Quelle ggraph.xml Sprache: XML

<?xml version="1.0" encoding="UTF-8"?>

<Chapter Label="chap-ggraph">
<Heading>Graphs of Groups and Groupoids</Heading>

This package was originally designed to implement <E>graphs of groups</E>,
a notion introduced by Serre in <Cite Key="Serre" />.
It was only when this was extended to <E>graphs of groupoids</E>
that the functions for groupoids, described in the previous chapters,
were required.
The methods described here are based on Philip Higgins' paper
<Cite Key="HiJLMS" />.
For further details see Chapter 2 of <Cite Key="emma-thesis" />.

Since a graph of groups involves a directed graph, with a group
associated to each vertex and arc, we first define digraphs
with edges weighted by the generators of a free group.

<Section Label="sec-dgph">
<Heading>Digraphs</Heading>

<ManSection>
   <Attr Name="FpWeightedDigraph"
         Arg="verts, arcs" />
   <Attr Name="IsFpWeightedDigraph"
         Arg="dig" />
   <Attr Name="InvolutoryArcs"
         Arg="dig" />
<Description>
A <E>weighted digraph</E> is a record with two components:
<E>vertices</E>, which are usually taken to be positive integers
(to distinguish them from the objects in a groupoid);
and <E>arcs</E>, which take the form of 3-element lists
<C>[weight,tail,head]</C>.
The <E>tail</E> and <E>head</E> are the two vertices of the arc.
The <E>weight</E> is taken to be an element of a finitely presented group,
so as to produce digraphs of type <C>IsFpWeightedDigraph</C>.
<P/>
</Description>
</ManSection>
<Example>
<![CDATA[
gap> V1 := [ 5, 6 ];;
gap> fg1 := FreeGroup( "y" );;
gap> y := fg1.1;;
gap> A1 := [ [ y, 5, 6 ], [ y^-1, 6, 5 ] ];
gap> D1 := FpWeightedDigraph( fg1, V1, A1 );
weighted digraph with vertices: [ 5, 6 ]
and arcs: [ [ y, 5, 6 ], [ y^-1, 6, 5 ] ]
gap> inv1 := InvolutoryArcs( D1 );
[ 2, 1 ]
]]>
</Example>

The example illustrates the fact that we require arcs to be defined
in involutory pairs, as though they were inverse elements in a groupoid.
We may in future decide just to give <C>[y,5,6]</C> as the data
and get the function to construct the reverse edge.
The attribute <C>InvolutoryArcs</C> returns a list of the positions
of each inverse arc in the list of arcs.
In the second example the graph is a complete digraph on three vertices.
<Example>
<![CDATA[
gap> fg3 := FreeGroup( 3, "z" );;
gap> z1 := fg3.1;;  z2 := fg3.2;;  z3 := fg3.3;;
gap> ob3 := [ 7, 8, 9 ];;
gap> A3 := [[z1,7,8],[z2,8,9],[z3,9,7],[z1^-1,8,7],[z2^-1,9,8],[z3^-1,7,9]];;
gap> D3 := FpWeightedDigraph( fg3, ob3, A3 );
weighted digraph with vertices: [ 7, 8, 9 ]
and arcs: [ [ z1, 7, 8 ], [ z2, 8, 9 ], [ z3, 9, 7 ], [ z1^-1, 8, 7 ],
  [ z2^-1, 9, 8 ], [ z3^-1, 7, 9 ] ]
[gap> inob3 := InvolutoryArcs( D3 );
[ 4, 5, 6, 1, 2, 3 ]
]]>
</Example>
</Section>

