <M>\gamma_i \in
            \pi_1(\mathbb{P}^1\mathbb{C} - B)</M>, that winds
        once around the point <M>b_i\in B</M>.

        Therefore,
        <Display>
            \langle \tau_1, ..., \tau_r \rangle = G
        </Display>
        and
        <Display>
            \tau_1 \cdots \tau_r = 1 
        </Display>
        Moreover by the Riemann-Hurwitz formula
        <Display>
            2(n+g-1) = \sum_1^r {ind}( \tau_i) 
        </Display>
        where the <M>{ind}(\tau_i)</M> is the minimal number of factors to
        express <M>\tau_i</M> as a product of  transpositions.

        A set <M>t_1, \ldots, t_r</M> of elements of <M>S_n</M> satisfying the
        Riemann-Hurwitz formula, the product-one condition, and generating some
        transitive subgroup <M>G</M> of <M>S_n</M> is called a <E>genus </E>
        <M> g </M> <E>generating system </E> for <M>G</M>. Therefore to the
        meromorphic function <M>(X, f)</M> there is an associated genus <M>g</M>
        system. In fact the conjugacy classes of the elements <M>\tau_i</M> are
        also determined by <M>f</M> -- the collection of conjugacy classes is
        sometimes called the <E> ramification type</E> of <M>f</M>.
        On the other hand for every genus <M> g</M> generating system, <M>t =
            (\tau_1, \ldots, \tau_n)</M> for <M>G</M> there is Riemann surface
        of genus <M>g</M> and a meromorphic function with associated generating
        system <M>t</M> -- this result is known as <E> Riemann's Existence
            Theorem</E>.<P />

        The question we hope to use our package to answer is: For a given finite
        group <M>G</M> how many meromorphic maps with monodromy group <M>G</M> are
        there? It can be shown -- see <Cite Key="Volklein96" /> for example -- that determining whether
        two genus <M>g</M> coverings are equivalent corresponds to determining
        whether their associated genus <M>g</M> systems are in the same mapping
        class orbit ( most literature would refer to mapping class orbits as
        braid orbits in this case - this is because of the equivalence between
        the mapping class group of a punctured disc and the braid groups <Cite
            Key="Birman75" />). <P />

        Thus for a finite group <M>G</M> we can answer the above principal
        question using the following process:
        <List>
            <Item>
                For a given finite group <M>G</M> the work of Breuer <Cite
                    Key="Breuer00" /> can be used to calculate all possible
                ramification types.
            </Item>
            <Item>
                Pick a tuple, <M>C = (c_1,...,c_r)</M>, of representative elements
                of the conjugacy classes which correspond to a chosen ramification
                type as calculated in the previous step.
            </Item>
            <Item>
                Use the function <Code>GeneratingMCOrbits(G, 0, [c1,...,cr])</Code>
                to calculate the number of mapping class orbits. Note that the
                genus argument is <M>0</M> because this is the genus of
                <M>\mathbb{P}^1\mathbb{C}</M>.
            </Item>
        </List>
        For more information on this process and the underlying theory see <Cite
            Key="MSV" /> and <Cite Key="Volklein96" />.
    </Section>
</Chapter>

