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## #W examples.gi Laurent Bartholdi ## ## #Y Copyright (C) 2006, Laurent Bartholdi ## ############################################################################# ## ## All interesting examples of Mealy machines and groups I came through ## ############################################################################# BindGlobal("SETGENERATORNAMES@", function(G,n) local i; for i in [1..Length(n)] do if IsGroup(G) then SetName(GeneratorsOfGroup(G)[i],n[i]); elif IsMonoid(G) then SetName(GeneratorsOfMonoid(G)[i],n[i]); elif IsSemigroup(G) then SetName(GeneratorsOfSemigroup(G)[i],n[i]); fi; od; end); BindGlobal("LPGROUPIMAGE@", function(G,F,Ggens,Fgens,Sgens,Scoord) local knows, Gtop, Ftop, Ptop, init, recur, bootstrap; bootstrap := true; recur := function(g,seen) local d, w, p, x, h, todoh; p := LookupDictionary(knows,g); if p<>fail then return p; fi; if KnowsDictionary(seen,g) then if bootstrap then AddDictionary(knows,g,MAPPEDWORD@(ShortGroupWordInSet(Group(Ggens),g,infinity)[2 Info(InfoFR,3,"Added ",g,"=",LookupDictionary(knows,g)); fi; # we reached a recurring state not yet known return fail; fi; AddDictionary(seen,g); w := Position(Ptop,ActivityPerm(g)); if w=fail then return fail; fi; # even activity is impossible w := Ftop[w]; h := LeftQuotient(MAPPEDWORD@(w,Ggens),g); todoh := []; while not IsOne(h) do if h in todoh then return fail; fi; # stuck in a loop AddSet(todoh,h); d := DecompositionOfFRElement(h)[1]; # start by hardest coordinate, i.e. one with largest norm x := List(d,x->NormOfBoundedFRElement(x)); p := Maximum(x); if x[Scoord]=p then p := Scoord; else p := PositionProperty(x,n->n=p); fi; p := PositionProperty(Ptop,s->p^s=Scoord); x := recur(State(h^Gtop[p],Scoord),seen); if x=fail then return fail; fi; x := MAPPEDWORD@(x,Sgens)^(Ftop[p]^-1); w := w*x; h := LeftQuotient(MAPPEDWORD@(x,Ggens),h); Assert(1,MAPPEDWORD@(w,Ggens)*h=g); od; AddDictionary(knows,g,w); return w; end; init := function() local x, todo, i, y; knows := NewDictionary(Representative(G),true); AddDictionary(knows,One(G),One(F)); Ptop := AsList(TopVertexTransformations(G)); Ftop := []; for x in Ptop do Add(Ftop,MAPPEDWORD@(ShortGroupWordInSet(Group(Ggens),g->ActivityPerm(g)=x,infini od; Gtop := List(Ftop,x->MAPPEDWORD@(x,Ggens)); todo := NewFIFO(TransposedMat([Ggens,Fgens])); for x in todo do y := false; while recur(x[1],NewDictionary(x[1],false))=fail do y := true; Info(InfoFR,3,"Bootstrapping recognizer with ",x[2]); # AddDictionary(knows,x[1],x[2]); # AddDictionary(knows,x[1]^-1,x[2]^-1); od; if y then for i in [1..Length(Ggens)] do Add(todo,[x[1]*Ggens[i],x[2]*Fgens[i]]); od; fi; od; bootstrap := false; end; return function(g) if bootstrap then init(); fi; return recur(g,NewDictionary(g,false)); end; end); BindGlobal("LPGROUPPREIMAGE@", function(Fgens,Sgens,Ggens,depth,Scoord) local Sletter; Sletter := Length(Ggens)+1; if Sgens=fail then return fail; fi; if depth=infinity then return function(w) local up, down, g, i, j; up := 0; down := 0; g := One(Ggens[1]); for i in LetterRepAssocWord(UnderlyingElement(w)) do if i=Sletter then # down in tree if up>0 then up := up-1; else down := down+1; g := VertexElement(Scoord,g); fi; elif i=-Sletter then if down>0 and ActivityPerm(g)=() then down := down-1; g := State(g,Scoord); else up := up+1; fi; else i := Fgens[AbsInt(i)]; for j in [1..up] do i := MAPPEDWORD@(i,Sgens); od; g := g*MAPPEDWORD@(i,Ggens); fi; od; if up<>down then return fail; Error("Element ",w," has non-trivial translation ",down-up); elif up>0 then return fail; Error("Element ",w," does not fix the root vertex"); fi; return g; end; else return w->MAPPEDWORD@(w,Ggens); fi; end); ############################################################################# ## #E AddingMachine(n) #E AddingGroup(n) ## InstallMethod(AddingMachine, "(FR) for a degree", [IsPosInt], function(n) local E; E := MealyMachine([List([1..n],i->1),List([1..n],i->1+QuoInt(i,n))], [[1..n],List([1..n],i->1+RemInt(i,n))]); SetName(E,Concatenation("AddingMachine(",String(n),")")); return E; end); InstallMethod(AddingElement, "(FR) for a degree", [IsPosInt], function(n) local E; E := MealyElement([List([1..n],i->1),List([1..n],i->1+QuoInt(i,n))], [[1..n],List([1..n],i->1+RemInt(i,n))],2); SetName(E,Concatenation("AddingElement(",String(n),")")); return E; end); InstallGlobalFunction(AddingGroup, function(n) local G; G := SCGroup(AddingMachine(n)); SetName(G,Concatenation("AddingGroup(",String(n),")")); return G; end); BindGlobal("BinaryAddingMachine",AddingMachine(2)); BinaryAddingMachine!.Name := "BinaryAddingMachine"; BindGlobal("BinaryAddingElement",AddingElement(2)); BinaryAddingElement!.Name := "BinaryAddingElement"; BindGlobal("BinaryAddingGroup",AddingGroup(2)); BinaryAddingGroup!.Name := "BinaryAddingGroup"; ############################################################################# ############################################################################# ## #E FiniteDepthBinaryGroup(l) #E FinitaryBinaryGroup #E BoundedBinaryGroup #E PolynomialStateGrowthBinaryGroup #E FiniteStateBinaryGroup #E FullBinaryGroup ## InstallGlobalFunction(FiniteDepthBinaryGroup, l->FullSCGroup([1..2],l)); BindGlobal("FinitaryBinaryGroup",FullSCGroup([1..2],IsFinitaryFRSemigroup)); BindGlobal("BoundedBinaryGroup",FullSCGroup([1..2],IsBoundedFRSemigroup)); BindGlobal("PolynomialGrowthBinaryGroup", FullSCGroup([1..2],IsPolynomialGrowthFRSemigroup)); BindGlobal("FiniteStateBinaryGroup",FullSCGroup([1..2],IsFiniteStateFRSemigroup) BindGlobal("FullBinaryGroup",FullSCGroup([1..2])); ############################################################################# ############################################################################# ## #E MixerMachine #E MixerGroup ## InstallGlobalFunction(MixerMachine, function(arg) local A, B, f, g, a, b, d, i, out, trans, r, t, corr; if not (Length(arg) in [3,4] and IsGroup(arg[1]) and IsGroup(arg[2])) then Error("MixerMachine: requires <group> <group> <list> [<endomorphism>]"); fi; A := arg[1]; B := arg[2]; f := arg[3]; if not ForAll(f,r->ForAll(r,x->IsGroupHomomorphism(x) and Source(x)=B and Range(x)=A)) th Error("MixerMachine: third argument should be list of lists of endomorphisms B->A"); fi; if Length(arg)=4 then g := arg[4]; if not IsGroupHomomorphism(g) and Source(g)=B and Range(g)=B then Error("MixerMachine: fourth argument should