<Section Label="sec-gphgps">
<Heading>Graphs of Groups</Heading>

<ManSection>
   <Oper Name="GraphOfGroups"
         Arg="dig, gps, isos" />
   <Attr Name="DigraphOfGraphOfGroups"
         Arg="gg" />
   <Attr Name="GroupsOfGraphOfGroups"
         Arg="gg" />
   <Attr Name="IsomorphismsOfGraphOfGroups"
         Arg="gg" />
   <Filt Name="IsGraphOfGroups"
         Arg='dig' Type='Category'/>
<Description>
A graph of groups is traditionally defined as consisting of:
<List>
<Item>
a digraph with involutory pairs of arcs;
</Item>
<Item>
a <E>vertex group</E> associated to each vertex;
</Item>
<Item>
a group associated to each pair of arcs;
</Item>
<Item>
an injective homomorphism from each arc group to the group at the head
of the arc.
</Item>
</List>
We have found it more convenient to associate to each arc:
<List>
<Item>
a subgroup of the vertex group at the tail;
</Item>
<Item>
a subgroup of the vertex group at the head;
</Item>
<Item>
an isomorphism between these subgroups,
such that each involutory pair of arcs determines inverse isomorphisms.
</Item>
</List>
These two viewpoints are clearly equivalent.
<P/>
In this implementation we require that all subgroups are of finite index
in the vertex groups.
<P/>
The three attributes provide a means of calling the three items
of data in the construction of a graph of groups.
<P/>
We shall be representing free products with amalgamation of groups
and HNN extensions of groups in Section <Ref Sect="sec-fpahnn"/>.
So we take as our first example the trefoil group with generators
<M>a,b</M> and relation <M>a^3=b^2</M>.
For this we take digraph <C>D1</C> above
with an infinite cyclic group at each vertex,
generated by <M>a</M> and <M>b</M> respectively.
The two subgroups will be generated by <M>a^3</M> and <M>b^2</M>
with the obvious isomorphisms.
<P/>
</Description>
</ManSection>
<Example>
<![CDATA[
gap> ## free vertex group at 5
gap> fa := FreeGroup( "a" );;
gap> a := fa.1;;
gap> SetName( fa, "fa" );
gap> hy := Subgroup( fa, [a^3] );;
gap> SetName( hy, "hy" );
gap> ## free vertex group at 6
gap> fb := FreeGroup( "b" );;
gap> b := fb.1;;
gap> SetName( fb, "fb" );
gap> hybar := Subgroup( fb, [b^2] );;
gap> SetName( hybar, "hybar" );
gap> ## isomorphisms between subgroups
gap> homy := GroupHomomorphismByImagesNC( hy, hybar, [a^3], [b^2] );;
gap> homybar := GroupHomomorphismByImagesNC( hybar, hy, [b^2], [a^3] );;
gap> ## defining graph of groups G1
gap> G1 := GraphOfGroups( D1, [fa,fb], [homy,homybar] );
Graph of Groups: 2 vertices; 2 arcs; groups [ fa, fb ]
gap> Display( G1 );
Graph of Groups with :-
    vertices: [ 5, 6 ]
        arcs: [ [ y, 5, 6 ], [ y^-1, 6, 5 ] ]
      groups: [ fa, fb ]
isomorphisms: [ [ [ a^3 ], [ b^2 ] ], [ [ b^2 ], [ a^3 ] ] ]
gap> IsGraphOfGroups( G1 );
true
]]>
</Example>

<ManSection>
   <Prop Name="IsGraphOfFpGroups"
         Arg="gg" />
   <Prop Name="IsGraphOfPcGroups"
         Arg="gg" />
   <Prop Name="IsGraphOfPermGroups"
         Arg="gg" />
<Description>
This is a list of properties to be expected of a graph of groups.
In principle any type of group known to &GAP;
may be used as vertex groups, though these types are not
normally mixed in a single structure.
</Description>
</ManSection>
<Example>
<![CDATA[
gap> IsGraphOfFpGroups( G1 );
true
gap> IsomorphismsOfGraphOfGroups( G1 );
[ [ a^3 ] -> [ b^2 ], [ b^2 ] -> [ a^3 ] ]

