<p>Let <span class="SimpleMath">\(S\)</span> be an alphabet, <span class="SimpleMath">\(Q\)</span> and <span class="SimpleMath">\(R\)</span> be subsets of the free group <span class="SimpleMath">\(F_S\)</span> over this alphabet, and <span class="SimpleMath">\(\Phi\)</span> be a set of free group endomorphisms <span class="SimpleMath">\(\varphi\colon F_S\to F_S\)</span>. An <em>L-presentation</em> is a quadruple <span class="SimpleMath">\((S,Q,\Phi,R)\)</span> and it is called <em>finite</em> if the sets <span class="SimpleMath">\(S\)</span>, <span class="SimpleMath">\(Q\)</span>, <span class="SimpleMath">\(\Phi\)</span>, and <span class="SimpleMath">\(R\)</span> are finite. A (finite) L-presentation <span class="SimpleMath">\((S,Q,\Phi,R)\)</span> defines the (<em>finitely</em>) <em>L-presented group</em></p>
<p class="center">\[ G=\left\langle S \left| Q\cup \bigcup_{\varphi\in\Phi^*}R^\varphi\right.\right\rangle\]</p>
<p>where <span class="SimpleMath">\(\Phi^*\)</span> denotes the free monoid generated by <span class="SimpleMath">\(\Phi\)</span>; that is, the closure of <span class="SimpleMath">\(\Phi\cup\{\rm id\}\)</span> under composition.</p>
<p>The elements in <span class="SimpleMath">\(Q\)</span> are the <em>fixed relators</em> and the elements in <span class="SimpleMath">\(R\)</span> are the <em>iterated relators</em> of the L-presentation <span class="SimpleMath">\((S,Q,\Phi,R)\)</span>. An L-presentation of the form <span class="SimpleMath">\((S,\emptyset,\Phi,R)\)</span> is an <em>ascending L-presentation</em> and it is an <em>invariant L-presentation</em> if the normal subgroup</p>
<p>is <span class="SimpleMath">\(\varphi\)</span>-invariant for each <span class="SimpleMath">\(\varphi\in\Phi\)</span>; that is, if <span class="SimpleMath">\(K\)</span> satisfies <span class="SimpleMath">\(K^\varphi\subset K\)</span> for each <span class="SimpleMath">\(\varphi\in\Phi\)</span>. Note that every ascending L-presentation is invariant and for each L-presentation <span class="SimpleMath">\((S,Q,\Phi,R)\)</span> there is a unique <em>underlying ascending L-presentation</em> <span class="SimpleMath">\((S,\emptyset,\Phi,R)\)</span> which is invariant. In general it is not decidable whether or not a given L-presentation is invariant as this would require a solution to the word-problem.</p>
<p>In the remainder of this manual, an L-presented group is always finitely L-presented.</p>
<h4>2.2 <span class="Heading">Creating an L-presented group</span></h4>
<p>The construction of an L-presented group is similar to the construction of a finitely presented group (see Chapter <a href="../../../doc/ref/chap47_mj.html#X7AA982637E90B35A"><span class="RefLink">Reference: Finitely Presented Groups</span></a> of the <strong class="pkg">GAP</strong> Reference manual for further details).</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LPresentedGroup</code>( <var class="Arg">F</var>, <var class="Arg">frels</var>, <var class="Arg">endos</var>, <var class="Arg">irels</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>returns the <strong class="pkg">GAP</strong> object of an L-presented group with the underlying free group <var class="Arg">F</var>, the fixed relators <var class="Arg">frels</var>, the set of endomorphisms <var class="Arg">endos</var>, and the iterated relators <var class="Arg">irels</var>. The input variables <var class="Arg">frels</var> and <var class="Arg">irels</var> need to be finite subsets of the underlying free group <var class="Arg">F</var> and <var class="Arg">endos</var> needs to be a finite list of homomorphisms <span class="SimpleMath">\(F\to F\)</span>.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ExamplesOfLPresentations</code>( <var class="Arg">n</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>returns some well-known examples of finitely L-presented groups. The input of this function needs to be a positive integer at most <span class="SimpleMath">\(10\)</span>.</p>
<dl>
<dt><strong class="Mark">n=1</strong></dt>
<dd><p>The Grigorchuk group on 4 generators; cf. <a href="chapBib_mj.html#biBGrigorchuk80">[Gri80]</a>, <a href="chapBib_mj.html#biBLysenok85">[Lys85]</a>, and <a href="chapBib_mj.html#biBBartholdi03">[Bar03, Theorem 4.6]</a>,</p>
</dd>
<dt><strong class="Mark">n=2</strong></dt>
<dd><p>the Grigorchuk group on 3 generators; cf. <a href="chapBib_mj.html#biBGrigorchuk80">[Gri80]</a>, <a href="chapBib_mj.html#biBLysenok85">[Lys85]</a>, and <a href="chapBib_mj.html#biBBartholdi03">[Bar03, Theorem 4.6]</a>,</p>
</dd>
<dt><strong class="Mark">n=10</strong></dt>
<dd><p>Baumslag's finitely generated, infinitely related group with a trivial multiplier; cf. [Bau71].
