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##########################################################################
#0
#F SL2ZTree
##
## This function will compute a G-equivalent CW-space for
## group G=SL2Z[1/(m*p)] and a tree X associated with
## amalgamated product SL2Z[1/m]*SL2z[1/m] over
## congruence subgroup gamma 0 of level p.
## Input: A pair of positive integers (m,p)
##
## Output: A G-equivarient CW-space where G is the free
## product of two copiesof SL(1/m) amalgamated by
## its congruence group of level p
##
InstallGlobalFunction(SL2ZTree,
function(m,p)
local t1,t2,
Elts,H,K,G,Gamma,
Id,ID,Idcoset,BoundaryList,
Boundary,Dimension,Action,Stabilizer,Homotopy,StabGrps,
pos,RemoveLoops,HtpyRec;
Elts:=[[[1,0],[0,1]]];
## In this case G=SL2Z
if p=0 then
H:=Group([[0,-1],[1,0]]);
K:=Group([[0,-1],[1,1]]);
G:=SL(2,Integers);
Gamma:=Group([[-1,0],[0,-1]]);
Append(Elts,Elements(H));
Append(Elts,Elements(K));
Append(Elts,Elements(Gamma));
Elts:=SSortedList(Elts);
else
if m=1 then
H:=SL(2,Integers);
K:=SL2Z(p);
Gamma:=CongruenceSubgroupGamma0(p);
else
H:=SL2Z(1/m);
K:=ConjugateSL2ZGroup(H,[[1,0],[0,p]]);
Gamma:=CongruenceSubgroup(m,p);
fi;
G:=SL2Z(1/(m*p));
ID:=Group(One(G));
Append(Elts,GeneratorsOfGroup(H));
Append(Elts,GeneratorsOfGroup(K));
Append(Elts,GeneratorsOfGroup(Gamma));
Elts:=SSortedList(Elts);
fi;
SetName(Gamma,"Gamma");
Id:=Position(Elts,[[1,0],[0,1]]);
###################################################################
#1
#F pos
## Find the position of an element in a list
## Input: A list Elts and a matrix g
## Output: If g in Elts return the position of g in the list,
## otherwise, add g to Elts and return the position.
##
pos:=function(Elts,g)
local posit;
posit:=Position(Elts,g);
if posit=fail then
Add(Elts,g);
return Length(Elts);
else
return posit;
fi;
end;
###################################################################
BoundaryList:=[];
t1:=pos(Elts,CanonicalRightCountableCosetElement(H,Elts[Id]^-1)^-1);
t2:=pos(Elts,CanonicalRightCountableCosetElement(K,Elts[Id]^-1)^-1);
Append(BoundaryList,[[[1,t1],[-2,t2]]]);
###################################################################
#1
#F Boundary
##
## This function presents the boundary map d_n: C_n -> C_{n-1}
##
## Input: a pair of integers (n,k) where n is the dimension and
## k is the position of the generator.
## Output: a list of words [g,f]
##
Boundary:=function(n,k)
local w;
if not n=1 then
return [];
fi;
w:=BoundaryList[AbsInt(k)];
if k>0 then
return w;
else
return NegateWord(w);
fi;
end;
###################################################################
#1
#F Dimension
##
## This function computes the G-rank of ZG-module C_n
##
## Input: a positive integer n
##
## Output: the G-rank of ZG-module C_n
##
Dimension:=function(n)
if not n in [0,1] then return 0;fi;
if n=0 then return 2;fi;
if n=1 then return 1;fi;
end;
###################################################################
###################################################################
#1
#F Action
##
## Input: a triple (n,k,l) of integers
##
## Output: 1 or -1
##
Action:=function(n,k,l);
return 1;
end;
###################################################################
StabGrps:=[];
Add(StabGrps,[H,K]);
Add(StabGrps,[Gamma]);
###################################################################
#1
#F Stabilizer
##
## Input: a pair of integers (n,k)
##
## Output: the kith stabiliser subgroup in dimension n
##
Stabilizer:=function(n,k);
return StabGrps[n+1][k];
end;
###################################################################
Idcoset:=pos(Elts,
CanonicalRightCountableCosetElement(Gamma,Elts[Id]^-1)^-1);