be endomorphism B->B"); fi; else g := IdentityMapping(B); fi; if not IsPeriodicList(f) then f := PeriodicList([],f); fi; d := Maximum(LargestMovedPoint(A),1+Maximum(List(f,Length))); b := ShallowCopy(GeneratorsOfGroup(B)); for i in b do i := i^g; if not i in b then Add(b,i); fi; od; a := Unique(Concatenation([()],GeneratorsOfGroup(A))); for i in Unique(f) do for i in i do a := Unique(Concatenation(a,List(b,x->x^i))); od; od; out := []; trans := []; for i in a do Add(trans,List([1..d],i->1)); Add(out,i); od; corr := [[2..Length(out)]]; for r in [1..Length(PrePeriod(f))+Length(Period(f))] do for i in b do t := List(f[r],pi->Position(a,i^pi)); while Length(t)<d-1 do Add(t,1); od; if r=Length(PrePeriod(f))+Length(Period(f)) then Add(t,Length(a)+Length(PrePeriod(f))*Length(b)+Position(b,i^g)); else Add(t,Length(a)+r*Length(b)+Position(b,i^g)); fi; Add(trans,t); Add(out,()); od; Add(corr,[Length(out)-Length(b)+1..Length(out)]); od; i := MealyMachine(trans,out); SetCorrespondence(i,corr); return i; end); InstallGlobalFunction(MixerGroup, function(arg) return SCGroup(CallFuncList(MixerMachine,arg)); end); ############################################################################# ############################################################################# ## #E GrigorchukGroup #E GrigorchukGroups ## InstallGlobalFunction(GrigorchukMachines, function(f) local a, b, pi; a := Group((1,2)); b := Group((1,2),(3,4),(1,2)(3,4)); pi := [GroupHomomorphismByImagesNC(b,a,[(1,2),(3,4)],[(1,2),()]), GroupHomomorphismByImagesNC(b,a,[(1,2),(3,4)],[(),(1,2)]), GroupHomomorphismByImagesNC(b,a,[(1,2),(3,4)],[(1,2),(1,2)])]; if IsPeriodicList(f) then return MixerMachine(a,b,PeriodicList(f,x->[pi[x]])); else return MixerMachine(a,b,List(f,x->[pi[x]])); fi; end); InstallGlobalFunction(GrigorchukGroups, function(f) local a, m; m := GrigorchukMachines(f); a := Group(Concatenation( List(Correspondence(m)[1],i->FRElement(m,i)), List(Correspondence(m)[2],i->FRElement(m,i)))); SetName(a,Concatenation("GrigorchukGroups(",String(f),")")); SetUnderlyingFRMachine(a,m); return a; end); BindGlobal("GrigorchukMachine", MealyMachine([[5,5],[1,3],[1,4],[5,2],[5,5]],[[2,1],[1,2],[1,2],[1,2],[1,2]])); BindGlobal("GrigorchukGroup",SCGroup(GrigorchukMachine)); SetName(GrigorchukGroup,"GrigorchukGroup"); SETGENERATORNAMES@(GrigorchukGroup,["a","b","c","d"]); CallFuncList(function(a,b,c,d) local x; x := Comm(a,b); SetBranchingSubgroup(GrigorchukGroup,Group(x,x^c,x^(c*a))); end, GeneratorsOfGroup(GrigorchukGroup)); BindGlobal("ITERATEMAP@", function(s,n,w) local r, i; r := [w]; for i in [1..n] do w := w^s; Add(r,w); od; return r; end); BindGlobal("GRIGP_IMAGE@", function(nuke,nukeimg,Fgens,Sgens,tau,reduce) local image, knows, i; knows := NewDictionary(nuke[1],true); for i in [1..Length(nuke)] do AddDictionary(knows,nuke[i],nukeimg[i]); od; return function(g) local todo, recur; todo := NewDictionary(g,false); recur := function(g) local i, x, y; i := LookupDictionary(knows,g); if i<>fail then return i; fi; i := DecompositionOfFRElement(g); if not i[2] in [[1,2],[2,1]] then return fail; fi; if KnowsDictionary(todo,g) then return fail; # we reached a recurring state not in the nucleus fi; AddDictionary(todo,g); x := recur(i[1][2]); if x=fail then return fail; fi; y := LeftQuotient(tau(i[1][2]),i[1][1]); if not IsOne(tau(y)) then return fail; fi; y := recur(y); if y=fail then return fail; fi; x := MAPPEDWORD@(x,Sgens)*Fgens[1]*MAPPEDWORD@(y,Sgens); if ISONE@(i[2]) then x:=x*Fgens[1]; fi; x := reduce(x); AddDictionary(knows,g,x); return x; end; return recur(g); end; end); SetFRGroupPreImageData(GrigorchukGroup, function(depth) local tau, reduce, creator, nuke, nukeimg, Fgens, Sgens, Ggens, F, a, b, c, d, s, rels; if depth=infinity then F := FreeGroup("a","b","c","d","s"); a := F.1; b := F.2; c := F.3; d := F.4; s := F.5; F := F / [a^2,b^2,c^2,d^2,b*c*d,(a*d)^4,(a*d*a*c*a*c)^4, a^s/c^a,b^s/d,c^s/b,d^s/c]; F := Subgroup(F,GeneratorsOfGroup(F){[1..4]}); creator := ElementOfFpGroup; else F := FreeGroup("a","b","c","d"); a := F.1; b := F.2; c := F.3; d := F.4; s := GroupHomomorphismByImagesNC(F,F,[a,b,c,d],[c^a,d,b,c]); rels := [a^2,b^2,c^2,d^2,b*c*d,(a*d)^4,(a*d*a*c*a*c)^4]; if depth>=0 then F := F / Concatenation(rels{[1..5]}, ITERATEMAP@(s,depth+1,rels[6]), ITERATEMAP@(s,depth,rels[7])); creator := ElementOfFpGroup; else F := LPresentedGroup(F,[],[s],rels); creator := ElementOfLpGroup; fi; fi; tau := function(g) local p, x; p := Portrait(g,2); x := One(GrigorchukGroup); if p[2][1]*p[3][2][1]*p[3][2][2]=(1,2) then x := x*GrigorchukGroup.1; fi; if p[2][2]*p[3][1][1]*p[3][1][2]=(1,2) then x := x*GrigorchukGroup.1^GrigorchukGroup. if p[1]=(1,2) then x := x*GrigorchukGroup.4; fi; return x; end; reduce := function(g) local i, w, x, changed; w := UnderlyingElement(g); x := LetterRepAssocWord(w); changed := false; for i in [1..Length(x)] do if x[i]<0 then x[i] := -x[i]; changed := true; fi; od; i := 1; while i<Length(x) do if x[i]=x[i+1] then changed := true; Remove(x,i); Remove(x,i); if i>1 then i := i-1; fi; elif x[i]<>1 and x[i+1]<>1 then changed := true; x[i] := 9-x[i]-x[i+1]; Remove(x,i+1); else i := i+1; fi; od; if changed then return creator(FamilyObj(g),AssocWordByLetterRep(FamilyObj(w),x)); else return g; fi; end; Fgens := [F.1, F.2,F.3,F.4]; Sgens := [F.1*F.3*F.1,F.4,F.2,F.3]; Ggens := [GrigorchukGroup.1,GrigorchukGroup.2, GrigorchukGroup.3,GrigorchukGroup.4]; nuke := [One(GrigorchukGroup),Ggens[1],Ggens[2],Ggens[3],Ggens[4]]; nukeimg := [One(F),F.1,F.2,F.3,F.4]; SortParallel(nuke,nukeimg); return rec(F:=F, image:=GRIGP_IMAGE@(nuke,nukeimg,Fgens,Sgens,tau,reduce), preimage:=LPGROUPPREIMAGE@(Fgens,Sgens,Ggens,depth,2), reduce:=reduce); end); BindGlobal("GrigorchukOverGroup", MixerGroup(Group((1,2)),Group((1,2)), [[IdentityMapping(Group((1,2)))],[],[]])); SetName(GrigorchukOverGroup,"GrigorchukOverGroup"); SETGENERATORNAMES@(GrigorchukOverGroup,["a","bb","cc","dd"]); # growth of PermGroup(GrigorchukOverGroup,5), generated by nucleus, is # [1, 8, 14, 56, 89, 248, 416, 1160, 1804, 3816, 5871, 13400, 20344, 42248, 64020, 134072, 189600 SetFRGroupPreImageData(GrigorchukOverGroup, function(depth) local tau, reduce, nuke, nukeimg, Fgens, Sgens, Ggens, F, a, b, c, d, s, rels, creator; if depth=infinity then F := FreeGroup("a","bb","cc","dd","s"); a := F.1; b := F.2; c := F.3; d := F.4; s := F.5; else F := FreeGroup("a","bb","cc","dd"); a := F.1; b := F.2; c := F.3; d := F.4; s := GroupHomomorphismByImagesNC(F,F,[a,b,c,d],[b^a,d,b,c]); fi; rels := [a^2,b^2,c^2,d^2,Comm(b,c),Comm(b,d),Comm(c,d), (a*c)^4,(a*d)^4,(a*c*a*d)^2,(a*b)^8, (a*b*a*b*a*c)^4,(a*b*a*b*a*d)^4,(a*b*a*b*a*c*a*b*a*b*a*d)^2]; if depth=infinity then F := F / Concatenation(rels,[a^s/b^a,b^s/d,c^s/b,d^s/c]); F := Subgroup(F,GeneratorsOfGroup(F){[1..4]}); creator := ElementOfFpGroup; elif depth=-1 then F := LPresentedGroup(F,[],[s],rels); creator := ElementOfLpGroup; else F := F / Concatenation(rels{[1..7]}, Concatenation(List(rels{[8..11]},x->ITERATEMAP@(s,depth+1,x))), Concatenation(List(rels{[12..14]},x->ITERATEMAP@(s,depth,x)))); creator := ElementOfFpGroup; fi; tau := function(g) local p, x; p := Portrait(g,3); x := One(GrigorchukOverGroup); if Product(Flat(p[4][2]))=(1,2) then x := x*GrigorchukOverGroup.1; fi; if Product(Flat(p[4][1]))=(1,2) then x := x*GrigorchukOverGroup.1^GrigorchukOverGroup if p[1]=(1,2) then x := x*GrigorchukOverGroup.3; fi; return x; end; reduce := function(g) local i, w, x, changed; w := UnderlyingElement(g); x := LetterRepAssocWord(w); changed := false; for i in [1..Length(x)] do if x[i]<0 then x[i] := -x[i]; changed := true; fi; od; i := 1; while i<Length(x) do if x[i]=x[i+1] then changed := true; Remove(x,i); Remove(x,i); if i>1 then i := i-1; fi; elif x[i]>x[i+1] and x[i+1]<>1 then changed := x[i]; x[i] := x[i+1]; x[i+1] := changed; changed := true; if i>1 then i := i-1; fi; else i := i+1; fi; od; if changed then return creator(FamilyObj(g),AssocWordByLetterRep(FamilyObj(w),x)); else return g; fi; end; Fgens := [F.1, F.2,F.3,F.4]; Sgens := [F.1*F.2*F.1,F.4,F.2,F.3]; Ggens := [GrigorchukOverGroup.1,GrigorchukOverGroup.2, GrigorchukOverGroup.3,GrigorchukOverGroup.4]; nuke := [One(GrigorchukOverGroup),Ggens[1],Ggens[2],Ggens[3], Ggens[4],Ggens[2]*Ggens[3],Ggens[2]*Ggens[4], Ggens[3]*Ggens[4],Ggens[2]*Ggens[3]*Ggens[4]]; nukeimg := [One(F),F.1,F.2,F.3,F.4,F.2*F.3,F.2*F.4,F.3*F.4,F.2*F.3*F.4]; SortParallel(nuke,nukeimg); return rec(F:=F, image:=GRIGP_IMAGE@(nuke,nukeimg,Fgens,Sgens,tau,reduce), preimage:=LPGROUPPREIMAGE@(Fgens,Sgens,Ggens,depth,2), reduce:=reduce); end); BindGlobal("GrigorchukTwistedTwin", SCGroup(MealyMachine( [[5,5],[3,1],[1,4],[5,2],[5,5]], [(1,2),(),(),(),()]))); SETGENERATORNAMES@(GrigorchukTwistedTwin,["a","x","y","z"]); SetFRGroupPreImageData(GrigorchukTwistedTwin, function(depth) local F, a, x, y, z, s, rels, Fgens, Ggens, Sgens; Ggens := GeneratorsOfGroup(GrigorchukTwistedTwin); if depth=infinity then F := FreeGroup("a","x","y","z","s"); a := F.1; x := F.2; y := F.3; z := F.4; s := F.5; Fgens := [a,x,y,z]; Sgens := [y^a,z,x^a,y]; else F := FreeGroup("a","x","y","z"); a := F.1; x := F.2; y := F.3; z := F.4; Fgens := [a,x,y,z]; Sgens := [y^a,z,x^a,y]; s := GroupHomomorphismByImagesNC(F,F,Fgens,Sgens); fi; rels := [a^2, x^2, y^2, z^2, Comm(z,y^a*x), Comm(z,Comm(z,a)), Comm(Comm(z,y),y^a*x), Comm(y^a*x,Comm(y^a*x,a))]; if depth=infinity then F := F / Concatenation(rels,List([1..4],i->Fgens[i]^s/Sgens[i])); F := Subgroup(F,Fgens); elif depth=-1 then F := LPresentedGroup(F,[],[s],rels); else F := F / Concatenation(List(rels,x->ITERATEMAP@(s,depth,x))); fi; Fgens := GeneratorsOfGroup(F){[1..4]}; if depth=-1 then Sgens := List(Sgens,x->ElementOfLpGroup(FamilyObj(F.1),x)); else Sgens := List(Sgens,x->ElementOfFpGroup(FamilyObj(F.1),x)); fi; return rec(F:=F, image:=LPGROUPIMAGE@(GrigorchukTwistedTwin,F,Ggens,GeneratorsOfGroup(F){[1..4]}, preimage:=LPGROUPPREIMAGE@(Fgens,Sgens,Ggens,depth,2), reduce:=w->w); end); GermData(GrigorchukTwistedTwin).init := function(data) local F; F := FreeGroup("x","y","z","c"); F := PcGroupFpGroup(F/[F.1^2,F.2^2,F.3^2,F.4^2, Comm(F.2,F.1)/F.4, Comm(F.3,F.1)/F.4, Comm(F.3,F.2)/F.4, Comm(F.4,F.1), Comm(F.4,F.2), Comm(F.4,F.3)]); data.group := F; data.endo := GroupHomomorphismByImages(F,F,[F.1,F.2,F.3],[F.3,F.1,F.2]); data.map := [One(F),F.3,F.1,F.2,One(F)]; data.eval := function(elm,data,h) local x, y, z, f1, f2, f3, i, recur; x := List(h,g->ExponentOfPcElement(Pcgs(data.group),g,1)); y := List(h,g->ExponentOfPcElement(Pcgs(data.group),g,2)); z := List(h,g->ExponentOfPcElement(Pcgs(data.group),g,3)); f1 := []; f2 := []; f3 := []; recur := function(s) local i, t; if not IsBound(f1[s]) then f1[s] := 0; f2[s] := 0; f3[s] := 0; t := elm!.transitions[s]; for i in t do recur(i); od; f1[s] := x[t[1]] + y[t[2]] + y[t[1]]*y[t[2]] + Sum([1..2],i->f2[t[i]]+x[t[i]]*z[t[3-i]]+y f2[s] := z[t[1]] + y[t[2]] + y[t[1]]*y[t[2]] + Sum([1..2],i->f3[t[i]]); f3[s] := z[t[1]] + y[t[2]] + z[t[1]]*z[t[2]] + Sum([1..2],i->f1[t[i]]+y[t[i]]*x[t[3-i]]+z fi; end; i := InitialState(elm); recur(i); return F.1^x[i]*F.2^y[i]*F.3^z[i]*F.4^(f3[i]+z[Transition(elm,i,1)]+y[Transition( end; Unbind(data.init); end; GermData(GrigorchukTwistedTwin).init(GermData(GrigorchukTwistedTwin)); ############################################################################# ############################################################################# ## #E SunicMachine #E SunicGroup ## InstallGlobalFunction(SunicMachine, function(phi) local k, p, A, B, d, f, g, gB, i, j; k := Field(CoefficientsOfUnivariatePolynomial(phi)); p := Size(k); A := ElementaryAbelianGroup(IsPermGroup,p); d := DegreeOfUnivariateLaurentPolynomial(phi); B := ElementaryAbelianGroup(p^d); gB := GeneratorsOfGroup(B); f := GroupHomomorphismByImages(B,A,gB,Concatenation(ListWithIdenticalEntries((d-1) i := gB{[Dimension(k)+1..d*Dimension(k)]}; for j in Basis(k) do j := Concatenation(List(CoefficientsOfUnivariatePolynomial(phi){[1..d]},x->Coeffici Add(i,Product([1..Length(j)],i->gB[i]^IntFFE(j[i]),One(B))); od; g := GroupHomomorphismByImages(B,B,GeneratorsOfGroup(B),i); i := MixerMachine(A,B,[[f]],g); SetName(i,Concatenation("SunicMachine(",String(phi),")")); return i; end); InstallGlobalFunction(SunicGroup, function(phi) local g; g := SCGroup(SunicMachine(phi)); SetName(g,Concatenation("SunicGroup(",String(phi),")")); return g; end); ############################################################################# ############################################################################# ## #E AleshinMachine #E AleshinGroup #E BabyAleshinMachine #E BabyAleshinGroup ## InstallGlobalFunction(AleshinMachines, function(n) local trans, out; trans := Concatenation([[3,2],[2,3]],List([3..n-1],s->[s+1,s+1]),[[1,1]]); out := Concatenation([(1,2),(1,2)],List([3..n],s->())); return MealyMachine(trans,out); end); InstallGlobalFunction(AleshinGroups, function(n) local g; g := SCGroup(AleshinMachines(n)); SetName(g,Concatenation("AleshinGroups(",String(n),")")); return g; end); BindGlobal("AleshinMachine", AleshinMachines(3)); BindGlobal("AleshinGroup", SCGroup(AleshinMachine)); # the main example AleshinGroup!.Name := "AleshinGroup"; SETGENERATORNAMES@(AleshinGroup,["a","b","c"]); BindGlobal("BabyAleshinMachine", MealyMachine([[2,3],[3,2],[1,1]],[(),(),(1,2)])); BindGlobal("BabyAleshinGroup", SCGroup(BabyAleshinMachine)); SetName(BabyAleshinGroup,"BabyAleshinGroup"); SETGENERATORNAMES@(BabyAleshinGroup,["a","b","c"]); BindGlobal("SidkiFreeGroup", FRGroup("a=<a^2,a^t>","t=<,t>(1,2)")); SetName(SidkiFreeGroup,"SidkiFreeGroup"); # F := FreeGroup("a","t"); # a := F.1; t := F.2; # H := Subgroup(F,[a,a^t,t^2]); # K := Subgroup(F,[a^2,t^a,t]); # phi := GroupHomomorphismByImages(H,K,[a,a^t,t^2],[a^2,t^a,t]); # psi := GroupHomomorphismByImages(K,H,[a^2,t^a,t],[a,a^t,t^2]); # # # Group([ a^-4, t^-1, a^2*t^-1*a^-2, t*a^-1*t*a*t^-1*a^-2, # # t*a^-1*t^-1*a*t^-1*a^-2 ]) # # F := FreeGroup("a","b","c"); # a := F.1; b := F.2; c := F.3; # H := Subgroup(F,[a^2,b^2,a*b,c,c^a]); # K0 := Subgroup(F,[b*c,c*b,b^2,a,a^c]); # K1 := Subgroup(F,[c*b,b*c,c^2,a,a^b]); # phi := GroupHomomorphismByImages(H,K0,GeneratorsOfGroup(H),GeneratorsOfGroup(K0)) # psi := GroupHomomorphismByImages(K0,H,GeneratorsOfGroup(K0),GeneratorsOfGroup(H)) # # S := [H]; # Append(S,[Image(psi,Intersection(K0,S[Size(S)]))]); # Append(S,[Image(psi,Intersection(K0,S[Size(S)]))]); # Append(S,[Image(psi,Intersection(K0,S[Size(S)]))]); # Append(S,[Image(psi,Intersection(K0,S[Size(S)]))]); # W := Subgroup(F,[b^-1*a*c,b^-1*c*a]); ############################################################################# ############################################################################# ## #E BrunnerSidkiVieiraMachine #E BrunnerSidkiVieiraGroup ## BindGlobal("BrunnerSidkiVieiraMachine", MealyMachine([[5,1],[5,3],[2,5],[4,5],[5,5]],[(1,2),(1,2),(1,2),(1,2),()])); BindGlobal("BrunnerSidkiVieiraGroup", SCGroup(BrunnerSidkiVieiraMachine)); SetName(BrunnerSidkiVieiraGroup,"BrunnerSidkiVieiraGroup"); SETGENERATORNAMES@(BrunnerSidkiVieiraGroup,["tau","mu"]); SetFRGroupPreImageData(BrunnerSidkiVieiraGroup, function(depth) local F, rels, sigma, tau, lambda, mu, Fgens, Ggens, Sgens; if depth=infinity then F := FreeGroup("tau","mu","s"); sigma := F.3; else F := FreeGroup("tau","mu"); fi; tau := F.1; mu := F.2; lambda := tau/mu; rels := [Comm(lambda,lambda^tau),Comm(lambda,lambda^(tau^3))]; Sgens := [tau^2,tau^-1*mu^-1]; if depth=infinity then F := F / Concatenation(rels,[tau^sigma/Sgens[1],mu^sigma/Sgens[2]]); Fgens := GeneratorsOfGroup(F){[1..2]}; F := Subgroup(F,Fgens); else sigma := GroupHomomorphismByImagesNC(F,F,[tau,mu],Sgens); if depth>=0 then F := F / Concatenation(List(rels,r->ITERATEMAP@(sigma,depth,r))); else F := LPresentedGroup(F,[],[sigma],rels); fi; Fgens := GeneratorsOfGroup(F); fi; Ggens := GeneratorsOfGroup(BrunnerSidkiVieiraGroup); if depth=-1 then Sgens := List(Sgens,x->ElementOfLpGroup(FamilyObj(Representative(F)),x)); else Sgens := List(Sgens,x->ElementOfFpGroup(FamilyObj(Representative(F)),x)); fi; return rec(F:=F, image:=LPGROUPIMAGE@(BrunnerSidkiVieiraGroup,F,Ggens,Fgens,Sgens,2), preimage:=LPGROUPPREIMAGE@(Fgens,Sgens,Ggens,depth,2), reduce:=w->w); end); #growth of H: #[ 1, 4, 12, 36, 100, 276, 760, 2020, 5306, 13828, 35832 ] #SEEMS TO BE EXPONENTIAL! # #growth of semigroup generated by {t,m}: #[2, 4, 8, 16, 32, 64, 120, 225, 420, 784, 1456, 2704, 4992, 9216, 16992, 31329, 57702, 106276] # #H / H' = Z^2 = <t,l> #H' / H" = Z^5 = <[l,t^i]: i=1..5> #H / <<l>> = Z = <t> #action of l on H'/H" is trivial #action of t on H'/H" has eigenvals 1,-1,I,-I: #[-1 -1 -1 -1 -1] #[ 1 1] #[ 1 ] #[ 1 1] #[ 1 ] #log_2 of size of H_n: #[1 2 4 7 13 24 46 89] = (2^(n+1) + 3n - 2 + (1-(-1)^n)/2) / 6 #weakly branched on H', w/ branch structure #1 --> IxI --> H --> <t> --> 1 with H = <<lt^-2>> #1 --> H'xH' --> I --> <[l,t],lt^-2|abelian> --> 1 #1 --> H'xH' --> H' --> <[l,t]> --> 1 #lower central series: #[ 32*64, ] #[ 16*64, 16, 8, 8, 2*8, 2*4 (6x), 4, 2*4 (6x), 4 (46x), 2 (192x) ]@10 #[ 16*32, 16, 8, 8, 8, 4, 2*4, 2*4, 2*4, 4 (23x), 2 (96x) ]@9 #[ 8*32, 8, 8, 8, 4 (12x), 2 (48x) ]@8 #OUT(H)=V_4? ############################################################################# ############################################################################# ## #E GuptaSidkiMachines #E GuptaSidkiGroups #E GuptaSidkiGroup #E FabrykowskiGuptaGroup #E ZugadiSpinalGroup ## InstallGlobalFunction(GuptaSidkiMachines, function(n) local P; P := CyclicGroup(IsPermGroup,n); return MixerMachine(P,P,[[IdentityMapping(P),GroupHomomorphismByImages(P,P,[P.1], end); InstallGlobalFunction(GuptaSidkiGroups, function(n) local G, a, t; G := SCGroup(GuptaSidkiMachines(n)); SETGENERATORNAMES@(G,["a","t"]); a := G.1; t := G.2; SetBranchingSubgroup(G,GroupByGenerators(ListX([0..n-1],[0..n-1],function(x,y) re SetName(G,Concatenation("GuptaSidkiGroups(",String(n),")")); return G; end); BindGlobal("GUPTASIDKIGROUPIMAGE@", function(g,f,Ggens,Fgens,Sgens,Scoord) local nuke, knows, x, y, Gtop, Ftop, Ptop, GENREDUCE; nuke := NucleusOfFRSemigroup(g); knows := NewDictionary(nuke[1],true); for x in nuke do AddDictionary(knows,x,MAPPEDWORD@(ShortGroupWordInSet(Group(Ggens),x,infinity)[2 od; Ptop := AsList(TopVertexTransformations(g)); Ftop := []; for x in Ptop do Add(Ftop,MAPPEDWORD@(ShortGroupWordInSet(Group(Ggens),g->ActivityPerm(g)=x,infini od; Gtop := List(Ftop,x->MAPPEDWORD@(x,Ggens)); GENREDUCE := function(h,w) local n, i, j, x; n := NormOfBoundedFRElement(h); for i in [1..Length(Ggens)] do for j in [1..Length(Ftop)-1] do x := LeftQuotient(Ggens[i]^j,h); if NormOfBoundedFRElement(x)<n then return [x,w*Fgens[i]^j]; fi; od; od; return fail; end; return function(g) local todo, recur; todo := NewDictionary(g,false); recur := function(g) local d, w, p, x, h; p := LookupDictionary(knows,g); if p<>fail then return p; fi; if KnowsDictionary(todo,g) then return fail; # we reached a recurring state not in the nucleus fi; AddDictionary(todo,g); w := Position(Ptop,ActivityPerm(g)); if w=fail then return fail; fi; w := Ftop[w]; h := LeftQuotient(MAPPEDWORD@(w,Ggens),g); while not IsOne(h) do x := GENREDUCE(h,w); if x<>fail then h := x[1]; w := x[2]; continue; fi; d := DecompositionOfFRElement(h)[1]; # start by hardest coordinate, i.e. one with largest norm x := List(d,x->NormOfBoundedFRElement(x)); p := Maximum(x); p := PositionProperty(x,n->n=p); p := PositionProperty(Ptop,s->p^s=Scoord); x := recur(State(h^Gtop[p],Scoord)); if x=fail then return fail; fi; x := MAPPEDWORD@(x,Sgens)^(Ftop[p]^-1); w := w*x; h := LeftQuotient(MAPPEDWORD@(x,Ggens),h); Assert(1,MAPPEDWORD@(w,Ggens)*h=g); od; AddDictionary(knows,g,w); return w; end; return recur(g); end; end); BindGlobal("GUPTASIDKIFRDATA@", function(G,p,depth,fullgroup) local F, rels, rels0, sigma, a, t, tt, Fgens, Ggens, Sgens, creator, i, j, k, l, e, image; if depth=infinity then Error("Do not know yet any 'subgroup of FP group' for GeneralizedGuptaSidkiGroups()"); else if fullgroup then a := ["a"]; else a := []; fi; F := FreeGroup(Concatenation(a,List([1..p],i->Concatenation("t",String(i))))); if fullgroup then a := F.1; t := GeneratorsOfGroup(F){[2..p+1]}; rels0 := List([1..p],i->t[1]^(a^(i-1))/t[i]); rels0[1] := a^p; else t := GeneratorsOfGroup(F); rels0 := []; fi; rels := List(t,x->x^p); tt := List(t,x->[x]); for i in GF(p) do for l in [1..p-1] do j := First(Cartesian(GF(p),GF(p)),p->p[1]<>p[2] and p[1]<>i and p[2]<>i and IntFFE((p[1]-i)*(i- k := j[2]; j := j[1]; e := (2*(j-k))^-1; tt[1+IntFFE(i)][1+l] := t[1+IntFFE(i)] / Comm(t[1+IntFFE(i)]^IntFFESymm(e*(j-k))*t[1 od; od; # tt[i][n] is a word in the t[*], equal to t[i]^(a[i]^n) for i in [1..p] do for j in Difference([1..p],[i]) do for k in [0..p-1] do for l in [0..p-1] do Add(rels,tt[i][1+RemInt(k+i,p)]^-1*tt[j][1+RemInt(l+i,p)]^-1*tt[i][1+RemInt(k+j, od; od; od; od; if fullgroup then sigma := GroupHomomorphismByImagesNC(F,F,Concatenation([a],t),Concatenation([t[p]] else sigma := GroupHomomorphismByImagesNC(F,F,t,tt[1]); fi; if depth>=0 then F := F / Flat([rels0,List(rels,r->ITERATEMAP@(sigma,depth,r))]); creator := x->ElementOfFpGroup(FamilyObj(Representative(F)),x); elif fullgroup then F := LPresentedGroup(F,rels0,[sigma],rels); creator := x->ElementOfLpGroup(FamilyObj(Representative(F)),x); else F := LPresentedGroup(F,rels0,[sigma,GroupHomomorphismByImagesNC(F,F,t,t{Concatenat creator := x->ElementOfLpGroup(FamilyObj(Representative(F)),x); fi; Fgens := GeneratorsOfGroup(F); fi; Ggens := List([0..p-1],i->G.2^(G.1^i)); if fullgroup then Ggens := Concatenation([G.1],Ggens); fi; Sgens := List(MappingGeneratorsImages(sigma)[2],creator); if fullgroup then return rec(F:=F, image:=GUPTASIDKIGROUPIMAGE@(G,F,Ggens,Fgens,Sgens,p), preimage:=LPGROUPPREIMAGE@(Ggens,Fgens,Sgens,depth,p), reduce:=w->w); else return rec(F:=F); fi; end); InstallGlobalFunction(GeneralizedGuptaSidkiGroups, function(p) local P, G, a, t; P := CyclicGroup(IsPermGroup,p); P := MixerMachine(P,P,[List([1..p-1],i->GroupHomomorphismByImages(P,P,[P.1],[P.1^i] G := Group(FRElement(P,2),FRElement(P,p+1)); SETGENERATORNAMES@(G,["a","t"]); SetName(G,Concatenation("GeneralizedGuptaSidkiGroups(",String(p),")")); SetUnderlyingFRMachine(G,P); SetIsStateClosed(G,true); a := G.1; t := G.2; SetBranchingSubgroup(G,GroupByGenerators(ListX([0..p-1],[0..p-1],function(x,y) re SetFRGroupPreImageData(G,function(depth) local r, s; r := GUPTASIDKIFRDATA@(G,p,depth,true); if depth=-1 then s := GUPTASIDKIFRDATA@(G,p,depth,false); SetEmbeddingOfAscendingSubgroup(r.F,GroupHomomorphismByImagesNC( s.F,r.F,GeneratorsOfGroup(s.F),List([1..p],i->r.F.2^(r.F.1^(i-1))))); fi; return r; end); return G; end); BindGlobal("GuptaSidkiMachine", GuptaSidkiMachines(3)); BindGlobal("GuptaSidkiGroup", GeneralizedGuptaSidkiGroups(3)); GuptaSidkiGroup!.Name := "GuptaSidkiGroup"; # automorphisms: # u := MealyElement([[1,1,1]],[(1,2)],1); # v := MealyElement([[2,2,2],[1,1,1]],[(),(1,2)],1); # x := GuptaSidkiGroup.1; # N3 := ClosureGroup(GuptaSidkiGroup,[DiagonalElement(0,x), # DiagonalElement([0,0],x),DiagonalElement([0,0,0],x)]); # N := ClosureGroup(GuptaSidkiGroup,[DiagonalElement(0,x), # DiagonalElement([0,0],x),DiagonalElement([0,0,0],x)],u,v]); # NN := ClosureGroup(N,[DiagonalElement([1,0],x)]); # NNN := ClosureGroup(NN,[A(C(a))]); # # N := ClosureGroup(GuptaSidkiGroup,[DiagonalElement([0],x), # DiagonalElement([0,0],x),DiagonalElement([0,0,0],x), # DiagonalElement([0,0,0,0],x)]); # M := ClosureGroup(N,[DiagonalElement([1],x),DiagonalElement([1,0],x), # DiagonalElement([1,0,0],x),DiagonalElement([1,0,0,0],x)]); # L := ClosureGroup(M,[DiagonalElement([2],x),DiagonalElement([2,0],x), # DiagonalElement([2,0,0],x),DiagonalElement([2,0,0,0],x), # DiagonalElement([0,1],x),DiagonalElement([0,1,0],x), # DiagonalElement([0,1,0,0],x)]); # K := ClosureGroup(L,[DiagonalElement([1,1],x),DiagonalElement([1,1,0],x), # DiagonalElement([1,1,0,0],x)]); # J := ClosureGroup(K,[DiagonalElement([2,1],x),DiagonalElement([2,1,0],x), # DiagonalElement([2,1,0,0],x)]); # I := ClosureGroup(J,[DiagonalElement([0,2],x),DiagonalElement([0,2,0],x), # DiagonalElement([0,2,0,0],x),DiagonalElement([0,0,1],x), # DiagonalElement([0,0,1,0],x)]); # H := ClosureGroup(I,[DiagonalElement([1,2],x),DiagonalElement([1,2,0],x), # DiagonalElement([1,2,0,0],x)]); # Y := ClosureGroup(H,[DiagonalElement([2,2],x),DiagonalElement([2,2,0],x), # DiagonalElement([2,2,0,0],x)]); # W := ClosureGroup(Y,[DiagonalElement([1,0,1],x),DiagonalElement([1,0,1,0],x)]); # # T := [W,Y,H,I,J,K,L,M,N,G]; # TST := g->1+Size(T)-First([Size(T),Size(T)-1..1],n->T[n]^g=T[n]); # #------------------- # lower central series growth: can be computed by # #alpha := [1,2]; #for i in [3..10] do alpha[i] := 2*alpha[i-1]+alpha[i-2]; od; #h := Indeterminate(Rationals); #P := [h]; #for i in [1..9] do P[i+1] := P[i+1-1]*(1+h^alpha[i]+h^(2*alpha[i])); od; #Q := []; #for i in [1..9] do # Q[i] := h+Sum([0..i],j->P[j+1]*h^alpha[j+1])+Sum([0..i-1],j->P[j+1]*h^(2*alpha[j+1] #od; # #then Q[n] tends to the Poincare series. In particular, Corollary 3.9 in #[Bartholdi:LCS] is wrong, and should read # \begin{align*} # Q_1&=0,\\ # Q_2&=\hbar+\hbar^2,\\ # Q_3&=\hbar+\hbar^2+2\hbar^3+\hbar^4+\hbar^5,\\ # Q_n&=(1+\hbar^{\alpha_n-\alpha_{n-1}})Q_{n-1} # +\hbar^{\alpha_{n-1}}(\hbar^{-\alpha_{n-3}}+1+\hbar^{\alpha_{n-3}})Q_{n-2} # \text{ for }n\ge4. # \end{align*} InstallGlobalFunction(NeumannMachine, function(P) return MixerMachine(P,P,[[IdentityMapping(P)]]); end); InstallGlobalFunction(NeumannGroup, function(P) local G, M; M := NeumannMachine(P); G := SCGroup(M); SetName(G,Concatenation("NeumannGroup(",STRINGGROUP@(P),")")); G!.Correspondence := [GroupHomomorphismByImages(P,G,GeneratorsOfGroup(P), GeneratorsOfGroup(G){Correspondence(G){Correspondence(M)[1]}}), GroupHomomorphismByImages(P,G,GeneratorsOfGroup(P), GeneratorsOfGroup(G){Correspondence(G){Correspondence(M)[2]}})]; return G; end); InstallGlobalFunction(FabrykowskiGuptaGroups, function(p) local G; G := NeumannGroup(CyclicGroup(IsPermGroup,p)); G!.Name := Concatenation("FabrykowskiGuptaGroups(",String(p),")"); SETGENERATORNAMES@(G,["a","r"]); SetFRGroupPreImageData(G,function(depth) local F, rels, sigma, a, r, Fgens, Ggens, Sgens, j, k, l; if depth=infinity then F := FreeGroup("a","r","s"); sigma := F.3; else F := FreeGroup("a","r"); fi; a := F.1; r := List([0..p-1],i->F.2^(a^i)); rels := [a^p]; for j in [3..p-1] do for k in [0..p-1] do for l in [0..p-1] do Add(rels,Comm(r[2]^(r[1]^l),r[j+1]^(r[j]^k))); od; od; od; for k in [0..p-1] do for l in [1..p-1] do Add(rels,Comm(r[3]^(r[2]^k),Comm(r[1]^l,r[2]^-1))); od; od; Sgens := [r[1]^(a^-1),r[1]]; if depth=infinity then F := F / Concatenation(rels,[a^sigma/Sgens[1],r[1]^sigma/Sgens[2]]); Fgens := GeneratorsOfGroup(F){[1..2]}; F := Subgroup(F,Fgens); else sigma := GroupHomomorphismByImagesNC(F,F,[a,r[1]],Sgens); if depth>=0 then F := F / Flat([rels[1],r[1]^p,List(rels{[2..Length(rels)]},r->ITERATEMAP@(sigma,depth else F := LPresentedGroup(F,[],[sigma],rels); fi; Fgens := GeneratorsOfGroup(F); fi; Ggens := GeneratorsOfGroup(G); if depth=-1 then Sgens := List(Sgens,x->ElementOfLpGroup(FamilyObj(Representative(F)),x)); else Sgens := List(Sgens,x->ElementOfFpGroup(FamilyObj(Representative(F)),x)); fi; return rec(F:=F, image:=LPGROUPIMAGE@(G,F,Ggens,Fgens,Sgens,p), preimage:=LPGROUPPREIMAGE@(Ggens,Fgens,Sgens,depth,p), reduce:=w->w); end); return G; end); BindGlobal("FabrykowskiGuptaGroup", FabrykowskiGuptaGroups(3)); FabrykowskiGuptaGroup!.Name := "FabrykowskiGuptaGroup"; BindGlobal("ZugadiSpinalGroup", MixerGroup(Group((1,2,3)),Group((1,2,3)), [[IdentityMapping(Group((1,2,3))),IdentityMapping(Group((1,2,3)))]])); SetName(ZugadiSpinalGroup,"ZugadiSpinalGroup"); SETGENERATORNAMES@(ZugadiSpinalGroup,["a","s"]); ############################################################################# ############################################################################# ## #E HanoiMachine #E HanoiGroup #E GuptaSidkiGroup #E FabrykowskiGuptaGroup ## InstallGlobalFunction(HanoiGroup, function(k) local G, trans, out, i; trans := [List([1..k],i->1)]; out := [()]; for i in Combinations([1..k],2) do Add(trans,List([1..k],i->Length(trans)+1)); trans[Length(trans)][i[1]] := 1; trans[Length(trans)][i[2]] := 1; Add(out,(i[1],i[2])); od; G := SCGroup(MealyMachine(trans,out)); SetName(G, Concatenation("HanoiGroup(",String(k),")")); if k=3 then SetFRGroupPreImageData(G,function(depth) local F, Fgens, Ggens, Sgens, a, b, c, d, e, f, g, h, i, tau, rels; if depth=infinity then F := FreeGroup("a","b","c","tau"); tau := F.4; else F := FreeGroup("a","b","c"); fi; a := F.1; b := F.2; c := F.3; d := Comm(a,b); e := Comm(b,c); f := Comm(c,a); g := d^c; h := e^a; i := f^b; Fgens := [a,b,c]; Sgens := [a,b^c,c^b]; rels := [a^2,b^2,c^2,d^-1*e*f/i*g/e,h/e/d*f*d/i,e^-1/g/f*e*g*f,e^-1*d*h/e^2/d*h^2] if depth=infinity then F := F / Concatenation(rels,List([1..3],i->Fgens[i]^tau/Sgens[i])); Fgens := GeneratorsOfGroup(F){[1..3]}; F := Subgroup(F,Fgens); else tau := GroupHomomorphismByImagesNC(F,F,Fgens,Sgens); if depth>=0 then F := F / Flat([rels{[1..3]},List(rels{[4..Length(rels)]},r->ITERATEMAP@(tau,depth,r)) else F := LPresentedGroup(F,[],[tau],rels); fi; Fgens := GeneratorsOfGroup(F); fi; Ggens := GeneratorsOfGroup(G); if depth=-1 then Sgens := List(Sgens,x->ElementOfLpGroup(FamilyObj(Representative(F)),x)); else Sgens := List(Sgens,x->ElementOfFpGroup(FamilyObj(Representative(F)),x)); fi; return rec(F:=F, image:=LPGROUPIMAGE@(G,F,Ggens,Fgens,Sgens,3), preimage:=LPGROUPPREIMAGE@(Ggens,Fgens,Sgens,depth,3), reduce:=w->w); end); fi; return G; end); BindGlobal("DahmaniGroup", SCGroup(MealyMachine([[3,1],[2,1],[2,3]],[(1,2),(1,2),()]))); SetName(DahmaniGroup,"DahmaniGroup"); BindGlobal("MamaghaniGroup", SCGroup(FRMachine(["a","b","c"],[[[],[2]],[[1],[3]],[[1],[-1]]],[(1,2),(),(1,2)] SetName(MamaghaniGroup,"MamaghaniGroup"); BindGlobal("WeierstrassGroup", SCGroup(MealyMachine([[1,1,1,1],[1,1,1,1],[1,1,1,1],[1,1,1,1],[5,2,3,4]],[(),(1, SetName(WeierstrassGroup,"WeierstrassGroup"); BindGlobal("StrichartzGroup", FRGroup("a=<b,b,c,c,a,a>","b=(1,2)(3,4)(5,6)","c=(1,6)(2,3)(4,5)")); SetName(StrichartzGroup,"StrichartzGroup"); ############################################################################# ############################################################################# ## #E FRAffineGroup #E CayleyMachine #E CayleyGroup InstallMethod(FRAffineGroup, "(FR) for a dimension, a ring, an element", [IsPosInt,IsRing,IsRingElement], function(dim,ring,unif) local trans; if IsIntegers(ring) then trans := List(ring mod AbsInt(unif),Int); elif IsUnivariatePolynomialRing(ring) and [unif]=IndeterminatesOfPolynomialRing(rin trans := Elements(CoefficientsRing(ring))*One(ring); else Error("FRAffineGroup: cannot handle ring ",ring," with uniformizer ",unif,"\n"); fi; return FRAffineGroup(dim,ring,unif,trans); end); InstallMethod(FRAffineGroup, "(FR) for a dimension, a ring, an element, a transversal", [IsPosInt,IsRing,IsRingElement,IsCollection], function(dim,ring,unif,transversal) local d, G, phi, t, i, fam, tval, eval, o, out, a; d := Length(transversal); phi := PermutationMat(PermList(Concatenation([dim],[1..dim-1])),dim+1,ring); phi[1][dim] := 1/unif; fam := FREFamily([1..d]); if IsIntegers(ring) then eval := x->x mod unif; elif IsUnivariatePolynomialRing(ring) and [unif]=IndeterminatesOfPolynomialRing(rin eval := x->Value(x,Zero(ring)); else Error("FRAffineGroup: cannot handle ring ",ring," with uniformizer ",unif,"\n"); fi; tval := List(transversal,eval); out := []; for i in transversal do for t in transversal do if Inverse(eval(t))<>fail then o := []; for a in transversal do Add(o,Position(tval,eval(i+a*t))); od; Add(out,PermList(o)); fi; od; od; G := Group(MinimalGeneratingSet(Group(out))); # vertex group G := FullSCGroup([1..d],G,IsFRObject); SetCorrespondence(G,GroupHomomorphismByFunction(MatrixAlgebra(ring,dim+1), G,function(mat) local i, j, states, trans, out, t, o, x, y, p, a; for i in [1..dim] do for j in [1..i-1] do if not IsZero(eval(mat[i][j])) then return fail; fi; od; if Inverse(eval(mat[i][i]))=fail then return fail; fi; if not IsZero(mat[i][dim+1]) then return fail; fi; od; if not IsOne(mat[dim+1][dim+1]) then return fail; fi; states := [mat]; trans := []; out := []; i := 1; while i <= Length(states) do t := []; o := []; for a in transversal do x := ShallowCopy(states[i]); x[dim+1] := x[dim+1]+a*states[i][1]; y := eval(x[dim+1][1]); p := Position(tval,y); if p=fail then return fail; fi; Add(o,p); x[dim+1][1] := x[dim+1][1]-transversal[p]; x := x^phi; p := Position(states,x); if p=fail then Add(states,x); Add(t,Length(states)); else Add(t,p); fi; od; Add(trans,t); Add(out,o); if not ISINVERTIBLE@(out[i]) then return fail; fi; i := i+1; if RemInt(i,10)=0 then Info(InfoFR, 2, "FRAffineGroup: at least ",i," states"); fi; od; i := MealyElementNC(fam,trans,out,1); return i; end)); return G; end); InstallGlobalFunction(CayleyMachine, function(g) local e, h; e := Elements(Range(RegularActionHomomorphism(g))); return MealyMachine(List(e,x->[1..Size(e)]),List(e,Inverse)); end); InstallGlobalFunction(CayleyGroup, function(g) local h, m, id, s; h := RegularActionHomomorphism(g); m := SCGroup(CayleyMachine(Range(h))); s := GeneratorsOfGroup(m); id := First(s,x->ActivityPerm(x)=()); m!.Correspondence := [GroupHomomorphismByImages(g,m,GeneratorsOfGroup(g),List(Gene SetName(m,Concatenation("CayleyGroup(",STRINGGROUP@(g),")")); return m; end); InstallMethod(LamplighterGroup, "(FR) yielding an FR group", [IsFRGroup,IsGroup], function(filter,G) local L; if IsAbelian(G) and IsFinite(G) then L := CayleyGroup(G); L!.Name := Concatenation("LamplighterGroup(",StructureDescription(G),")"); return L; else TryNextMethod(); fi; end); ############################################################################# ############################################################################# ## #E BinaryKneadingGroup #E BasilicaGroup ## BindGlobal("BINARYKNEADINGMACHINE@", function(arg) local dbl, i, s, G, M, gen, act, h0, h1, k, n, ksym, transition, output, name, kseq, preperiod, pe if arg=[] then arg := ["*"]; fi; kseq := ["",""]; ksym := function(c) if c='0' or c=0 then return '0'; elif c='1' or c=1 then return '1'; else Error("Kneading symbol should be 0,1,'0' or '1', but not ",c,"\n"); fi; end; name := "("; if IsRat(arg[1]) then # argument is theta dbl := function(a) if a>=1/2 then return 2*a-1; else return 2*a; fi; end; h0 := function(a) if a<arg[1] then return a/2; else return (a+1)/2; fi; end; h1 := function(a) if a<arg[1] then return (a+1)/2; else return a/2; fi; end; gen := function(a) if a in period then return Position(period,a); else return 1; fi; end; act := function(x) if x=arg[1] then return (1,2); else return (); fi; end; i := arg[1]; period := [666]; # out of the way; will correspond to id while not i in period do if i=arg[1]/2 or i=(arg[1]+1)/2 then k := '*'; elif i>arg[1]/2 and i<(arg[1]+1)/2 then k := '1'; else k := '0'; fi; if IsEvenInt(DenominatorRat(i)) then Add(kseq[1],k); else Add(kseq[2],k); fi; Add(period,i); i := dbl(i); od; transition := []; output := []; for i in period do Add(transition, [gen(h0(i)),gen(h1(i))]); Add(output, act(i)); od; M := MealyMachine(transition, output); G := SCGroup(M); G!.Correspondence := function(alpha) local p; p := Position(period,alpha); if p<>fail then return GeneratorsOfGroup(G)[p]; else return One(G); fi; end; Append(name,String(arg[1])); elif not ForAll(arg,IsList) then Error("Arguments should be lists\n"); elif (Length(arg)=2 and arg[1]<>[]) or (Length(arg)=1 and IsPeriodicList(arg[1])) then # argument is pair of lists w,v if Length(arg)=2 then preperiod := arg[1]; period := arg[2]; else preperiod := PrePeriod(arg[1]); period := Period(arg[2]); fi; k := Length(preperiod); n := Length(period); transition := [[n+k+1,n+k+1]]; # b1 output := [(1,2)]; Add(name,'"'); #" to fix font-lock for i in [1..k-1] do s := ksym(preperiod[i]); if s='1' then Add(transition,[n+k+1,i]); else Add(transition,[i,n+k+1]); fi; Add(name,s); Add(kseq[1],s); Add(output,()); od; s := ksym(preperiod[k]); if s=ksym(period[n]) then Error("Last symbols of w and v must differ\n"); fi; if s='1' then Add(transition,[n+k,k]); # a1 else Add(transition,[k,n+k]); fi; Add(name,s); Add(kseq[1],s); Add(output,()); Append(name,"\",\""); for i in [1..n-1] do s := ksym(period[i]); if s='1' then Add(transition,[n+k+1,k+i]); else Add(transition,[k+i,n+k+1]); fi; Add(name,s); Add(kseq[2],s); Add(output,()); od; Add(transition,[n+k+1,n+k+1]); # identity state Add(output,()); Add(name,ksym(period[n])); Add(kseq[2],ksym(period[n])); Add(name,'"'); M := MealyMachine(transition, output); G := SCGroup(M); G!.Correspondence := [GeneratorsOfGroup(G){[1..k]}, GeneratorsOfGroup(G){[1+k..n+k]}]; elif Length(arg)=1 or arg[1]=[] then # argument is list v period := Concatenation(arg); Add(name,'"'); if Length(period)=0 then n := 1; elif period[Length(period)]='*' then n := Length(period); else n := Length(period)+1; fi; transition := [[n+1,n]]; # a1 output := [(1,2)]; for i in [1..n-1] do s := ksym(period[i]); if s='1' then Add(transition,[n+1,i]); else Add(transition,[i,n+1]); fi; Add(name,s); Add(kseq[2],s); Add(output,()); od; Add(transition,[n+1,n+1]); # identity state Add(output,()); Append(name,"*\""); Add(kseq[2],'*'); M := MealyMachine(transition, output); G := SCGroup(M); G!.Correspondence := GeneratorsOfGroup(G); fi; SetKneadingSequence(G,PeriodicList(kseq[1],kseq[2])); SetKneadingSequence(M,PeriodicList(kseq[1],kseq[2])); Append(name,")"); return [M,G,name]; end); BindGlobal("PERIODICBKG_PREIMAGE@", function(G,depth) local a, s, t, kseq, i, j, n, d, epsilon, F, r, tau, image, knows, nuke, nukeimg, Ggens, Fgens, Sgens, preimage, makeSgens; kseq := KneadingSequence(G); a := ShallowCopy(Period(kseq)); n := Length(a); a[n] := '0'; d := n/Length(Period(CompressedPeriodicList("",a))); if d>1 then epsilon := 1; else epsilon := -1; a[n] := '1'; d := n/Length(Period(CompressedPeriodicList("",a))); fi; Ggens := GeneratorsOfGroup(G); makeSgens := function(Fgens) local i, Sgens; Sgens := []; for i in [1..n] do if kseq[i]='0' then Add(Sgens,Fgens[i+1]); elif kseq[i]='1' then Add(Sgens,Fgens[i+1]^(Fgens[1]^-1)); else Add(Sgens,Fgens[1]^2); fi; od; return Sgens; end; if depth=infinity then F := FreeGroup("a","t"); a := F.1; t := F.2; s := One(F); for i in [1..n-1] do if