]]>
</Example>

<ManSection>
   <Attr Name="RightTransversalsOfGraphOfGroups"
         Arg="gg" />
   <Attr Name="LeftTransversalsOfGraphOfGroups"
         Arg="gg" />
<Description>
Computation with graph of groups words will require,
for each arc subgroup <C>ha</C>, a set of representatives for the left cosets
of <C>ha</C> in the tail vertex group.
As already pointed out, we require subgroups of finite index.
Since &GAP; prefers to provide right cosets, we obtain the right
representatives first, and then invert them.
<P/>
When the vertex groups are of type <C>FpGroup</C>
we shall require normal forms for these groups,
so we assume that such vertex groups are provided with Knuth Bendix
rewriting systems using functions from the main &GAP; library,
(e.g. <C>IsomorphismFpSemigroup</C>).
<P/>
</Description>
</ManSection>
<Example>
<![CDATA[
gap> RTG1 := RightTransversalsOfGraphOfGroups( G1 );
[ [ <identity ...>, a, a^2 ], [ <identity ...>, b ] ]
gap> LTG1 := LeftTransversalsOfGraphOfGroups( G1 );
[ [ <identity ...>, a^-1, a^-2 ], [ <identity ...>, b^-1 ] ]
]]>
</Example>
</Section>

<Section Label="sec-words">
<Heading>Words in a Graph of Groups and their normal forms</Heading>

<ManSection>
   <Oper Name="GraphOfGroupsWord"
         Arg="gg, tv, list" />
   <Prop Name="IsGraphOfGroupsWord"
         Arg="w" />
   <Attr Name="GraphOfGroupsOfWord"
         Arg="w" />
   <Attr Name="WordOfGraphOfGroupsWord"
         Arg="w" />
   <Attr Name="TailOfGraphOfGroupsWord"
         Arg="w" />
   <Attr Name="HeadOfGraphOfGroupsWord"
         Arg="w" />
<Description>
If <C>G</C> is a graph of groups with underlying digraph <C>D</C>,
the following groupoids may be considered.
First there is the free groupoid or path groupoid on <C>D</C>.
Since we want each involutory pair of arcs to represent inverse elements
in the groupoid, we quotient out by the relations <C>y^-1 = ybar</C>
to obtain <C>PG(D)</C>.
Secondly, there is the discrete groupoid <C>VG(D)</C>, namely the union
of all the vertex groups.
Since these two groupoids have the same object set (the vertices of <C>D</C>)
we can form <C>A(G)</C>, the free product of <C>PG(D)</C> and <C>VG(D)</C>
amalgamated over the vertices.
For further details of this universal groupoid construction
see <Cite Key="emma-thesis" />.
(Note that these groupoids are not implemented in this package.)
<P/>
An element of <C>A(G)</C> is a graph of groups word which may be represented
by a list of the form <M>w = [g_1,y_1,g_2,y_2,...,g_n,y_n,g_{n+1}]</M>.
Here each <M>y_i</M> is an arc of <C>D</C>;
the head of <M>y_{i-1}</M> is a vertex <M>v_i</M>
which is also the tail of <M>y_i</M>;
and <M>g_i</M> is an element of the vertex group at <M>v_i</M>.
<P/>
So a graph of groups word requires as data the graph of groups;
the tail vertex for the word; and a list of arcs and group elements.
We may specify each arc by its position in the list of arcs.
<P/>
In the following example, where <C>gw1</C> is a word
in the trefoil graph of groups,
the <M>y_i</M> are specified by their positions in <C>A1</C>.
Both arcs are traversed twice,
so the resulting word is a loop at vertex <M>5</M>.
<P/>
</Description>
</ManSection>
<Example>
<![CDATA[
gap> L1 := [ a^7, 1, b^-6, 2, a^-11, 1, b^9, 2, a^7 ];;
gap> gw1 := GraphOfGroupsWord( G1, 5, L1 );
(5)a^7.y.b^-6.y^-1.a^-11.y.b^9.y^-1.a^7(5)
gap> IsGraphOfGroupsWord( gw1 );
true
gap> [ TailOfGraphOfGroupsWord(gw1), HeadOfGraphOfGroupsWord(gw1) ];
[ 5, 5 ]
gap> GraphOfGroupsOfWord(gw1);
Graph of Groups: 2 vertices; 2 arcs; groups [ fa, fb ]
gap> WordOfGraphOfGroupsWord( gw1 );
[ a^7, 1, b^-6, 2, a^-11, 1, b^9, 2, a^7 ]
]]>
</Example>

