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#
# Ugaly: Universal Groups Acting LocallY
#
# Implementations
#
##################################################################################################################
InstallMethod( SatisfiesD, "for a local action F", [IsLocalAction],
function(F)
local d, k, K, dir;
d:=LocalActionDegree(F);
k:=LocalActionRadius(F);
if k=0 then
return true;
elif k=1 then
return IsSemiRegular(F,[1..d]);
else
K:=Kernel(RestrictedMapping(Projection(AutBall(d,k)),F));
for dir in [1..d] do;
if not IsTrivial(Stabilizer(K,[(dir-1)*(d-1)^(k-1)+1..dir*(d-1)^(k-1)],OnTuples)) then
return false;
fi;
od;
return true;
fi;
end );
##################################################################################################################
InstallMethod( YieldsDiscreteUniversalGroup, "for a local action F", [IsLocalAction],
function(F)
local k, CF;
k:=LocalActionRadius(F);
if k=0 then
return true;
elif k=1 then
CF:=F;
else
CF:=MaximalCompatibleSubgroup(F);
fi;
return SatisfiesD(CF);
end );
##################################################################################################################
InstallGlobalFunction( IsCocycleViaGeneratorData,
function(F,c,pr,rel)
local d, r, dir;
d:=LocalActionDegree(F);
# suffices to check $z(r,dir)=1$ for all relations r and directions dir (?)
for r in rel do
for dir in [1..d] do
if not ()=EvaluateCocycleViaWords(F,c,pr,LetterRepAssocWord(r),dir) then return false; fi;
od;
od;
return true;
end );
##################################################################################################################
InstallGlobalFunction( IsInvolutiveViaGeneratorData,
function(F,c,pr)
local d, a, dir;
d:=LocalActionDegree(F);
# suffices to check on generators: z(z(ab,i),i)=z(z(a,bi)z(b,i),i)=z(z(a,bi),z(b,i)*i)z(z(b,i),i)=z(z(a,bi),bi)*b=a*b
for a in GeneratorsOfGroup(F) do
for dir in [1..d] do
if not EvaluateCocycleViaElements(F,c,pr,EvaluateCocycleViaElements(F,c,pr,a,dir),dir)=a then return false; fi;
od;
od;
return true;
end );
##################################################################################################################
InstallGlobalFunction( EvaluateCocycleViaWords,
function(F,c,pr,w,dir)
local d, k, gens, value, act, i;
d:=LocalActionDegree(F);
k:=LocalActionRadius(F);
gens:=GeneratorsOfGroup(F);
value:=();
act:=();
for i in [1..Length(w)] do
if SignInt(w[i])=1 then
value:=value*c[w[i]][dir^act];
else
# z(a^{-1},i)=z(a,a^{-1}i)^{-1}
value:=value*c[-w[i]][(dir^act)^(LocalAction(1,d,k,gens[-w[i]],[])^(-1))]^(-1);
fi;
act:=act*LocalAction(1,d,k,gens[AbsInt(w[i])],[])^SignInt(w[i]);
od;
return value;
end );
##################################################################################################################
InstallGlobalFunction( EvaluateCocycleViaElements,
function(F,c,pr,a,dir)
local d, k, gens, w, value, act, i;
d:=LocalActionDegree(F);
k:=LocalActionRadius(F);
gens:=GeneratorsOfGroup(F);
w:=LetterRepAssocWord(PreImagesRepresentative(pr,a));
value:=();
act:=();
for i in [1..Length(w)] do
if SignInt(w[i])=1 then
value:=value*c[w[i]][dir^act];
else
# z(a^{-1},i)=z(a,a^{-1}i)^{-1}
value:=value*c[-w[i]][(dir^act)^(LocalAction(1,d,k,gens[-w[i]],[])^(-1))]^(-1);
fi;
act:=act*LocalAction(1,d,k,gens[AbsInt(w[i])],[])^SignInt(w[i]);
od;
return value;
end );
##################################################################################################################
InstallGlobalFunction( CocycleMapFromGeneratorData,
function(F,c,pr)
local d, gens_free;
d:=LocalActionDegree(F);
gens_free:=MappingGeneratorsImages(pr)[1];
return MappingByFunction(Domain(Cartesian(F,Domain([1..d]))),F,
s->EvaluateCocycleViaElements(F,c,pr,s[1],s[2]));
end );
##################################################################################################################
InstallMethod( InvolutiveCompatibilityCocycle, "for a local action F", [IsLocalAction],
function(F)
local d, k, gens, C, a, comp_sets, dir, iter, pr, rel, c;
d:=LocalActionDegree(F);
k:=LocalActionRadius(F);
if k=0 or k=1 then
# trivial cocycle
gens:=GeneratorsOfGroup(F);
c:=[];
for a in gens do Add(c,ListWithIdenticalEntries(d,a)); od;
return MappingByFunction(Domain(Cartesian(F,Domain([1..d]))),F,s->s[1]);
else
# change to a small generating set of F
gens:=SmallGeneratingSet(F);
F:=LocalAction(d,k,GroupWithGenerators(gens));
# initialize compatibility sets
C:=[];
for a in gens do
comp_sets:=[];
for dir in [1..d] do
Add(comp_sets,CompatibilitySet(F,a,dir));
od;
Add(C,Cartesian(comp_sets));
od;
# for each possibility, check i.c.c.
iter:=IteratorOfCartesianProduct(C);
pr:=EpimorphismFromFreeGroup(F);
rel:=SmallGeneratingSet(Kernel(pr));
for c in iter do
if IsCocycleViaGeneratorData(F,c,pr,rel) and IsInvolutiveViaGeneratorData(F,c,pr) then
return CocycleMapFromGeneratorData(F,c,pr);
fi;
od;
return fail;
fi;
end );
##################################################################################################################
InstallMethod( AllInvolutiveCompatibilityCocycles, "for a local action F", [IsLocalAction],
function(F)
local d, k, iccs, gens, C, a, comp_sets, dir, iter, pr, rel, c;
d:=LocalActionDegree(F);
k:=LocalActionRadius(F);
if k=0 then
return [InvolutiveCompatibilityCocycle(F)];
else
iccs:=[];
# change to a small generating set of F
gens:=SmallGeneratingSet(F);
F:=LocalAction(d,k,GroupWithGenerators(gens));
# initialize compatibility sets
C:=[];
for a in gens do
comp_sets:=[];
for dir in [1..d] do
Add(comp_sets,CompatibilitySet(F,a,dir));
od;
Add(C,Cartesian(comp_sets));
od;
# for each possibility, check i.c.c.
iter:=IteratorOfCartesianProduct(C);
pr:=EpimorphismFromFreeGroup(F);
rel:=SmallGeneratingSet(Kernel(pr));
for c in iter do
if IsCocycleViaGeneratorData(F,c,pr,rel) and IsInvolutiveViaGeneratorData(F,c,pr) then
Add(iccs,CocycleMapFromGeneratorData(F,c,pr));
fi;
od;
return iccs;
fi;
end );
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