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]
\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
‣ 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
generated by GAPDoc2HTML