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5 Approximating the Schur multiplier

The algorithm in [Har10] approximates the Schur multiplier of an invariantly finitely L-presented group by the quotients in its Dwyer-filtration. This is implemented in the lpres-package and the following methods are available:

5.1 Methods

5.1-1 GeneratingSetOfMultiplier
 ‣ GeneratingSetOfMultiplier( lpgroup ) ( operation )

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.

5.1-2 FiniteRankSchurMultiplier
 ‣ FiniteRankSchurMultiplier( lpgroup, c ) ( operation )

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.

5.1-3 EndomorphismsOfFRSchurMultiplier
 ‣ EndomorphismsOfFRSchurMultiplier( lpgroup, c ) ( operation )

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

5.1-4 EpimorphismCoveringGroups
 ‣ EpimorphismCoveringGroups( lpgroup, d, c ) ( operation )

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

5.1-5 EpimorphismFiniteRankSchurMultiplier
 ‣ EpimorphismFiniteRankSchurMultiplier( lpgroup, d, c ) ( operation )

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.

5.1-6 ImageInFiniteRankSchurMultiplier
 ‣ ImageInFiniteRankSchurMultiplier( lpgroup, c, elm ) ( function )

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.

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 ) ) =
</A> Kernel( EndomorphismsOfFRSchurMultiplier( G, 5 )[1] );
true
gap> ImageInFiniteRankSchurMultiplier( G, 5, gens.FixedGens[1] );
g15
gap> ImageInFiniteRankSchurMultiplier(G,5,Image(gens.Endomorphisms[1],
</A> gens.IteratedGens[1] ) );
g16
gap> ImageInFiniteRankSchurMultiplier(G,5,gens.IteratedGens[1] );
g17

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