<html><head><title>[automgrp] 3 Properties and operations with group and semigroup elements</title></head>
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<h1>3 Properties and operations with group and semigroup elements</h1><p>
<P>
<H3>Sections</H3>
<oL>
<li> <A HREF="CHAP003.htm#SECT001">Creation of tree automorphisms and homomorphisms</a>
<li> <A HREF="CHAP003.htm#SECT002">Properties and attributes of tree automorphisms and homomorphisms</a>
<li> <A HREF="CHAP003.htm#SECT003">Operations with tree automorphisms and homomorphisms</a>
<li> <A HREF="CHAP003.htm#SECT004">Elements of groups and semigroups defined by wreath recursion</a>
<li> <A HREF="CHAP003.htm#SECT005">Elements of contracting groups</a>
</ol><p>
<p>
In this chapter we present the functionality applicable to elements of groups and semigroups.
<p>
<p>
<h2><a name="SECT001">3.1 Creation of tree automorphisms and homomorphisms</a></h2>
<p><p>
<a name = "SSEC001.1"></a>
<li><code>TreeAutomorphism( </code><var>states</var><code>, </code><var>perm</var><code> ) O</code>
<p>
Constructs the tree automorphism with states on the first level given by the
argument <var>states</var> and acting
on the first level as the permutation <var>perm</var>. The <var>states</var> must
belong to the same family.
<pre>
gap> L := AutomatonGroup("p=(p,q)(1,2), q=(p,q)");
< p, q >
gap> r := TreeAutomorphism([p, q, p, q^2],(1,2)(3,4));
(p, q, p, q^2)(1,2)(3,4)
gap> t := TreeAutomorphism([q, 1, p*q, q],(1,2));
(q, 1, p*q, q)(1,2)
gap> r*t;
(p, q^2, p*q, q^2*p*q)(3,4)
</pre>
<p>
<a name = "SSEC001.2"></a>
<li><code>TreeHomomorphism( </code><var>states</var><code>, </code><var>tr</var><code> ) O</code>
<p>
Constructs an homomorphism with states <var>states</var> and acting
on the first level with transformation <var>tr</var>. The <var>states</var> must
belong to the same family.
<pre>
gap> S := AutomatonSemigroup("a=(a,b)[1,1],b=(b,a)(1,2)");
< a, b >
gap> x := TreeHomomorphism([a,b^2,a,a*b],Transformation([3,1,2,2]));
(a, b^2, a, a*b)[3,1,2,2]
gap> y := TreeHomomorphism([a*b,b,b,b^2],Transformation([1,4,2,3]));
(a*b, b, b, b^2)[1,4,2,3]
gap> x*y;
(a*b, b^2*a*b, a*b, a*b^2)[2,1,4,4]
</pre>
<p>
<a name = "SSEC001.3"></a>
<li><code>Representative( </code><var>word</var><code>, </code><var>fam</var><code> ) O</code>
<li><code>Representative( </code><var>word</var><code>, </code><var>a</var><code> ) O</code>
<p>
Given an associative word <var>word</var> constructs the tree homomorphism from the family
<var>fam</var>, or to which homomorphism <var>a</var> belongs. This function is useful when
one needs to make some operations with associative words. See also <code>Word</code> (<a href="CHAP003.htm#SSEC002.12">Word</a>).
<pre>
gap> L := AutomatonGroup("p=(p,q)(1,2), q=(p,q)");
< p, q >
gap> F := UnderlyingFreeGroup(L);
<free group on the generators [ p, q ]>
gap> r := Representative(F.1*F.2^2, p);
p*q^2
gap> Decompose(r);
(p*q^2, q*p^2)(1,2)
gap> H := SelfSimilarGroup("x=(x*y,x)(1,2), y=(x^-1,y)");
< x, y >
gap> F := UnderlyingFreeGroup(H);
<free group on the generators [ x, y ]>
gap> r := Representative(F.1^-1*F.2, x);
x^-1*y
gap> Decompose(r);
(x^-1*y, y^-1*x^-2)(1,2)
</pre>
<p>
<p>
<h2><a name="SECT002">3.2 Properties and attributes of tree automorphisms and homomorphisms</a></h2>
<p><p>
<a name = "SSEC002.1"></a>
<li><code>IsSphericallyTransitive( </code><var>a</var><code> ) P</code>
<p>
Returns whether the action of <var>a</var> is spherically transitive (see <a href="CHAP001.htm#SECT001">Short math background</a>).