<ManSection>
   <Oper Name="ReducedGraphOfGroupsWord"
         Arg="w" />
   <Prop Name="IsReducedGraphOfGroupsWord"
         Arg="w" />
<Description>
A graph of groups word may be reduced in two ways, to give a normal form.
Firstly, if part of the word has the form <C>[yi, identity, yibar]</C>
then this subword may be omitted.
This is known as a length reduction.
Secondly there are coset reductions.
Working from the left-hand end of the word, subwords of the form
<M>[g_i,y_i,g_{i+1}]</M>  are replaced by  <M>[t_i,y_i,m_i(h_i)*g_{i+1}]</M>
where <M>g_i = t_i*h_i</M> is the unique factorisation of <M>g_i</M>
as a left coset representative times an element of the arc subgroup,
and <M>m_i</M> is the isomorphism associated to <M>y_i</M>.
Thus we may consider a coset reduction as passing a subgroup element
along an arc.
The resulting normal form (if no length reductions have taken place)
is then <M>[t_1,y_1,t_2,y_2,...,t_n,y_n,k]</M>
for some <M>k</M> in the head group of <M>y_n</M>.
For further details see Section 2.2 of <Cite Key="emma-thesis" />.
<P/>
The reduction of the word <C>gw1</C> in our example
includes one length reduction.
The four stages of the reduction are as follows:
<Display>
a^7b^{-6}a^{-11}b^9a^7 ~\mapsto~
a^{-2}b^0a^{-11}b^9a^7 ~\mapsto~
a^{-13}b^9a^7 ~\mapsto~
a^{-1}b^{-8}b^9a^7 ~\mapsto~
a^{-1}b^{-1}a^{10}.
</Display>
<P/>
</Description>
</ManSection>
<Example>
<![CDATA[
gap> nw1 := ReducedGraphOfGroupsWord( gw1 );
(5)a^-1.y.b^-1.y^-1.a^10(5)
]]>
</Example>
</Section>

<Section Label="sec-fpahnn">
<Heading>Free products with amalgamation and HNN extensions</Heading>

