6 Approximating the Schur multiplier

uses Tietze transformations for computing an equivalent set of relators for LpGroup so that a generating set for its Schur multiplier can be read off easily.

computes a finitely generated quotient of the Schur multiplier of LpGroup. The method computes the image of the Schur multiplier of LpGroup in the Schur multiplier of its class-c quotient.

computes a list of endomorphisms of the FiniteRankSchurMultiplier of LpGroup. These are the endomorphisms of the invariant L-presentation induced to FiniteRankSchurMultiplier.

computes an epimorphism of the covering group of the class-d quotient onto the covering group of the class-c quotient.

computes an epimorphism of the d-th FiniteRankSchurMultiplier of the invariant LpGroup onto the c-th FiniteRankSchurMultiplier. Its restricts the epimorphism EpimorphismCoveringGroups to the corresponding finite rank multipliers.

computes the image of the free group element elm in the c-th FiniteRankSchurMultiplier. Note that elm must be a relator contained in the Schur multiplier of LpGroup; otherwise, the function fails in computing the image.

vskip3ex

The following example tackels the Schur multiplier of the Grigorchuk group.

gap> G := ExamplesOfLPresentations( 1 );;
gap> gens := GeneratingSetOfMultiplier( G );
rec( FixedGens := [ b^-2*c^-2*d^-2*b*c*d*b*c*d ],
  IteratedGens := [ d^-1*a^-1*d^-1*a*d*a^-1*d*a,
      d^-1*a^-1*c^-1*a^-1*c^-1*a^-1*d^-1*a*c*a*c*a*d*a^-1*c^-1*a^-1*c^-1*a^
        -1*d*a*c*a*c*a ],
  BasisGens := [ a^2, b*c*d, b^-2*d^-2*b*c*d*b*c*d, b^-2*c^-2*b*c*d*b*c*d ],
  Endomorphisms := [ [ a, b, c, d ] -> [ a^-1*c*a, d, b, c ] ] )
gap> H := FiniteRankSchurMultiplier( G, 5 );
Pcp-group with orders [ 2, 2, 2 ] 
gap> GeneratorsOfGroup( H );
[ g15, g17, g16 ]
gap> EndomorphismsOfFRSchurMultiplier( G, 5 );
[ [ g15, g16, g17 ] -> [ g15, id, g16 ] ]
gap> Kernel( last[1] );
Pcp-group with orders [ 2 ]
gap> GeneratorsOfGroup( last );
[ g16 ]
gap> EpimorphismFiniteRankSchurMultipliers( G, 5, 2 );
[ g15, g16, g17 ] -> [ g10, id, g13 ]
gap> Range( last ) = FiniteRankSchurMultiplier( G, 2 );
true
gap> Kernel( EpimorphismFiniteRankSchurMultipliers( G, 5, 2 ) );
Pcp-group with orders [ 2 ]
gap> GeneratorsOfGroup( last );
[ g16 ]
gap> Kernel( EpimorphismFiniteRankSchurMultipliers( G, 5, 2 ) ) =
> Kernel( EndomorphismsOfFRSchurMultiplier( G, 5 )[1] );
true
gap> ImageInFiniteRankSchurMultiplier( G, 5, gens.FixedGens[1] );
g15
gap> ImageInFiniteRankSchurMultiplier(G,5,Image(gens.Endomorphisms[1],
> gens.IteratedGens[1] ) );
g16
gap> ImageInFiniteRankSchurMultiplier(G,5,gens.IteratedGens[1] );
g17

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