<p>
<a name = "SSEC002.2"></a>
<li><code>IsTransitiveOnLevel( </code><var>a</var><code>, </code><var>lev</var><code> ) O</code>
<p>
Returns whether <var>a</var> acts transitively on level <var>lev</var> of the tree.
<p>
<a name = "SSEC002.3"></a>
<li><code>IsOne( </code><var>a</var><code> ) O</code>
<p>
Returns whether an automorphism <var>a</var> acts trivially on the tree. For contracting groups see also
<code>UseContraction</code> (<a href="CHAP002.htm#SSEC005.6">UseContraction</a>) and <code>IsOneContr</code> (<a href="CHAP003.htm#SSEC002.4">IsOneContr</a>).
<pre>
gap> L := AutomatonGroup("p=(p,q)(1,2), q=(p,q)");
< p, q >
gap> IsOne(q*p^-1*q*p^-1);
true
</pre>
<p>
<a name = "SSEC002.4"></a>
<li><code>IsOneContr( </code><var>a</var><code> ) F</code>
<p>
Returns <code>true</code> if <var>a</var> is trivial automorphism and <code>false</code> otherwise. Works for
contracting groups only. Uses polynomial time algorithm.
<p>
<a name = "SSEC002.5"></a>
<li><code>Order( </code><var>a</var><code> ) O</code>
<p>
Computes the order of an automorphism <var>a</var>. In some cases it does not stop. Works always (modulo memory
restrictions) for groups generated by bounded automata.
<p>
If <code>InfoLevel</code> of <code>InfoAutomGrp</code> is greater than or equal to 3 (one can set it by
<code>SetInfoLevel( InfoAutomGrp, 3)</code>) and the element has infinite order, then the proof of this fact
is printed.
<p>
<pre>
gap> L := AutomatonGroup("p=(p,q)(1,2), q=(p,q)");
< p, q >
gap> Basilica := AutomatonGroup( "u=(v,1)(1,2), v=(u,1)" );
< u, v >
gap> Order(p*q^-1);
2
gap> SetInfoLevel( InfoAutomGrp, 3);
gap> Order( u^35*v^-12*u^2*v^-3 );
#I (u^35*v^-12*u^2*v^-3)^68719476736 has conjugate of u^2*v^-3*u^35*v^
-12 as a section at vertex [ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ]
infinity
</pre>
<p>
<a name = "SSEC002.6"></a>
<li><code>OrderUsingSections( </code><var>a</var><code>[, </code><var>max_depth</var><code>] ) O</code>
<p>
Tries to compute the order of the element <var>a</var> by looking at its sections
of depth up to <var>max_depth</var>-th level.
If <var>max_depth</var> is omitted it is assumed to be <code>infinity</code>, but then it may not stop. Also note,
that if <var>max_depth</var> is not given, it searches the tree in depth first and may be trapped
in some infinite ray, while specifying finite <var>max_depth</var> may produce a result by looking at
a section not in that ray.
For bounded automata it will always produce a result.
<p>
If <code>InfoLevel</code> of <code>InfoAutomGrp</code> is greater than
or equal to 3 (one can set it by <code>SetInfoLevel( InfoAutomGrp, 3)</code>)
and the element has infinite order, then the proof of this fact is printed.
<p>
<pre>
gap> Grigorchuk_Group := AutomatonGroup("a=(1,1)(1,2),b=(a,c),c=(a,d),d=(1,b)");
< a, b, c, d >
gap> OrderUsingSections( a*b*a*c*b );
16
gap> Basilica := AutomatonGroup( "u=(v,1)(1,2), v=(u,1)" );
< u, v >
gap> SetInfoLevel( InfoAutomGrp, 3);
gap> OrderUsingSections( u^23*v^-2*u^3*v^15, 10 );
#I v^13*u^15 acts transitively on levels and is obtained from (u^23*v^-2*u^3*v^15)^1
by taking sections and cyclic reductions at vertex [ 1 ]
infinity
gap> G := AutomatonGroup("a=(c,a)(1,2), b=(b,c), c=(b,a)");
< a, b, c >
gap> OrderUsingSections(b,10);
#I b*c*a^2*b^2*c*a acts transitively on levels and is obtained from (b)^8
by taking sections and cyclic reductions at vertex
[ 2, 2, 1, 1, 1, 1, 2, 2, 1, 1 ]
infinity
</pre>
<p>
<a name = "SSEC002.7"></a>
<li><code>Perm( </code><var>a</var><code>[, </code><var>lev</var><code>] ) O</code>
<p>
Returns the permutation induced by the tree automorphism <var>a</var> on the level <var>lev</var>
(or first level if <var>lev</var> is not given). See also
<code>TransformationOnLevel</code> (<a href="CHAP003.htm#SSEC002.10">TransformationOnLevel</a>).
