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<div class="ChapSects"><a href="chap2.html#X7ADCE68284FB4ACF">2 FR package</a>
<div <div class classSimpleMathW/> denote automorphismgroup ofthegraph<spanclass"SimpleMath">X^*<span.Given< class=SimpleMath> ∈W</pan, we restrict itsaction ⊂*/span> and> >/spanthe>activity/m>of< class""a<span. WeWemayalsoobtainforall =""x X/span,a automorphism<spanclass="">a_x W/>, called <emstate <spanclass"impleMath>a> <a href="chap2.html#X78DF4DE18260BD80">2.2 An example session</a>

</div>
</div>

<h3>2 FR package</h3>

<a id="X80C332C686212786

<>.1

 the <emWreath</m>The of span="SimpleMath"></pan the permutational product span="SimpleMath">^ ⋊(X)</span

We shall here be interested in groups G defined by their action on a regular rooted tree. Let X be a finite set; and let Let W denote the automorphism group of the graph X^*. Given a ∈ W, we may restrict its action to X ⊂ X^*(v){a_x} = w \quad\textrm{if}\quad (xv)a = x^{\pi_a}w.

The data (a_x,π_a)java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

\phi: W \to W\wr \mathop{Sym}(X),

called the emWreath</em.The of the permutational wreath <span="SimpleMath"WX ⋊ Sym(X)<span</>

<e <spanclassSimpleMatha_xspan should beinterpretedas the of action < class"SimpleMath">a/span on subtree <panclass"SimpleMath">xX^*</>; theautomorphismspanclass"SimpleMath">axX*</pan> by itsrespective, and thenpermuting these according tospanclass"impleMath>πa. The wreath recursion can be on the of ">a, to define states< class="SimpleMath<>foranyspan""v∈^<>The automorphism a ∈ W mayp> followingimportantproperties have also considered.A subgroup <spanclass"SimpleMath>G Wis all the G-subsets .It is weakly branched/> ifit is level-transitive,andforevery>∈X*/span there non-trivial<spanclassSimpleMath>a_v G/spanthat span=SimpleMath>^ ∖ vX*/> It em/>if eachspan=SimpleMath> ∈</> the generatedbyallsuchspanclass"SimpleMath"a_vspan allspanclassSimpleMathv/>of spanclassSimpleMath>n/span has < classSimpleMath">G./>

<>his may beconvenientlyencodedinwhatis a <emMoore</em:itconsistsofa setspanclass"">Q/>,the setof graph;analphabet <span="SimpleMath">X</span; a`transition' function span class="impleMath"ϕQ X-&t; Q,where ϕq,) is the endpoint oftheedgestarting span class="SimpleMath"> andlabeled x labeling< class"SimpleMath>"< group on We willuse theequivalent machinesϕQ× X-> X× Q/span> encoding both ϕ and together.

Of particular interest are finite-state automorphisms: these are automorphisms whose Mealy machine has finitely many states. The product and inverse of finite-state automorphisms is again finite-state.

A subgroup G ≤ W is self-similar if G^ϕ ⊂ G≀Sym(X). This is equivalent to asking

pThe importantproperties have also considered.A subgroup< class=SimpleMath> ≤W</> <m>level-transitive/> action transitiveonallthespan"">G span=SimpleMath>^ifis,and everyspan="">v∈X^</spanthereanon-trivial<spanclass"SimpleMath"a_v Gthatfixesspan=""X*∖vX</pan>. Itis<embranched class="SimpleMath">n ∈ Na_vv/span oflength< class"SimpleMath"></pan finitespan=">G.

A self-similar finitely generated group G ≤ W

java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

We assumed for simplicity that the elements a were invertiblejava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<a id="X78DF4DE18260BD80>gap> LoadPackage("fr");

<h4>2.2 An example session</h4>

ThisLoading FR085714p5Functionallyrecursive automata groups

The package---------------------------------

<<pre></iv
<span java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
---------------------------------------------------
Loading FR 0.857142p5 (Functionally recursive and automata groups)
by Laurent Bartholdi (http://www.uni-math.gwdg.de/laurent)
----------------------------------------------------------------
true
/pre></div>

Many FR groups are predefined by FR, see Chapter <a href="chap9.html#X7A489A5D79DA9E5C">We

div=">
< classgap<> < classB:=("&;1,b&;1,2""=;1,a&;",IsFRMealyElement>
< class>>span="">AssignGeneratorVariablesgap Assigned global variables a,b ]
#I Assigned the global variables [ a, b ]
</pre</iv

We have just created the<>Note that group is ascode="code">BasilicaGroupcode.We compute the of the:

<>Note this group predefined <code classcode>asilicaGroupcode. We now the decompositions the generators</>

