GAP (lpres) - Chapter 5: Approximating the Schur multiplier

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5 Approximating the Schur multiplier
5.1 Methods
5.1-1 GeneratingSetOfMultiplier
5.1-2 FiniteRankSchurMultiplier
5.1-3 EndomorphismsOfFRSchurMultiplier
5.1-4 EpimorphismCoveringGroups
5.1-5 EpimorphismFiniteRankSchurMultiplier
5.1-6 ImageInFiniteRankSchurMultiplier

### 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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