This chapter contains functions that are related to the discreteness property (D) presented in Proposition 3.12 of [Tor20].
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.
‣ 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
‣ 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
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.
‣ InvolutiveCompatibilityCocycle( F ) | ( attribute ) |
Returns: an involutive compatibility cocycle of F, which is a mapping F\(\times\)[1..d]\(\to\)F 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
‣ 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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