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products/sources/formale Sprachen/GAP/pkg/ugaly/doc/ (Algebra von RWTH Aachen Version 4.15.1^©) Datei vom 7.6.2023 mit Größe 5 kB

Quelle _Chapter_Discreteness.xml Sprache: XML

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


<Chapter Label="Chapter_Discreteness">
<Heading>Discreteness</Heading>

<P/>
This chapter contains functions that are related to the discreteness property (D) presented in Proposition 3.12 of <Cite Key="Tor20"/>.
<Section Label="Section_condition_D">
<Heading>The discreteness condition (D)</Heading>

Said proposition shows that for a given <Math>F\le \mathrm{Aut}(B_{d,k})</Math> the group <Math>\mathrm{U}_{k}(F)</Math> is discrete if and only if the maximal compatible subgroup <Math>C(F)</Math> of <Math>F</Math> satisfies condition (D): <Display>\forall \omega \in \Omega: F_{T_{\omega}}=\{\mathrm{id}\},</Display> where <Math>T_{\omega}</Math> is the <Math>k-1</Math>-neighbourhood of the edge <Math>(b,b_{\omega})</Math> inside <Math>B_{d,k}</Math>. In other words, <Math>F</Math> satisfies (D) if and only if the compatibility set <Math>C_{F}(\mathrm{id},\omega)=\{\mathrm{id}\}</Math>.
We distinguish between <Math>F</Math> satisfying condition (D) and <Math>\mathrm{U}_{k}(F)</Math> being discrete with the methods <Ref Prop="SatisfiesD" Label="for IsLocalAction"/> and <Ref Prop="YieldsDiscreteUniversalGroup" Label="for IsLocalAction"/> below.
</Section>

<Section Label="Chapter_Discreteness_Section_Discreteness">
<Heading>Discreteness</Heading>

<ManSection>
  <Prop Arg="F" Name="SatisfiesD" Label="for IsLocalAction"/>
<Returns> <K>true</K> if <A>F</A> satisfies the discreteness condition (D), and <K>false</K> otherwise.

</Returns>
<Description>
The argument of this attribute is a local action <A>F</A> <Math>\le\mathrm{Aut}(B_{d,k})</Math> (see <Ref Filt="IsLocalAction" Label="for IsPermGroup"/>).
<P/>
</Description>
</ManSection>

<P/>
<Example><![CDATA[
gap> G:=LocalActionGamma(3,SymmetricGroup(3));
Group([ (1,4,5)(2,3,6), (1,3)(2,4)(5,6) ])
gap> SatisfiesD(G);
true
]]></Example>

<ManSection>
  <Prop Arg="F" Name="YieldsDiscreteUniversalGroup" Label="for IsLocalAction"/>
<Returns> <K>true</K> if <Math>\mathrm{U}_{k}(F)</Math> is discrete, and <K>false</K> otherwise.

</Returns>
<Description>
The argument of this attribute is a local action <A>F</A> <Math>\le\mathrm{Aut}(B_{d,k})</Math> (see <Ref Filt="IsLocalAction" Label="for IsPermGroup"/>).
<P/>
</Description>
</ManSection>

<P/>
<Example><![CDATA[
gap> G:=LocalActionGamma(3,SymmetricGroup(3));
Group([ (1,4,5)(2,3,6), (1,3)(2,4)(5,6) ])
gap> YieldsDiscreteUniversalGroup(G);
true
]]></Example>

<P/>
<Example><![CDATA[
gap> F:=LocalAction(3,2,Group((1,2)));
Group([ (1,2) ])
gap> YieldsDiscreteUniversalGroup(F);
true
gap> SatisfiesD(F);
false
gap> C:=MaximalCompatibleSubgroup(F);
Group(())
gap> SatisfiesD(C);
true
]]></Example>

</Section>

<Section Label="Chapter_Discreteness_Section_Cocycles">
<Heading>Cocycles</Heading>

Subgroups <Math>F\le\mathrm{Aut}(B_{d,k})</Math> that satisfy both (C) and (D) admit an involutive compatibility cocycle, i.e. a map <Math>z:F\times\{1,\ldots,d\}\to F</Math> that satisfies certain properties, see <Cite Key="Tor20" Where="Section 3.2.2"/>. When <Math>F</Math> satisfies just (C), it may still admit an involutive compatibility cocycle. In this case, F admits an extension <Math>\Gamma_{z}(F)\le\mathrm{Aut}(B_{d,k})</Math> that satisfies both (C) and (D). Involutive compatibility cocycles can be searched for using <Ref Attr="InvolutiveCompatibilityCocycle" Label="for IsLocalAction"/> and <Ref Attr="AllInvolutiveCompatibilityCocycles" Label="for IsLocalAction"/> below.
<ManSection>
  <Attr Arg="F" Name="InvolutiveCompatibilityCocycle" Label="for IsLocalAction"/>
<Returns>an involutive compatibility cocycle of <A>F</A>, which is a mapping <A>F</A><Math>\times</Math><C>[1..d]</C><Math>\to</Math><A>F</A> with certain properties, if it exists, and <K>fail</K> otherwise. When <A>k</A> <Math>=1</Math>, the standard cocycle is returned.

</Returns>
<Description>
The argument of this attribute is a local action <A>F</A> <Math>\le\mathrm{Aut}(B_{d,k})</Math> (see <Ref Filt="IsLocalAction" Label="for IsPermGroup"/>), which is compatible (see <Ref Prop="SatisfiesC" Label="for IsLocalAction"/>).
<P/>
</Description>
</ManSection>

<P/>
<Example><![CDATA[
gap> F:=LocalAction(3,1,AlternatingGroup(3));;
gap> z:=InvolutiveCompatibilityCocycle(F);;
gap> mt:=RandomSource(IsMersenneTwister,1);;
gap> a:=Random(mt,F);; dir:=Random(mt,[1..3]);;
gap> a; Image(z,[a,dir]);
(1,2,3)
(1,2,3)
]]></Example>

<P/>
<Example><![CDATA[
gap> G:=LocalActionGamma(3,AlternatingGroup(3));
Group([ (1,4,5)(2,3,6) ])
gap> InvolutiveCompatibilityCocycle(G) <> fail;
true
gap> InvolutiveCompatibilityCocycle(AutBall(3,2));
fail
]]></Example>

<ManSection>
  <Attr Arg="F" Name="AllInvolutiveCompatibilityCocycles" Label="for IsLocalAction"/>
<Returns>the list of all involutive compatibility cocycles of <Math>F</Math>.

</Returns>
<Description>
The argument of this attribute is a local action <A>F</A> <Math>\le\mathrm{Aut}(B_{d,k})</Math> (see <Ref Filt="IsLocalAction" Label="for IsPermGroup"/>), which is compatible (see <Ref Prop="SatisfiesC" Label="for IsLocalAction"/>).
<P/>
</Description>
</ManSection>

<P/>
<Example><![CDATA[
gap> S3:=LocalAction(3,1,SymmetricGroup(3));;
gap> Size(AllInvolutiveCompatibilityCocycles(S3));
4
gap> Size(AllInvolutiveCompatibilityCocycles(LocalActionGamma(3,SymmetricGroup(3))));
1
]]></Example>

</Section>

</Chapter>

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