<[ [ <2|identity..>, <2|>], 2, 1]]
gap> DecompositionOfFRElement(a); DecompositionOfFRElement(b);
 &;2|identity ...&;, <|bgt][2 ]java.lang.StringIndexOutOfBoundsException: Index 53 out of bounds for length 53
[&;2|identity ..&;, <2agt, [1 2]]
</pre></div>

Elements are described as words in the generators; theyjava.lang.StringIndexOutOfBoundsException: Range [0, 59) out of bounds for length 0

The optional argument <code class="func">IsFRElement</code> (<a href="chap10.html#X7966F9B982B1DFE1">10.2-11</a>) tells FR to store elements in this way. This representation is always

Elements act on sequences

<div classexample<>
gap> 1^a; [1]^a; [1,1]^a;
2<spanclass"GAPprompt"gapgt/span> </div>

Periodic sequences are also implemented in FR; they are constructed by giving the period and preperiod. The period is printed by preceding it with a "/":

<div class="example"
<span="GAPprompt">gapgt; <spanclass"GAPinput">v := PeriodicList[[];</pan
[ 1java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
gap> v^a; v^java.lang.StringIndexOutOfBoundsException: Range [69, 70) out of bounds for length 26
/]
[/ 1, 2 ]
gap> last{[1..10]};
[ 1, 2, [ 1 /2]
</pre></div>

Most much efficient <span="">B/span>s elementsare to> representation/m,/

<div class="example">/1 ]
< class>ap;</> <span=""> =IsomorphismMealyGroup>
<recursive group over [ 1 . 1 2,1 2, 1, 2 ,2,1,2]
gap> a := Bm<pre</div>
&;Mealy elementon [ 1, 2] withstates;
<Mealy element on alphabet [ 1, 2java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
</pre>/div>

<pan classGAPprompt>ap;a : Bm;b : Bm;</pan

The group;Mealy element on [ 1, 2 with 3 states;

<div class="example"><pre>
<span="GAPprompt">gap; span="GAPinput">IsBoundedFRSemigroup);gap> Order(a); Order(b);
infinity
infinity
gap class=GAPpromptgapgt; IsBoundedFRSemigroupBmgapgt/> <span="GAPinput">Orderg) timespan
infinity
2599
</pre></div>

pThe < classSimpleMath>B isweakly; derived <spanclass="">B'/span>contains Toprovethat it to check[,] 1∈B'/span> and 1 ×[a,]∈B'/>. These are constructed using><pre> <span=GAPprompt&;</> <span="">c: (ab;/> < </div ver. with 3generators; gap> <java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0 true true <span;Mealy on [ 1 9 states; true <n ="">>> spanG"VertexElement(c=Commba^ true </pre></div> <ptrue We may also check the relations [b^p,(b^p)^a^p]=1 and java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4 <We check < classSimpleMath^(^^^]1/> and="SimpleMath">2(^p^^]/span /> power class<>:/> gap> ForAll([0..10],i->IsOne(Comm(b^(2^i),(b^(2^i))^((a^(2^i)) true 1361 </pre></div> p group> , compute <div1 <pan =""gap =>(B);/> [ <2|identity </pre/> java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 0 <div class="example"><pre> gap> N := NucleusMachine(B); &[ <|identity..>&;2|b>,&;2|b^-> &;2|agt<|^-1gtlt^-*&;, <^1b&; ] <spanclass="APprompt>> GAPinput">(N); </pre></div> We may also draw powers of the dual automaton: these are approximations to the Schreier graph of gap> Draw(N)^3;/pan classGAPinput =(FRMachine1 < classDraw(M^)</div> These Schreier may powers the dual automaton: these are approximations to theSchreier ofspan="B/>. ,also construct smallerMealy spanclass="a/>andSimpleMath> givep> <div < class"example"<re span=GAPprompt>> span">(Bpoint=1111,); pjava.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 12 More class>>> spanG"DualMachineM^) <div class="example"><pre> gap> </> We may>ore properties of BB</pan length < ="ode">PermGroup(Bn)</code> constructsa permutation group which the quotient of class"SimpleMath"B2n : <><pre <class"gapgtGAPinput">J := JenningsLieAlgebra(;time; <Lie algebra of dimension 88 over3948092567715 18035 The " <div classspanclass=GAPpromptgapspan=>([1..1]>((J.(i));/> <div class="chlinkbot">Goto Chapter: <a href="chap0.html">Top</a> <a href="chap1.html">1</a>java.lang.StringIndexOutOfBoundsException: Index 124 out of bounds for length 12 <hr /> quality99% ¤ Dauer der Verarbeitung: 0.8 Sekunden ¤ *© Formatika GbR, Deutschland

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Cephes Mathematical Library

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