<p>
<a name = "SSEC002.8"></a>
<li><code>PermOnLevel( </code><var>a</var><code>, </code><var>k</var><code> ) O</code>
<p>
Does the same thing as <code>Perm</code> (<a href="CHAP003.htm#SSEC002.7">Perm</a>).
<p>
<a name = "SSEC002.9"></a>
<li><code>PermOnLevelAsMatrix( </code><var>g</var><code>, </code><var>lev</var><code> ) F</code>
<p>
Computes the action of the element <var>g</var> of a group on the <var>lev</var>-th level as a permutational matrix, in
which the i-th row contains 1 at the position i^<var>g</var>.
<pre>
gap> L := AutomatonGroup("p=(p,q)(1,2), q=(p,q)");
< p, q >
gap> PermOnLevel(p*q,2);
(1,4)(2,3)
gap> PermOnLevelAsMatrix(p*q, 2);
[ [ 0, 0, 0, 1 ], [ 0, 0, 1, 0 ], [ 0, 1, 0, 0 ], [ 1, 0, 0, 0 ] ]
</pre>
<p>
<a name = "SSEC002.10"></a>
<li><code>TransformationOnLevel( </code><var>a</var><code>, </code><var>lev</var><code> ) O</code>
<a name = "SSEC002.10"></a>
<li><code>TransformationOnFirstLevel( </code><var>a</var><code> ) O</code>
<p>
The first function returns the transformation induced by the tree homomorphism
<var>a</var> on the level <var>lev</var>. See also <code>PermOnLevel</code> (<a href="CHAP003.htm#SSEC002.8">PermOnLevel</a>).
<p>
If the transformation is invertible then it returns a permutation, and
<code>Transformation</code> otherwise.
<p>
<code>TransformationOnFirstLevel</code>(<var>a</var>) is equivalent to
<code>TransformationOnLevel</code>(<var>a</var>, <code>1</code>).
<p>
<a name = "SSEC002.11"></a>
<li><code>TransformationOnLevelAsMatrix( </code><var>g</var><code>, </code><var>lev</var><code> ) F</code>
<p>
Computes the action of the element <var>g</var> on the <var>lev</var>-th level as a permutational matrix, in
which the i-th row contains 1 at the position i^<var>g</var>.
<pre>
gap> L := AutomatonSemigroup("p=(p,q)(1,2), q=(p,q)[1,1]");
< p, q >
gap> TransformationOnLevel(p*q,2);
Transformation( [ 1, 1, 2, 2 ] )
gap> TransformationOnLevelAsMatrix(p*q,2);
[ [ 1, 0, 0, 0 ], [ 1, 0, 0, 0 ], [ 0, 1, 0, 0 ], [ 0, 1, 0, 0 ] ]
</pre>
<p>
<a name = "SSEC002.12"></a>
<li><code>Word( </code><var>a</var><code> ) O</code>
<p>
Returns <var>a</var> as an associative word (an element of the underlying free group) in
the generators of the self-similar group (semigroup) to which <var>a</var> belongs.
<pre>
gap> L := AutomatonGroup("p=(p,q)(1,2), q=(p,q)");
< p, q >
gap> w := Word(p*q^2*p^-1);
p*q^2*p^-1
gap> Length(w);
4
</pre>
<p>
<p>
<h2><a name="SECT003">3.3 Operations with tree automorphisms and homomorphisms</a></h2>
<p><p>
The multiplication of tree homomorphisms is defined in the standard way
<p>
<a name = "SSEC003.1"></a>
<li><code></code><var>a</var><code> * </code><var>b</var><code></code>
<p>
The following operations allow computation of the actions of tree
homomorphisms on letters and vertices
<p>
<a name = "SSEC003.2"></a>
<li><code></code><var>letter</var><code> ^ </code><var>a</var><code></code>
<a name = "SSEC003.2"></a>
<li><code></code><var>vertex</var><code> ^ </code><var>a</var><code></code>
<p>
<pre>
gap> L := AutomatonGroup("p=(p,q)(1,2), q=(p,q)");
< p, q >
gap> 1^p;
2
gap> [1, 2, 2, 1, 2, 1]^(p*q^2);
[ 2, 1, 2, 2, 1, 2 ]
</pre>
<p>
The operations below describe how to work with sections of tree homomorphisms.