<ManSection>
   <Oper Name="FreeProductWithAmalgamation"
         Arg="gp1, gp2, iso" />
   <Attr Name="FreeProductWithAmalgamationInfo"
         Arg="fpa" />
   <Prop Name="IsFreeProductWithAmalgamation"
         Arg="fpa" />
   <Attr Name="GraphOfGroupsRewritingSystem"
         Arg="fpa" />
   <Attr Name="NormalFormGGRWS"
         Arg="fpa, word" />
<Description>
As we have seen with the trefoil group example in Section
<Ref Sect="sec-gphgps"/>, graphs of groups can be used to obtain
a normal form for free products with amalgamation <M>G_1 *_H G_2</M>
when <M>G_1, G_2</M> both have rewrite systems,
and <M>H</M> is of finite index in both <M>G_1</M> and <M>G_2</M>.
<P/>
When <C>gp1</C> and <C>gp2</C> are fp-groups, the operation
<C>FreeProductWithAmalgamation</C> constructs the required fp-group.
When the two groups are permutation groups,
the <C>IsomorphismFpGroup</C> operation is called on both
<C>gp1</C> and <C>gp2</C>,  and the resulting isomorphism
is transported to one between the two new subgroups.
<P/>
The attribute <C>GraphOfGroupsRewritingSystem</C> of <C>fpa</C>
is the graph of groups which has underlying digraph <C>D1</C>,
with two vertices and two arcs;
the two groups as vertex groups;
and the specified isomorphisms on the arcs.
Despite the name, graphs of groups constructed in this way
<E>do not</E> belong to the category <C>IsRewritingSystem</C>.
This anomaly may be dealt with when time permits.
<P/>
The example below shows a computation in the the free product
of the symmetric <C>s3</C> and the alternating <C>a4</C>,
amalgamated over a cyclic subgroup <C>c3</C>.
<P/>
</Description>
</ManSection>
<Example>
<![CDATA[
gap> ## set up the first group s3 and a subgroup c3=<a1>
gap> fg2 := FreeGroup( 2, "a" );;
gap> rel1 := [ fg2.1^3, fg2.2^2, (fg2.1*fg2.2)^2 ];;
gap> s3 := fg2/rel1;;
gap> gs3 := GeneratorsOfGroup(s3);;
gap> SetName( s3, "s3" );
gap> a1 := gs3[1];;  a2 := gs3[2];;
gap> H1 := Subgroup(s3,[a1]);;
gap> ## then the second group a4 and subgroup c3=<b1>
gap> f2 := FreeGroup( 2, "b" );;
gap> rel2 := [ f2.1^3, f2.2^3, (f2.1*f2.2)^2 ];;
gap> a4 := f2/rel2;;
gap> ga4 := GeneratorsOfGroup(a4);;
gap> SetName( a4, "a4" );
gap> b1 := ga4[1];  b2 := ga4[2];;
gap> H2 := Subgroup(a4,[b1]);;
gap> ## form the isomorphism and the fpa group
gap> iso := GroupHomomorphismByImages(H1,H2,[a1],[b1]);;
gap> inv := InverseGeneralMapping(iso);;
gap> fpa := FreeProductWithAmalgamation( s3, a4, iso );
<fp group on the generators [ f1, f2, f3, f4 ]>
gap> RelatorsOfFpGroup( fpa );
[ f1^2, f2^3, (f2*f1)^2, f3^3, f4^3, (f4*f3)^2, f2*f3^-1 ]
gap> gg1 := GraphOfGroupsRewritingSystem( fpa );;
gap> Display( gg1 );
Graph of Groups with :-
    vertices: [ 5, 6 ]
        arcs: [ [ y, 5, 6 ], [ y^-1, 6, 5 ] ]
      groups: [ s3, a4 ]
isomorphisms: [ [ [ a1 ], [ b1 ] ], [ [ b1 ], [ a1 ] ] ]
gap> LeftTransversalsOfGraphOfGroups( gg1 );
[ [ <identity ...>, a2^-1 ], [ <identity ...>, b2^-1, b1^-1*b2^-1, b1*b2^-1 ]
]
gap> gfpa := GeneratorsOfGroup( fpa );;
gap> w2 := (gfpa[1]*gfpa[2]*gfpa[3]^gfpa[4])^3;
(f1*f2*f4^-1*f3*f4)^3
gap> n2 := NormalFormGGRWS( fpa, w2 );
f2*f3*(f4^-1*f2)^2*f4^-1*f3
]]>
</Example>

<ManSection>
   <Oper Name="ReducedImageElm"
         Arg="hom, eml" />
   <Prop Name="IsMappingToGroupWithGGRWS"
         Arg="map" />
   <Meth Name="Embedding"
         Arg="fpa, num" Label="for fpa-groups" />
<Description>
All fpa-groups are provided with a record attribute,
<C>FreeProductWithAmalgamationInfo(fpa)</C>
which is a record storing the groups, subgroups and isomorphism
involved in their construction.
This information record also contains the embeddings of the two groups
into the product.
The operation <C>ReducedImageElm</C>, applied to a homomorphism <M>h</M>
of type <C>IsMappingToGroupWithGGRWS</C> and an element
<M>x</M> of the source, finds the usual <C>ImageElm(h,x)</C>
and then reduces this to its normal form using the
graph of groups rewriting system.
<P/>
</Description>
</ManSection>
<Example>
<![CDATA[
gap> fpainfo;
rec( embeddings := [ [ a2, a1 ] -> [ f1, f2 ], [ b1, b2 ] -> [ f3, f4 ] ],
  groups := [ s3, a4 ], isomorphism := [ a1 ] -> [ b1 ],
  positions := [ [ 1, 2 ], [ 3, 4 ] ],
  subgroups := [ Group([ a1 ]), Group([ b1 ]) ] )
gap> emb2 := Embedding( fpa, 2 );
[ b1, b2 ] -> [ f3, f4 ]
gap> ImageElm( emb2, b1^b2 );
f4^-1*f3*f4
gap> ReducedImageElm( emb2, b1^b2 );
f4*f3^-1
]]>
</Example>

