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5 Discreteness
 5.1 The discreteness condition (D)
 5.2 Discreteness
 5.3 Cocycles

5 Discreteness

This chapter contains functions that are related to the discreteness property (D) presented in Proposition 3.12 of [Tor20].

5.1 The discreteness condition (D)

Said proposition shows that for a given F\le \mathrm{Aut}(B_{d,k}) the group \mathrm{U}_{k}(F) is discrete if and only if the maximal compatible subgroup C(F) of F satisfies condition (D):

\forall \omega \in \Omega: F_{T_{\omega}}=\{\mathrm{id}\},

where T_{\omega} is the k-1-neighbourhood of the edge (b,b_{\omega}) inside B_{d,k}. In other words, F satisfies (D) if and only if the compatibility set C_{F}(\mathrm{id},\omega)=\{\mathrm{id}\}. We distinguish between F satisfying condition (D) and \mathrm{U}_{k}(F) being discrete with the methods SatisfiesD (5.2-1) and YieldsDiscreteUniversalGroup (5.2-2) below.

5.2 Discreteness

5.2-1 SatisfiesD
‣ SatisfiesD( F )( property )

Returns: true if F satisfies the discreteness condition (D), and false otherwise.

The argument of this attribute is a local action F \le\mathrm{Aut}(B_{d,k}) (see IsLocalAction (2.1-1)).

gap> G:=LocalActionGamma(3,SymmetricGroup(3));
Group([ (1,4,5)(2,3,6), (1,3)(2,4)(5,6) ])
gap> SatisfiesD(G);
true

5.2-2 YieldsDiscreteUniversalGroup
‣ YieldsDiscreteUniversalGroup( F )( property )

Returns: true if \mathrm{U}_{k}(F) is discrete, and false otherwise.

The argument of this attribute is a local action F \le\mathrm{Aut}(B_{d,k}) (see IsLocalAction (2.1-1)).

gap> G:=LocalActionGamma(3,SymmetricGroup(3));
Group([ (1,4,5)(2,3,6), (1,3)(2,4)(5,6) ])
gap> YieldsDiscreteUniversalGroup(G);
true
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

5.3 Cocycles

Subgroups F\le\mathrm{Aut}(B_{d,k}) that satisfy both (C) and (D) admit an involutive compatibility cocycle, i.e. a map z:F\times\{1,\ldots,d\}\to F that satisfies certain properties, see [Tor20, Section 3.2.2]. When F satisfies just (C), it may still admit an involutive compatibility cocycle. In this case, F admits an extension \Gamma_{z}(F)\le\mathrm{Aut}(B_{d,k}) that satisfies both (C) and (D). Involutive compatibility cocycles can be searched for using InvolutiveCompatibilityCocycle (5.3-1) and AllInvolutiveCompatibilityCocycles (5.3-2) below.

5.3-1 InvolutiveCompatibilityCocycle
‣ InvolutiveCompatibilityCocycle( F )( attribute )

Returns: an involutive compatibility cocycle of F, which is a mapping F\times[1..d]\toF with certain properties, if it exists, and fail otherwise. When k =1, the standard cocycle is returned.

The argument of this attribute is a local action F \le\mathrm{Aut}(B_{d,k}) (see IsLocalAction (2.1-1)), which is compatible (see SatisfiesC (3.3-2)).

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

5.3-2 AllInvolutiveCompatibilityCocycles
‣ AllInvolutiveCompatibilityCocycles( F )( attribute )

Returns: the list of all involutive compatibility cocycles of F.

The argument of this attribute is a local action F \le\mathrm{Aut}(B_{d,k}) (see IsLocalAction (2.1-1)), which is compatible (see SatisfiesC (3.3-2)).

gap> S3:=LocalAction(3,1,SymmetricGroup(3));;
gap> Size(AllInvolutiveCompatibilityCocycles(S3));
4
gap> Size(AllInvolutiveCompatibilityCocycles(LocalActionGamma(3,SymmetricGroup(3))));
1
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