<p>
<a name = "SSEC003.3"></a>
<li><code>Section( </code><var>a</var><code>, </code><var>v</var><code> ) O</code>
<p>
Returns the section of the automorphism (homomorphism) <var>a</var> at the vertex <var>v</var>.
The vertex <var>v</var> can be a list representing the vertex, or a positive integer
representing a vertex of the first level of the tree.
<pre>
gap> L := AutomatonGroup("p=(p,q)(1,2), q=(p,q)");
< p, q >
gap> Section(p*q*p^2, [1,2,2,1,2,1]);
p^2*q^2
</pre>
<p>
<a name = "SSEC003.4"></a>
<li><code>Sections( </code><var>a</var><code> [, </code><var>lev</var><code>] ) O</code>
<p>
Returns the list of sections of <var>a</var> at the <var>lev</var>-th level. If <var>lev</var> is omitted
it is assumed to be 1.
<pre>
gap> L := AutomatonGroup("p=(p,q)(1,2), q=(p,q)");
< p, q >
gap> Sections(p*q*p^2);
[ p*q^2*p, q*p^2*q ]
</pre>
<p>
<a name = "SSEC003.5"></a>
<li><code>Decompose( </code><var>a</var><code>[, </code><var>k</var><code>] ) O</code>
<p>
Returns the decomposition of the tree homomorphism <var>a</var> on the <var>k</var>-th level of the tree, i.e. the
representation of the form <br clear="all" /><table border="0" width="100%"><tr><td><table align="center" cellspacing="0" cellpadding="2"><tr><td nowrap="nowrap" align="center"><i>a</i> = (<i>a</i><sub>1</sub>, <i>a</i><sub>2</sub>, …, <i>a</i><sub><i>d</i><sub>1</sub>×·.·×<i>d</i><sub><i>k</i></sub></sub>)σ</td></tr></table></td></tr></table>
where <i>a</i><sub><i>i</i></sub> are the sections of <var>a</var> at the <var>k</var>-th level, and σ is the
transformation of the <var>k</var>-th level. If <var>k</var> is omitted it is assumed to be 1.
<pre>
gap> L := AutomatonGroup("p=(p,q)(1,2), q=(p,q)");
< p, q >
gap> Decompose(p*q^2);
(p*q^2, q*p^2)(1,2)
gap> Decompose(p*q^2,3);
(p*q^2, q*p^2, p^2*q, q^2*p, p*q*p, q*p*q, p^3, q^3)(1,8,3,5)(2,7,4,6)
</pre>
<p>
<a name = "SSEC003.6"></a>
<li><code></code><var>a</var><code> in </code><var>G</var><code></code>
<p>
Returns whether the automorphism <var>a</var> belongs to the group <var>G</var>. In some cases it does not stop.
<pre>
gap> L := AutomatonGroup("p=(p,q)(1,2), q=(p,q)");
< p, q >
gap> H := Group([p^2, q^2]);
< p^2, q^2 >
gap> p in H;
false
</pre>
<p>
<a name = "SSEC003.7"></a>
<li><code>OrbitOfVertex( </code><var>ver</var><code>, </code><var>g</var><code>[, </code><var>n</var><code>] ) O</code>
<p>
Returns the list of vertices in the orbit of the vertex <var>ver</var> under the
action of the semigroup generated by the automorphism <var>g</var>.
If <var>n</var> is specified, it returns only the first <var>n</var> elements of the orbit.
Vertices are defined either as lists with entries from {1,…,<i>d</i>}, or as
strings containing characters 1,…,<i>d</i>, where <i>d</i>
is the degree of the tree.
