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:
‣ 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.
‣ 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.
‣ 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'.
‣ 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.
‣ 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.
‣ 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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