<ManSection>
   <Oper Name="HnnExtension"
         Arg="gp, iso" />
   <Attr Name="HnnExtensionInfo"
         Arg="gp, iso" />
   <Prop Name="IsHnnExtension"
         Arg="hnn" />
<Description>
For <E>HNN extensions</E>, the appropriate graph of groups
has underlying digraph with just one vertex and one pair of loops,
weighted with <C>FpGroup</C> generators <M>z,z^{-1}</M>.
There is one vertex group <C>G</C>, two isomorphic subgroups
<C>H1,H2</C> of <C>G</C>, with the isomorphism and its inverse on the loops.
The presentation of the extension has one more generator than that of <C>G</C>
and corresponds to the generator <M>z</M>.
<P/>
The functions <C>GraphOfGroupsRewritingSystem</C> and <C>NormalFormGGRWS</C>
may be applied to hnn-groups as well as to fpa-groups.
<P/>
In the example we take <C>G=a4</C> and the two subgroups are cyclic groups
of order 3.
</Description>
</ManSection>
<Example>
<![CDATA[
gap> H3 := Subgroup(a4,[b2]);;
gap> i23 := GroupHomomorphismByImages( H2, H3, [b1], [b2] );;
gap> hnn := HnnExtension( a4, i23 );
<fp group on the generators [ fe1, fe2, fe3 ]>
gap> phnn := PresentationFpGroup( hnn );;
gap> TzPrint( phnn );
#I  generators: [ fe1, fe2, fe3 ]
#I  relators:
#I  1.  3  [ 1, 1, 1 ]
#I  2.  3  [ 2, 2, 2 ]
#I  3.  4  [ 1, 2, 1, 2 ]
#I  4.  4  [ -3, 1, 3, -2 ]
gap> gg2 := GraphOfGroupsRewritingSystem( hnn );
Graph of Groups: 1 vertices; 2 arcs; groups [ a4 ]
gap> LeftTransversalsOfGraphOfGroups( gg2 );
[ [ <identity ...>, b2^-1, b1^-1*b2^-1, b1*b2^-1 ],
  [ <identity ...>, b1^-1, b1, b2^-1*b1 ] ]
gap> gh := GeneratorsOfGroup( hnn );;
gap> w3 := (gh[1]^gh[2])*gh[3]^-1*(gh[1]*gh[3]*gh[2]^2)^2*gh[3]*gh[2];
fe2^-1*fe1*fe2*fe3^-1*(fe1*fe3*fe2^2)^2*fe3*fe2
gap> n3 := NormalFormGGRWS( hnn, w3 );
(fe2*fe1*fe3)^2
]]>
</Example>

As with fpa-groups, hnn-groups are provided with a record attribute,
<C>HnnExtensionInfo(hnn)</C>,
storing the group, subgroups and isomorphism involved in their construction.
<P/>
<Example>
<![CDATA[
gap> hnninfo := HnnExtensionInfo( hnn );
rec( embeddings := [ [ b1, b2 ] -> [ fe1, fe2 ] ], group := a4,
  isomorphism := [ b1 ] -> [ b2 ],
  subgroups := [ Group([ b1 ]), Group([ b2 ]) ] )
gap> emb := Embedding( hnn, 1 );
[ b1, b2 ] -> [ fe1, fe2 ]
gap> ImageElm( emb, b1^b2 );
fe2^-1*fe1*fe2
gap> ReducedImageElm( emb, b1^b2 );
fe2*fe1^-1
]]>
</Example>

</Section>

<Section Label="sec-gphgpds">
<Heading>GraphsOfGroupoids and their Words</Heading>