<pre>
gap> T := AutomatonGroup("t=(1,t)(1,2)");
< t >
gap> OrbitOfVertex([1,1,1], t);
[ [ 1, 1, 1 ], [ 2, 1, 1 ], [ 1, 2, 1 ], [ 2, 2, 1 ], [ 1, 1, 2 ],
[ 2, 1, 2 ], [ 1, 2, 2 ], [ 2, 2, 2 ] ]
gap> OrbitOfVertex("11111111111", t, 6);
[ [ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ], [ 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ],
[ 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1 ], [ 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1 ],
[ 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1 ], [ 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1 ] ]
</pre>
<p>
<a name = "SSEC003.8"></a>
<li><code>PrintOrbitOfVertex( </code><var>ver</var><code>, </code><var>g</var><code>[, </code><var>n</var><code>] ) O</code>
<p>
Prints the orbit of the vertex <var>ver</var> under the action of the semigroup generated by
<var>g</var>. Each vertex is printed as a string containing characters 1,…,<i>d</i>, where <i>d</i>
is the degree of the tree. In case of binary tree the symbols `` '' and ``<code>x</code>''
are used to represent <code>1</code> and <code>2</code>.
If <var>n</var> is specified only the first <var>n</var> elements of the orbit are printed.
Vertices are defined either as lists with entries from {1,…,<i>d</i>}, or as
strings. See also <code>OrbitOfVertex</code> (<a href="CHAP003.htm#SSEC003.7">OrbitOfVertex</a>).
<pre>
gap> L := AutomatonGroup("p=(p,q)(1,2), q=(p,q)");
< p, q >
gap> PrintOrbitOfVertex("2222222222222222222222222222222", p*q^-2, 6);
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
x x x x x x x x x x x x x x x
x xx xx xx xx xx xx xx
x x x x x x x
xxx xxxx xxxx xxxx
x x x x x x x
gap> H := AutomatonGroup("t=(s,1,1)(1,2,3), s=(t,s,t)(1,2)");
< t, s >
gap> PrintOrbitOfVertex([1,2,1], s^2);
121
132
123
131
122
133
</pre>
<p>
<a name = "SSEC003.9"></a>
<li><code>PermActionOnLevel( </code><var>perm</var><code>, </code><var>big_lev</var><code>, </code><var>sm_lev</var><code>, </code><var>deg</var><code> ) F</code>
<p>
Given a permutation <var>perm</var> on the <var>big_lev</var>-th level of the tree of degree
<var>deg</var> returns the permutation induced by <var>perm</var> on a smaller level
<var>sm_lev</var>.
<pre>
gap> PermActionOnLevel((1,4,2,3), 2, 1, 2);
(1,2)
gap> PermActionOnLevel((1,13,5,9,3,15,7,11)(2,14,6,10,4,16,8,12), 4, 2, 2);
(1,4,2,3)
</pre>
<p>
<p>
<h2><a name="SECT004">3.4 Elements of groups and semigroups defined by wreath recursion</a></h2>
<p><p>
<a name = "SSEC004.1"></a>
<li><code>IsFiniteState( </code><var>a</var><code> ) P</code>
<p>
Returns <code>true</code> if <var>a</var> has finitely many different sections.
It will never stop if the free reduction of words is not sufficient
to establish the finite-state property or if <var>a</var> is not finite-state (has
infinitely many different sections).
<p>
See also <code>AllSections</code> (<a href="CHAP003.htm#SSEC004.2">AllSections</a>) for the list of all sections and
<code>MealyAutomaton</code> (<a href="CHAP004.htm#SSEC001.1">MealyAutomaton</a>), which allows to construct
a Mealy automaton whose states are the sections of <var>a</var> and which
encodes its action on the tree.
<pre>
gap> D := SelfSimilarGroup("x=(1,y)(1,2), y=(z^-1,1)(1,2), z=(1,x*y)");
< x, y, z >
gap> IsFiniteState(x*y^-1);
true
</pre>
<p>
<a name = "SSEC004.2"></a>
<li><code>AllSections( </code><var>a</var><code> ) A</code>
<p>
Returns the list of all sections of <var>a</var> if there are finitely many of them and
this fact can be established using free reduction of words in sections. Otherwise
will never stop. Note, that in the case when <var>a</var> is an element of a self-similar
(semi)group defined by wreath recurion it does not check whether all elements of the list
are actually different automorphisms (homomorphisms) of the tree. If <var>a</var> is a element of
of a (semi)group generated by finite automaton, it will always return the list of
all distinct sections of <var>a</var>.