###################################################################
#1
#F RemoveLoops
##
## This function will remove loops in a path.
##
## Input: a list of integers d
##
## Output: a list of integers with no loop.
##
RemoveLoops:=function(d)
local i,h,j,l;
l:=StructuralCopy(d);
h:=[[1,0],[0,1]];
i:=1;
while i<Length(d) do
h:=h*d[i];
if h in H or h in K then
for j in [1..i-1] do
Remove(l,1);
od;
l[1]:=h;
fi;
i:=i+1;
od;
return l;
end;
###################################################################
## Create a record for the homotopy
HtpyRec:=[];
HtpyRec[1]:=[];
HtpyRec[2]:=[];
###################################################################
#1
#F Homotopy
##
## This function presents the homotopy map h_n: C_n -> C_{n+1}
##
## Input: a positive integer n and a word w.
##
## Output: a list of words.
##
Homotopy:=function(n,w)
local d,path,i,h,k,g,pk,r,t;
if not n=0 then
return [];
fi;
k:=w[1];
g:=w[2];
pk:=AbsInt(k);
if not IsBound(HtpyRec[pk][g]) then
d:=SL2ZmElementsDecomposition(Elts[g],p);
r:=[];
Add(r,d[1]);
for i in [2..Length(d)] do
if d[i]*d[i-1] in H then
r[Length(r)]:=r[Length(r)]*d[i];
else
Add(r,d[i]);
fi;
od;
d:=r;
r:=[];
Add(r,d[1]);
for i in [2..Length(d)] do
if d[i]*d[i-1] in K then
r[Length(r)]:=r[Length(r)]*d[i];
else
Add(r,d[i]);
fi;
od;
d:=StructuralCopy(r);
## kill all the loops in the path d
d:=RemoveLoops(d);
if (d[1] in K) and (not d[1] in Gamma) then
r:=[[[1,0],[0,1]]];
Append(r,d);
d:=StructuralCopy(r);
fi;
h:=[[1,0],[0,1]];
path:=[];
if d[Length(d)] in Gamma and Length(d)>1 then
Remove(d,Length(d));
fi;
if pk=1 then
if d[Length(d)] in H then
Remove(d,Length(d));fi;
for i in [1..Length(d)] do
h:=h*d[i];
t:=CanonicalRightCountableCosetElement(Gamma,h^-1)^-1;
Add(path,[(-1)^(i),pos(Elts,t)]);
od;
else
if Elts[g] in Gamma then
Add(path,[-1,Idcoset]);
fi;
if d[Length(d)] in K then
Remove(d,Length(d));
fi;
for i in [1..Length(d)] do
h:=h*d[i];
t:=CanonicalRightCountableCosetElement(Gamma,h^-1)^-1;
Add(path,[(-1)^(i),pos(Elts,t)]);
od;
fi;
HtpyRec[pk][g]:=path;
fi;
if k>0 then
return HtpyRec[pk][g];
else
return NegateWord(HtpyRec[pk][g]);
fi;
end;
#######################################################################
return Objectify(HapNonFreeResolution,
rec(
dimension:=Dimension,
boundary:=Boundary,
homotopy:=Homotopy,
elts:=Elts,
group:=G,
stabilizer:=Stabilizer,
action:=Action,
properties:=
[["length",100],
["characteristic",0],
["type","resolution"]] ));
end);
################### end of SL2ZTree ######################################
##########################################################################
#0
#F TreeOfResolutionsToSL2Zcomplex
## Input: A list of resolutions D and an arithmetic group G=SL2Z[1/m]
##
## Output: A G-equivarient CW-space
##
##
InstallGlobalFunction(TreeOfResolutionsToSL2Zcomplex,
function(D,G)
local RH,RK,RGamma,
H,K,Gamma,
NameH,
i,j,m,p,
C,Resolutions,NamesOfGroups;
RH:=D[1];
RK:=D[2];
RGamma:=D[3];
H:=RH!.group;
K:=RK!.group;
Gamma:=RGamma!.group;
NameH:=H!.Name;
if H=SL(2,Integers) then
m:=1;
p:=Gamma!.LevelOfCongruenceSubgroup;
else
i:=Position(NameH,'/');
j:=Position(NameH,']');
m:=Int(NameH{[i+1..j-1]});
p:=Gamma!.levels[2];
fi;
NamesOfGroups:=[Name(H),Name(K),Name(Gamma)];
Resolutions:=[RH,RK,RGamma];
C:=SL2ZTree(m,p);
C!.resolutions:=[Resolutions,NamesOfGroups];
return C;
end);
################### end of TreeOfResolutionsToSL2Zcomplex ################
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