<ManSection>
   <Oper Name="GraphOfGroupoids"
         Arg="dig, gpds, subgpds, isos" />
   <Prop Name="IsGraphOfPermGroupoids"
         Arg="gg" />
   <Prop Name="IsGraphOfFpGroupoids"
         Arg="gg" />
   <Attr Name="GroupoidsOfGraphOfGroupoids"
         Arg="gg" />
   <Attr Name="DigraphOfGraphOfGroupoids"
         Arg="gg" />
   <Attr Name="SubgroupoidsOfGraphOfGroupoids"
         Arg="gg" />
   <Attr Name="IsomorphismsOfGraphOfGroupoids"
         Arg="gg" />
   <Attr Name="RightTransversalsOfGraphOfGroupoids"
         Arg="gg" />
   <Attr Name="LeftTransversalsOfGraphOfGroupoids"
         Arg="gg" />
   <Filt Name="IsGraphOfGroupoids"
         Arg='dig' Type='Category'/>
<Description>
Graphs of groups generalise naturally to graphs of groupoids,
forming the class <C>IsGraphOfGroupoids</C>.
There is now a groupoid at each vertex and the isomorphism on an arc
identifies wide subgroupoids at the tail and at the head.
Since all subgroupoids are wide,
every groupoid in a connected constituent of the graph has
the same number of objects, but there is no requirement that the
object sets are all the same.
<P/>
The example below generalises the trefoil group example in subsection 4.4.1,
taking at each vertex of <C>D1</C> a two-object groupoid with a free group
on one generator, and full subgroupoids with groups
<M>\langle a^3 \rangle</M> and <M>\langle b^2 \rangle</M>.
<P/>
</Description>
</ManSection>
<Example>
<![CDATA[
gap> Gfa := SinglePieceGroupoid( fa, [-2,-1] );;
gap> ofa := One( fa );;
gap> SetName( Gfa, "Gfa" );
gap> Uhy := Subgroupoid( Gfa, [ [ hy, [-2,-1] ] ] );;
gap> SetName( Uhy, "Uhy" );
gap> Gfb := SinglePieceGroupoid( fb, [-4,-3] );;
gap> ofb := One( fb );;
gap> SetName( Gfb, "Gfb" );
gap> Uhybar := Subgroupoid( Gfb, [ [ hybar, [-4,-3] ] ] );;
gap> SetName( Uhybar, "Uhybar" );
gap> gens := GeneratorsOfGroupoid( Uhy );;
gap> gensbar := GeneratorsOfGroupoid( Uhybar );;
gap> mory := GroupoidHomomorphismFromSinglePiece(
>                Uhy, Uhybar, gens, gensbar );
groupoid homomorphism : Uhy -> Uhybar
[ [ [a^3 : -2 -> -2], [<identity ...> : -2 -> -1] ],
  [ [b^2 : -4 -> -4], [<identity ...> : -4 -> -3] ] ]
gap> morybar := InverseGeneralMapping( mory );
groupoid homomorphism : Uhybar -> Uhy
[ [ [b^2 : -4 -> -4], [<identity ...> : -4 -> -3] ],
  [ [a^3 : -2 -> -2], [<identity ...> : -2 -> -1] ] ]
gap> gg3 := GraphOfGroupoids( D1, [Gfa,Gfb], [Uhy,Uhybar], [mory,morybar] );;
gap> Display( gg3 );
Graph of Groupoids with :-
    vertices: [ 5, 6 ]
        arcs: [ [ y, 5, 6 ], [ y^-1, 6, 5 ] ]
   groupoids:
fp single piece groupoid: Gfa
  objects: [ -2, -1 ]
    group: fa = <[ a ]>
fp single piece groupoid: Gfb
  objects: [ -4, -3 ]
    group: fb = <[ b ]>
subgroupoids: single piece groupoid: Uhy
  objects: [ -2, -1 ]
    group: hy = <[ a^3 ]>
single piece groupoid: Uhybar
  objects: [ -4, -3 ]
    group: hybar = <[ b^2 ]>
isomorphisms: [ groupoid homomorphism : Uhy -> Uhybar
    [ [ [a^3 : -2 -> -2], [<identity ...> : -2 -> -1] ],
      [ [b^2 : -4 -> -4], [<identity ...> : -4 -> -3] ] ],
  groupoid homomorphism : Uhybar -> Uhy
    [ [ [b^2 : -4 -> -4], [<identity ...> : -4 -> -3] ],
      [ [a^3 : -2 -> -2], [<identity ...> : -2 -> -1] ] ] ]
gap> IsGraphOfGroupoids( gg3 );
true
]]>
</Example>