<pre>
gap> D := SelfSimilarGroup("x=(1,y)(1,2), y=(z^-1,1)(1,2), z=(1,x*y)");
< x, y, z >
gap> AllSections(x*y^-1);
[ x*y^-1, z, 1, x*y, y*z^-1, z^-1*y^-1*x^-1, y^-1*x^-1*z*y^-1, z*y^-1*x*y*z,
y*z^-1*x*y, z^-1*y^-1*x^-1*y*z^-1, x*y*z, y, z^-1, y^-1*x^-1, z*y^-1 ]
</pre>
<p>
See also operation <code>MealyAutomaton</code> (<a href="CHAP004.htm#SSEC001.1">MealyAutomaton</a>), which allows to construct
a Mealy automaton whose states are the sections of given tree homomorphism and which
encodes its action on the tree.
<p>
<p>
<h2><a name="SECT005">3.5 Elements of contracting groups</a></h2>
<p><p>
<a name = "SSEC005.1"></a>
<li><code>AutomPortrait( </code><var>a</var><code> ) F</code>
<a name = "SSEC005.1"></a>
<li><code>AutomPortraitBoundary( </code><var>a</var><code> ) F</code>
<a name = "SSEC005.1"></a>
<li><code>AutomPortraitDepth( </code><var>a</var><code> ) F</code>
<p>
Constructs the portrait of an element <var>a</var> of a
contracting group <i>G</i>. The portrait of <var>a</var> is defined recursively as follows.
For <i>g</i> in the nucleus of <i>G</i> the portrait is just [<i>g</i>]. For any other
element <i>g</i>=(<i>g</i><sub>1</sub>,<i>g</i><sub>2</sub>,…,<i>g</i><sub><i>d</i></sub>)σ the portrait of <i>g</i> is
[σ, <tt>AutomPortrait</tt>(<i>g</i><sub>1</sub>),…, <tt>AutomPortrait</tt>(<i>g</i><sub><i>d</i></sub>)], where <i>d</i> is
the degree of the tree. This structure describes a finite tree whose inner vertices
are labelled by permutations from <i>S</i><sub><i>d</i></sub> and the leaves are labelled by
elements from the nucleus. The contraction in <i>G</i> guarantees that the
portrait of any element is finite.
<p>
The portraits may be considered as ``normal forms''
of the elements of <i>G</i>, since different elements have different portraits.
<p>
One also can be interested only in the boundary of a portrait, which consists
of all leaves of the portrait. This boundary can be described by an ordered set of
pairs [<i>level</i><sub><i>i</i></sub>, <i>g</i><sub><i>i</i></sub>], <i>i</i>=1,…,<i>r</i> representing the leaves of the tree ordered from left
to right (where <i>level</i><sub><i>i</i></sub> and <i>g</i><sub><i>i</i></sub> are the level and the label of the <i>i</i>-th leaf
correspondingly, <i>r</i> is the number of leaves). The operation <code>AutomPortraitBoundary</code>(<var>a</var>)
computes this boundary.
<p>
<code>AutomPortraitDepth</code>( <var>a</var> ) returns the depth of the portrait, i.e. the minimal
level such that all sections of <var>a</var> at this level belong to the nucleus of <i>G</i>.
<p>
<pre>
gap> Basilica := AutomatonGroup("u=(v,1)(1,2), v=(u,1)");
< u, v >
gap> AutomPortrait(u^3*v^-2*u);
[ (), [ (), [ (), [ v ], [ v ] ], [ 1 ] ],
[ (), [ (), [ v ], [ u^-1*v ] ], [ v^-1 ] ] ]
gap> AutomPortrait(u^3*v^-2*u^3);
[ (), [ (), [ (1,2), [ (), [ (), [ v ], [ v ] ], [ 1 ] ], [ v ] ], [ 1 ] ],
[ (), [ (1,2), [ (), [ (), [ v ], [ v ] ], [ 1 ] ], [ u^-1*v ] ], [ v^-1 ]
] ]
gap> AutomPortraitBoundary(u^3*v^-2*u^3);
[ [ 5, v ], [ 5, v ], [ 4, 1 ], [ 3, v ], [ 2, 1 ], [ 5, v ], [ 5, v ],
[ 4, 1 ], [ 3, u^-1*v ], [ 2, v^-1 ] ]
gap> AutomPortraitDepth(u^3*v^-2*u^3);
5
</pre>
<p>
<p>
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<P>
<address>automgrp manual<br>January 2025
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