kseq[i]='1' then s := a*s; fi; s := s^t; od; r := [a^(t^n)/(a^2)^s]; for i in [1..n-1] do for j in [1..n-1] do Add(r,Comm(a^(t^i),a^(t^j*a))); Add(r,Comm(a^(t^i),a^(t^j*a^3))); od; od; F := F / r; a := F.1; t := F.2; r := a^-1; Fgens := [r]; for i in [1..n-1] do r := r^t; if kseq[i]='1' then r := r^(a^-1); fi; Add(Fgens,r); od; F := Subgroup(F,Fgens); else F := FreeGroup(n,"a"); a := GeneratorsOfGroup(F); s := GroupHomomorphismByImagesNC(F,F,a,makeSgens(a)); r := []; if depth=-1 then depth := 0; knows := true; # knows that we want an L presentation else knows := false; fi; for i in [2..n] do for j in [2..n] do if kseq[i-1]=kseq[j-1] then Append(r,ITERATEMAP@(s,depth,Comm(a[i],a[j]^a[1]))); else Append(r,ITERATEMAP@(s,depth,Comm(a[i],a[j]))); Append(r,ITERATEMAP@(s,depth,Comm(a[i],a[j]^(a[1]^2)))); fi; od; od; if knows then F := LPresentedGroup(F,[],[s],r); else F := F / r; fi; Fgens := GeneratorsOfGroup(F); fi; tau := function(g) local x, t; t := 0; for x in Germs(g) do if Output(g,ConfinalityClass(x[2])[n],1)=2 then t := t+2*ConfinalityClass(x[1])[n]-3; fi; od; return Ggens[n]^t; end; nuke := [One(G)]; Append(nuke,Ggens); Append(nuke,List(Ggens,Inverse)); nukeimg := [One(F)]; Append(nukeimg,Fgens); Append(nukeimg,List(Fgens,Inverse)); for j in [1..d-1] do for i in [1..n] do r := RemInt(i+(n/d)*j-1,n)+1; Add(nuke,Ggens[i]^epsilon/Ggens[r]^epsilon); Add(nukeimg,Fgens[i]^epsilon/Fgens[r]^epsilon); od; od; SortParallel(nuke,nukeimg); if depth=infinity then knows := NewDictionary(nuke[1],true); for i in [1..Length(nuke)] do AddDictionary(knows,nuke[i],nukeimg[i]); od; image := function(g) local todo, recur; todo := NewDictionary(g,false); recur := function(g) local i, x, y; i := LookupDictionary(knows,g); if i<>fail then return i; fi; i := DecompositionOfFRElement(g); if not i[2] in [[1,2],[2,1]] then return fail; fi; if KnowsDictionary(todo,g) then return fail; # we reached a recurring state not in the nucleus fi; AddDictionary(todo,g); x := recur(i[1][1]); y := recur(LeftQuotient(tau(i[1][1]),i[1][2])); if x=fail or y=fail then return fail; fi; x := x^t*a*y^t; if ISONE@(i[2]) then x := x/a; fi; AddDictionary(knows,g,x); return x; end; return recur(g); end; r := FreeGroup(n,"a"); Fgens := GeneratorsOfGroup(r); Sgens := makeSgens(Fgens); preimage := function(w) local up, down, g, i, j; up := 0; down := 0; g := One(Ggens[1]); for i in LetterRepAssocWord(UnderlyingElement(w)) do if AbsInt(i)=1 then i := r.1^(-SignInt(i)); for j in [1..up] do i := MappedWord(i,Fgens,Sgens); od; g := g*MappedWord(i,Fgens,Ggens); elif i=2 then if up>0 then up := up-1; else down := down+1; g := VertexElement(1,g); fi; elif i=-2 then if down>0 and ActivityPerm(g)=() then down := down-1; g := State(g,1); else up := up+1; fi; fi; od; if up<>down then return fail; Error("Element ",w," has non-trivial translation ",down-up,"\n"); elif up>0 then return fail; Error("Element ",w," does not fix the root vertex\n"); fi; return g; end; else Sgens := makeSgens(Fgens); knows := NewDictionary(nuke[1],true); for i in [1..Length(nuke)] do AddDictionary(knows,nuke[i],nukeimg[i]); od; image := function(g) local todo, recur; todo := NewDictionary(g,false); recur := function(g) local i, x, y; i := LookupDictionary(knows,g); if i<>fail then return i; fi; i := DecompositionOfFRElement(g); if not i[2] in [[1,2],[2,1]] then return fail; fi; if KnowsDictionary(todo,g) then return fail; # we reached a recurring state not in the nucleus fi; AddDictionary(todo,g); x := recur(i[1][1]); y := recur(LeftQuotient(tau(i[1][1]),i[1][2])); if x=fail or y=fail then return fail; fi; x := MappedWord(x,Fgens,Sgens)/Fgens[1]* MappedWord(y,Fgens,Sgens); if ISONE@(i[2]) then x := x*Fgens[1]; fi; if MappedWord(x,Fgens,Ggens)<>g then return fail; fi; AddDictionary(knows,g,x); return x; end; return recur(g); end; preimage := w->MappedWord(w,Fgens,Ggens); fi; return rec(F:=F, image:=image, preimage:=preimage, reduce:=w->w); end); BindGlobal("PREPERIODICBKG_PREIMAGE@", function(G,depth) local kseq, k, n, d, a, b, i, j, rel, sigma, t, w, glob_t, glob_s, glob_m, glob_u, makeSgens, image, knows, preimage, dihedralimage, reduce, tau, F, Fgens, Ggens, Sgens, Fnuke, Gnuke, O, creator; kseq := KneadingSequence(G); k := Length(PrePeriod(kseq)); n := Length(Period(kseq)); d := n/Length(Period(CompressedPeriodicList(kseq))); Ggens := GeneratorsOfGroup(G); makeSgens := function(Fgens) # returns images of Fgens under sigma # also sets globals glob_t, glob_s, glob_m, glob_u # local i, s, t, Sgens, ob, ooob; Sgens := []; for i in [1..k+n] do if i<k+n then s := i+1; else s := i+1-n; fi; if kseq[i]='0' then Add(Sgens,Fgens[s]); else Add(Sgens,Fgens[s]^Fgens[1]); fi; od; ob := Sgens[1]^(Fgens[1]^-1); if k>=2 and n>=2 then if kseq[k+n-1]=kseq[k-1] then s := Fgens[1]; t := ob; else s := One(Fgens[1]); t := One(Fgens[1]); fi; glob_m := 1; elif k>=3 and n=1 then ooob := Fgens[3]; if kseq[2]='0' then ooob := ooob^Fgens[1]; fi; if kseq[1]='0' then ooob := ooob^ob; fi; if kseq[k-1]='1' and kseq[k-2]='1' then s := ob; t := ooob; elif kseq[k-1]='0' and kseq[k-2]='0' then s := ob^(Fgens[1]^-1); t := ooob^(ob^-1); else s := One(Fgens[1]); t := One(Fgens[1]); fi; if kseq[k]<>kseq[k-1] then s := Fgens[1]*s; t := ob^-1*t; fi; glob_m := 1; elif k=2 and n=1 then if kseq[1]<>kseq[2] then s := Fgens[1]; t := ob; else s := One(Fgens[1]); t := One(Fgens[1]); fi; glob_m := 2; elif k=1 and n>=2 then s := One(Fgens[1]); t := One(Fgens[1]); glob_m := 2; else s := One(Fgens[1]); t := One(Fgens[1]); glob_m := infinity; fi; if kseq[k]='1' then glob_u := t^Fgens[1]; t := Fgens[1]*t; else glob_u := t; fi; glob_s := Fgens[k+n]^s; glob_t := Fgens[k+1]^t; Sgens[k] := glob_t; return Sgens; end; if depth=infinity then F := FreeGroup("a","b","t"); a := F.1; b := F.2; t := F.3; w := One(F); Fgens := [b]; for i in [1..k] do w := w^t; if kseq[i]='1' then w := b*w; fi; Add(Fgens,b^(t^i/w)); od; rel := [a^2,b^2,b^(t^k)/a^w]; w := One(F); for i in [k+1..k+n] do w := w^t; if kseq[i]='1' then w := b*w; fi; od; Remove(Fgens); Append(Fgens,ListWithIdenticalEntries(n,One(F))); # not needed makeSgens(Fgens); # to compute glob_u Add(rel,a^(glob_u^-1*t^n)/a^(glob_u^-1*w)); if glob_m<>infinity then Add(rel,(a*b)^(2^(glob_m+1))); fi; --> -------------------- --> maximum size reached --> -------------------- [ Verzeichnis aufwärts0.64unsichere Verbindung Übersetzung europäischer Sprachen durch Browser ] |
2026-03-28