<ManSection>
   <Oper Name="GraphOfGroupoidsWord"
         Arg="gg, tv, list" />
   <Prop Name="IsGraphOfGroupoidsWord"
         Arg="w" />
   <Attr Name="GraphOfGroupoidsOfWord"
         Arg="w" />
   <Attr Name="WordOfGraphOfGroupoidsWord"
         Arg="w" />
   <Oper Name="ReducedGraphOfGroupoidsWord"
         Arg="w" />
   <Prop Name="IsReducedGraphOfGroupoidsWord"
         Arg="w" />
<Description>
Having produced the graph of groupoids <C>gg3</C>,
we may construct left coset representatives;
choose a graph of groupoids word; and reduce this to normal form.
Analogous to the word <M>a^7b^{-6}a^{-11}b^9a^7</M> in subsection
<Ref Oper="ReducedGraphOfGroupsWord"/> we shall consider
<Display>
(a^7 : -1 \to -2)~(b^{-6} : -4 \to -4 )~(a^{-11} : -2 \to -1)~
(b^9 : -3 \to -4)~(a^7 : -2 \to -1).
</Display>
Compare the normal form <C>nw3</C> below with the normal form <C>nw1</C> above.
<P/>
</Description>
</ManSection>
<Example>
<![CDATA[
gap> f1 := Arrow( Gfa, a^7, -1, -2);;
gap> f2 := Arrow( Gfb, b^-6, -4, -4 );;
gap> f3 := Arrow( Gfa, a^-11, -2, -1 );;
gap> f4 := Arrow( Gfb, b^9, -3, -4 );;
gap> f5 := Arrow( Gfa, a^7, -2, -2 );;
gap> L3 := [ f1, 1, f2, 2, f3, 1, f4, 2, f5 ];
[ [a^7 : -1 -> -2], 1, [b^-6 : -4 -> -4], 2, [a^-11 : -2 -> -1], 1,
  [b^9 : -3 -> -4], 2, [a^7 : -2 -> -2] ]
gap> gw3 := GraphOfGroupoidsWord( gg3, 5, L3);
(5)[a^7 : -1 -> -2].y.[b^-6 : -4 -> -4].y^-1.[a^-11 : -2 -> -1].y.[b^9 :
-3 -> -4].y^-1.[a^7 : -2 -> -2](5)
gap> nw3 := ReducedGraphOfGroupoidsWord( gw3 );
(5)[a^-1 : -1 -> -1].y.[b^-1 : -3 -> -3].y^-1.[a^10 : -1 -> -2](5)
]]>
</Example>

The reduction proceeds as follows.
<List>
<Item>
<M> [a^7 : -1 \to -2] = [a^{-2} : -1 \to -1]*[a^9 : -1 \to -2]
    \stackrel{y}{\to} [a^{-2} : -1 \to -1]*[b^6 : -3 \to -4] </M>
</Item>
<Item>
<M> [b^6 : -3 \to -4]*[b^{-6} : -4 \to -4] = [\mathrm{id} : -3 \to -4]
    \stackrel{\bar{y}}{\to} [\mathrm{id} : -1 \to -2] </M>
</Item>
<Item>
<M> [a^{-2} : -1 \to -1]*[\mathrm{id} : -1 \to -2]*[a^{-11} : -2 \to -1]
    = [a^{-13} : -1 \to -1] </M>
</Item>
<Item>
<M> [a^{-13} : -1 \to -1] = [a^{-1} : -1 \to -1]*[a^{-12} : -1 \to -1]
    \stackrel{y}{\to} [a^{-1} : -1 \to -1]*[b^{-8} : -3 \to -3] </M>
</Item>
<Item>
<M> [b^{-8} : -3 \to -3]*[b^9 : -3 \to -4]
    = [b^{-1} : -3 \to -3]*[b^2 : -3 \to -4]
    \stackrel{\bar{y}}{\to} [b^{-1} : -3 \to -3]*[a^3 : -1 \to -2] </M>
</Item>
<Item>
<M> [a^3 := -1 \to -2]*[a^7 : -2 \to -2] = [a^{10} : -1 \to -2] </M>
</Item>
</List>

So the resulting word is
<M>~(a^{-1} : -1, -1)(b^{-1} : -3, -3)(a^{10} : -1, -2)</M>.
Notice that all the arrows except the final one are loops.
</Section>

</Chapter>

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