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<title>GAP (lpres) - Chapter 5: Approximating the Schur multiplier</title>
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<p><a id="X7FBE94957D7ECCFC" name="X7FBE94957D7ECCFC"></a></p>
<div class="ChapSects"><a href="chap5.html#X7FBE94957D7ECCFC">5 <span class="Heading">Approximating the Schur multiplier</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap5.html#X8606FDCE878850EF">5.1 <span class="Heading">Methods</span></a>
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<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X83A5F95E84D3B662">5.1-1 GeneratingSetOfMultiplier</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X87A3D6C07D99C79A">5.1-2 FiniteRankSchurMultiplier</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X78084374873BDFE1">5.1-3 EndomorphismsOfFRSchurMultiplier</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X7CF92D9880A3687E">5.1-4 EpimorphismCoveringGroups</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X86EAE6457CE03B7B">5.1-5 EpimorphismFiniteRankSchurMultiplier</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X87182BC081DCA91E">5.1-6 ImageInFiniteRankSchurMultiplier</a></span>
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<h3>5 <span class="Heading">Approximating the Schur multiplier</span></h3>

<p>The algorithm in <a href="chapBib.html#biBMR2678952">[Har10]</a> approximates the Schur multiplier of an invariantly finitely L-presented group by the quotients in its Dwyer-filtration. This is implemented in the <strong class="pkg">lpres</strong>-package and the following methods are available:</p>

<p><a id="X8606FDCE878850EF" name="X8606FDCE878850EF"></a></p>

<h4>5.1 <span class="Heading">Methods</span></h4>

<p><a id="X83A5F95E84D3B662" name="X83A5F95E84D3B662"></a></p>

<h5>5.1-1 GeneratingSetOfMultiplier</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; GeneratingSetOfMultiplier</code>( <var class="Arg">lpgroup</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>uses Tietze transformations for computing an equivalent set of relators for <var class="Arg">lpgroup</var> so that a generating set for its Schur multiplier can be read off easily.</p>

<p><a id="X87A3D6C07D99C79A" name="X87A3D6C07D99C79A"></a></p>

<h5>5.1-2 FiniteRankSchurMultiplier</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; FiniteRankSchurMultiplier</code>( <var class="Arg">lpgroup</var>, <var class="Arg">c</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>computes a finitely generated quotient of the Schur multiplier of <var class="Arg">lpgroup</var>. The method computes the image of the Schur multiplier of <var class="Arg">lpgroup</var> in the Schur multiplier of its class-<var class="Arg">c</var> quotient.</p>

<p><a id="X78084374873BDFE1" name="X78084374873BDFE1"></a></p>

<h5>5.1-3 EndomorphismsOfFRSchurMultiplier</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; EndomorphismsOfFRSchurMultiplier</code>( <var class="Arg">lpgroup</var>, <var class="Arg">c</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>computes a list of endomorphisms of the `FiniteRankSchurMultiplier' of <var class="Arg">lpgroup</var>. These are the endomorphisms of the invariant L-presentation induced to `FiniteRankSchurMultiplier'.</p>

<p><a id="X7CF92D9880A3687E" name="X7CF92D9880A3687E"></a></p>

<h5>5.1-4 EpimorphismCoveringGroups</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; EpimorphismCoveringGroups</code>( <var class="Arg">lpgroup</var>, <var class="Arg">d</var>, <var class="Arg">c</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>computes an epimorphism of the covering group of the class-<var class="Arg">d</var> quotient onto the covering group of the class-<var class="Arg">c</var> quotient.</p>

<p><a id="X86EAE6457CE03B7B" name="X86EAE6457CE03B7B"></a></p>

<h5>5.1-5 EpimorphismFiniteRankSchurMultiplier</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; EpimorphismFiniteRankSchurMultiplier</code>( <var class="Arg">lpgroup</var>, <var class="Arg">d</var>, <var class="Arg">c</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>computes an epimorphism of the <span class="SimpleMath">d</span>-th `FiniteRankSchurMultiplier' of the invariant <var class="Arg">lpgroup</var> onto the <span class="SimpleMath">c</span>-th `FiniteRankSchurMultiplier'. Its restricts the epimorphism `EpimorphismCoveringGroups' to the corresponding finite rank multipliers.</p>

<p><a id="X87182BC081DCA91E" name="X87182BC081DCA91E"></a></p>

<h5>5.1-6 ImageInFiniteRankSchurMultiplier</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; ImageInFiniteRankSchurMultiplier</code>( <var class="Arg">lpgroup</var>, <var class="Arg">c</var>, <var class="Arg">elm</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>computes the image of the free group element <var class="Arg">elm</var> in the <var class="Arg">c</var>-th `FiniteRankSchurMultiplier'. Note that <var class="Arg">elm</var> must be a relator contained in the Schur multiplier of <var class="Arg">lpgroup</var>; otherwise, the function fails in computing the image.</p>

<p>The following example tackels the Schur multiplier of the Grigorchuk group.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">G := ExamplesOfLPresentations( 1 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">gens := GeneratingSetOfMultiplier( G );</span>
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 ] ] )
<span class="GAPprompt">gap></span> <span class="GAPinput">H := FiniteRankSchurMultiplier( G, 5 );</span>
Pcp-group with orders [ 2, 2, 2 ] 
<span class="GAPprompt">gap></span> <span class="GAPinput">GeneratorsOfGroup( H );</span>
[ g15, g17, g16 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">EndomorphismsOfFRSchurMultiplier( G, 5 );</span>
[ [ g15, g16, g17 ] -> [ g15, id, g16 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">Kernel( last[1] );</span>
Pcp-group with orders [ 2 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">GeneratorsOfGroup( last );</span>
[ g16 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">EpimorphismFiniteRankSchurMultipliers( G, 5, 2 );</span>
[ g15, g16, g17 ] -> [ g10, id, g13 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">Range( last ) = FiniteRankSchurMultiplier( G, 2 );</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">Kernel( EpimorphismFiniteRankSchurMultipliers( G, 5, 2 ) );</span>
Pcp-group with orders [ 2 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">GeneratorsOfGroup( last );</span>
[ g16 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">Kernel( EpimorphismFiniteRankSchurMultipliers( G, 5, 2 ) ) =</span>
</A> Kernel( EndomorphismsOfFRSchurMultiplier( G, 5 )[1] );
true
<span class="GAPprompt">gap></span> <span class="GAPinput">ImageInFiniteRankSchurMultiplier( G, 5, gens.FixedGens[1] );</span>
g15
<span class="GAPprompt">gap></span> <span class="GAPinput">ImageInFiniteRankSchurMultiplier(G,5,Image(gens.Endomorphisms[1],</span>
</A> gens.IteratedGens[1] ) );
g16
<span class="GAPprompt">gap></span> <span class="GAPinput">ImageInFiniteRankSchurMultiplier(G,5,gens.IteratedGens[1] );</span>
g17
</pre></div>


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