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#############################################################################
##
#W Whitehd.gi FGA package Christian Sievers
##
## Computations with Whitehead automorphisms
##
#Y 2004 - 2012
##
InstallMethod( FGA_WhiteheadAutomorphisms,
"for finitely generated free groups",
[ CanComputeWithInverseAutomaton ],
function( G )
local ngens, ngen, combs, auts, L, R;
ngens := [ 1 .. RankOfFreeGroup( G ) ];
auts := [];
for ngen in ngens do
combs := Combinations( Difference( ngens, [ngen] ));
for L in combs do
for R in combs do
if L <> [] or R <> [] then
Add( auts, FGA_WhiteheadAutomorphism( G, ngen, L, R ));
fi;
od;
od;
od;
return auts;
end );
InstallMethod( FGA_NielsenAutomorphisms,
"for finitely generated free groups",
[ CanComputeWithInverseAutomaton ],
G -> Filtered( f -> FGA_WhiteheadParams(f).isnielsen ) );
InstallGlobalFunction( FGA_WhiteheadAutomorphism,
function( G, ngen, L, R )
local gens, gen, ng, g, img, imginv, imgs, imgsinv, aut, autinv;
imgs := [];
imgsinv := [];
gens := GeneratorsOfGroup( G );
gen := gens[ngen];
for ng in [ 1 .. RankOfFreeGroup( G ) ] do
img := gens[ng];
imginv := img;
if ng in L then
img := LeftQuotient( gen, img );
imginv := gen * imginv;
fi;
if ng in R then
img := img * gen;
imginv := imginv / gen;
fi;
Add( imgs, img );
Add( imgsinv, imginv);
od;
aut := GroupHomomorphismByImagesNC( G, G, GeneratorsOfGroup(G), imgs );
autinv := GroupHomomorphismByImagesNC( G, G, GeneratorsOfGroup(G), imgsinv );
SetInverse( aut, autinv );
SetInverse( autinv, aut );
SetFGA_WhiteheadParams( aut , rec( gen := ngen, L := L, R := R,
isnielsen := Length(L)+Length(R)=1 ) );
SetFGA_WhiteheadParams( autinv, true );
return aut;
end );
InstallGlobalFunction( FGA_WhiteheadAnalyse,
function( whs, elm, act , len , val, comb , combrest )
# [w] * e * (e*w->e) * (e->Int) * v * (v*w->v) * (v*e->r) -> r
local l, newl, wh, bestwh , newelm, bestnewelm;
# Int , w , Maybe w, e
l := len( elm );
while true do
bestwh := fail;
for wh in whs do
newelm := act( elm, wh );
newl := len( newelm );
if newl < l then
l := newl;
bestwh := wh;
bestnewelm := newelm;
fi;
od;
if bestwh=fail then
return combrest( val, elm );
fi;
val := comb( val, bestwh );
elm := bestnewelm;
od;
# not reached
end );
########################################################################
# Equation numbers and pages refer to
# Jakob Nielsen: Die Isomorphismengruppe der freien Gruppen
# see ../doc/manual.bib
########################################################################
InstallGlobalFunction( FGA_WhiteheadToPQOU,
function ( w , p , q , o , u )
# w * g * g * g * g -> g
local n ,g, whp, word, sign, nik;
n := RankOfFreeGroup( Source ( w ) );
if FGA_WhiteheadParams(w) = true then
w := Inverse(w);
sign := -1;
else
sign := 1;
fi;
whp:= FGA_WhiteheadParams(w);
word := One(p);
for g in [ 1 .. n ] do
if g in whp.L or g in whp.R then
# using and possibly combining eq. (12) and (11)
# for V_{g,gen}^-1 and U_{g,gen}
nik := FGA_NikToPQ( g, whp.gen, p, q );
word := word * nik^-1;
if g in whp.L then
word := word * o * u^sign * o;
# eq. (7)
fi;
if g in whp.R then
word := word * u^sign;
fi;
word := word * nik;
fi;
od;
return word;
end );
InstallGlobalFunction( FGA_NikToPQ,
function( i , k , p , q )
# Int * g * g * g -> g
# eq. (8)
local l;
l := k-i;
if i<k then
l := l-1;
fi;
return (q*p)^l * q^(i-1);
end );
InstallGlobalFunction( FGA_TiToPQ,
function( i , p , q )
# Int * g * g -> g
# follows from eq. at the middle of page 171
return q^(2-i)*p*(q*p)^(i-2);
end );
InstallGlobalFunction( FGA_ExtSymListRepToPQO,
function( target, p , q , o )
# [Int] * g * g * g -> g
local rank, word1, word2, lastshift, i, t,
f2, P, Q, Pperm, Qperm, homperm, homrep, perm;
f2 := FreeGroup("P","Q");
P := f2.1; Q := f2.2;
word1 := One(p);
word2 := word1;
rank := Length( target );
Pperm := (1,2);
Qperm := PermList(Concatenation([2..rank],[1]));
homperm := GroupHomomorphismByImagesNC( f2,
SymmetricGroup( rank ),
GeneratorsOfGroup(f2),
[ Pperm, Qperm ] );
homrep := GroupHomomorphismByImagesNC( f2,
Group( p, q ),
GeneratorsOfGroup( f2 ),
[ p, q ] );
# first get rid of extendedness, using o and q
lastshift := 1;
for i in [ 1 .. rank ] do
if not IsPosInt( target[i] ) then
word1 := word1 * q^(lastshift-i) * o;
lastshift := i;
target[i] := AbsInt(target[i]);
fi;
od;
word1 := word1 * q^(lastshift-1);
# now target is a permutation, represent it as such
target := SortingPerm(target);
# decompose it as product of powers of T_i, compare p. 171
while not IsOne( target ) do
i := LargestMovedPoint( target );
t := i^target;
perm := FGA_TiToPQ( i, P, Q );
word2 := (perm^homrep)^(t-i) * word2;
target := target * (perm^homperm)^(i-t);
od;
return word1*word2;
end );
InstallGlobalFunction( FGA_CurryAutToPQOU,
function( p, q, o, u)
return
function( aut )
local fg, words, wh;
fg := Source( aut );
words := List( GeneratorsOfGroup( fg ), gen -> gen ^ aut );
wh := FGA_WhiteheadAutomorphisms( fg );
# use Nielsen generators only
wh := Filtered( wh, f -> FGA_WhiteheadParams(f).isnielsen );
wh := Concatenation( wh, List( wh, Inverse ));
return FGA_WhiteheadAnalyse( wh, words, OnTuples,
l -> Sum( l, Length ),
One( p ),
function( v, w )
return FGA_WhiteheadToPQOU( Inverse(w), p, q, o, u ) * v;
end,
function( v, e )
e := List( e, g -> LetterRepAssocWord(g)[1] );
return FGA_ExtSymListRepToPQO( e, p, q, o ) * v;
end );
end;
end );
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