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#############################################################################
##
#W betti.gi Karel Dekimpe
#W Bettina Eick
##
#############################################################################
##
## The following functions can be used to determine Betti-numbers of a
## torsion-free polycyclic group given by a pcp presentation. All
## Betti-numbers can be obtained if G has Hirsch length at most 5.
##
## The Betti-numbers B(G,m) are defined as the ranks of H_m(G,Z) for the
## trivial G-module Z. If M is the orientation G-module, then we can also
## characterise B(G,m) for n >= m >= n-2 as the ranks of H^n-m(G,M).
## Further, the alternating sum of all Betti-numbers is 0 using the
## Euler characteristic.
##
#############################################################################
##
#F OrientationModule( G )
##
InstallMethod( OrientationModule, "for pcp groups", true, [IsPcpGroup], 0,
function( G )
local pcps, gens, mats, acts, dets, i, pcp;
pcps := PcpsOfEfaSeries( G );
pcps := Filtered( pcps, x -> RelativeOrdersOfPcp(x)[1] = 0 );
gens := Igs(G);
mats := List( gens, x -> IdentityMat( 1 ) );
for pcp in pcps do
acts := LinearActionOnPcp( gens, pcp );
dets := List( acts, x -> Determinant( x ) );
for i in [1..Length(mats)] do
mats[i] := dets[i] * mats[i];
od;
od;
return mats;
end );
#############################################################################
##
#F IsOrientedMatGroup( G )
##
IsOrientedMatGroup := function( G )
return ForAll( GeneratorsOfGroup(G), x -> Determinant(x) = 1 );
end;
#############################################################################
##
#F BettiNumber( G, m )
##
BettiNumberPcpGroup := function(G,m)
local n, pcp, mats, CR, one, two;
if not IsTorsionFree( G ) then
Print("the input group must be torsion-free \n");
return fail;
fi;
# catch the trivial case
if IsFinite(G) then
if m = 0 then
return 1;
else
return 0;
fi;
fi;
# the hirsch length
n := HirschLength( G );
if m < 0 or m > n then return 0; fi;
if m = 0 then return 1; fi;
if m = 1 then
pcp := Pcp( G, DerivedSubgroup(G) );
return Length( Filtered( RelativeOrdersOfPcp( pcp ),x -> x=0 ));
fi;
if m = n then
mats := OrientationModule( G );
if ForAny( mats, x -> x[1][1] = -1 ) then
return 0;
else
return 1;
fi;
fi;
if m = 2 then
mats := List( Pcp(G), x -> IdentityMat(1) );
CR := CRRecordByMats( G, mats );
two := TwoCohomologyCR( CR ).factor.rels;
return Length( Filtered( two, x -> x = 0 ) );
fi;
if m = n-1 then
mats := OrientationModule( G );
CR := CRRecordByMats( G, mats );
one := OneCohomologyCR( CR ).factor.rels;
return Length( Filtered( one, x -> x = 0 ) );
fi;
if m = n-2 then
mats := OrientationModule( G );
CR := CRRecordByMats( G, mats );
two := TwoCohomologyCR( CR ).factor.rels;
return Length( Filtered( two, x -> x = 0 ) );
fi;
Print("Betti-number is out of range for our methods \n");
return fail;
end;
InstallMethod( BettiNumber, "for torsion-free pcp groups", true,
[IsPcpGroup, IsInt], 0,
function(G, m)
if not IsTorsionFree(G) then TryNextMethod(); fi;
if m in [3..HirschLength(G)-3] then TryNextMethod(); fi;
return BettiNumberPcpGroup(G,m);
end);
#############################################################################
##
#F BettiNumbers( G )
##
InstallMethod( BettiNumbers, "for torsion-free pcp groups", true,
[IsPcpGroup], 0,
function( G )
local n, betti;
n := HirschLength( G );
if not IsTorsionFree( G ) or n > 6 then TryNextMethod(); fi;
# set up the Betti-numbers
betti := [1];
if n = 0 then return betti; fi;
betti[2] := BettiNumber( G, 1 );
if n = 1 then return betti; fi;
betti[3] := BettiNumber( G, 2 );
if n = 2 then return betti; fi;
betti[4] := betti[1] - betti[2] + betti[3];
if n = 3 then return betti; fi;
if n > 3 then
betti[5] := BettiNumber( G, 4 );
betti[4] := betti[4] + betti[5];
fi;
if n > 4 then
betti[6] := BettiNumber( G, 5 );
betti[4] := betti[4] - betti[6];
fi;
if n > 5 then
betti[7] := BettiNumber( G, 6 );
betti[4] := betti[4] + betti[7];
fi;
return betti;
end );
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