</dd>
</dl>
<p>Furthermore, every free group in a variety of groups satisfying finitely many identities is finitely L-presented. Some of these groups are available from the <strong class="pkg">lpres</strong>-package using the following operations; for further details we refer to the diploma thesis <a href="chapBib_mj.html#biBH08">[Har08]</a>.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ FreeEngelGroup</code>( <var class="Arg">num</var>, <var class="Arg">n</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>returns an L-presentation for the free <var class="Arg">n</var>-Engel group on <var class="Arg">num</var> generators; that is, the free group in the variety of <var class="Arg">num</var>-generated groups satisfying the <var class="Arg">n</var>-Engel identity.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ FreeBurnsideGroup</code>( <var class="Arg">num</var>, <var class="Arg">exp</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>returns an L-presentation for the free Burnside group on <var class="Arg">num</var> generators with exponent <var class="Arg">exp</var>; that is, the free group on <var class="Arg">num</var> generators in the variety of groups with exponent <var class="Arg">exp</var>.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ FreeNilpotentGroup</code>( <var class="Arg">num</var>, <var class="Arg">c</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>returns an L-presentation for the free nilpotent group of class <var class="Arg">c</var> on <var class="Arg">num</var> generators; that is, the free group in the variety of <var class="Arg">num</var>-generated, nilpotent groups with nilpotency class <var class="Arg">c</var>.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ GeneralizedFabrykowskiGuptaLpGroup</code>( <var class="Arg">n</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>returns an L-presentation for the <var class="Arg">n</var>-th generalized Fabrykowski-Gupta group as constructed in <a href="chapBib_mj.html#biBBEH08">[BEH08]</a>.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LamplighterGroup</code>( <var class="Arg">filter</var>, <var class="Arg">int</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LamplighterGroup</code>( <var class="Arg">filter</var>, <var class="Arg">pcgroup</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>returns a finite L-presentation for the lamplighter group on <var class="Arg">int</var> lamp states in the first case, if <var class="Arg">filter</var> is the filter <code class="code">IsLpGroup</code>. In the second case, the group <var class="Arg">pcgroup</var> must be a finite cyclic group. Then the method returns a finite L-presentation for the lamplighter group on <code class="code">Size(pcgroup)</code> lamp states; for details on the L-presentation see <a href="chapBib_mj.html#biBBartholdi03">[Bar03]</a>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">LamplighterGroup( IsLpGroup, 2 );</span>
<L-presented group on the generators [ a, t, u ]>
<span class="GAPprompt">gap></span> <span class="GAPinput">LamplighterGroup( IsLpGroup, CyclicGroup(3) );</span>
<L-presented group on the generators [ a, t, u ]>
</pre></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ EmbeddingOfIASubgroup</code>( <var class="Arg">a</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>computes an L-presentation for the IA-automorphism group of a free group. This is the subgroup of automorphisms of a free group <span class="SimpleMath">\(f\)</span> that act trivially on the abelianization of <span class="SimpleMath">\(f\)</span>.</p>
<p>The L-presentation is taken from <a href="chapBib_mj.html#biBDayPutman">[DP]</a>.</p>
<p>An L-presented group is defined as an image of its underlying free group. Note that these are two different <strong class="pkg">GAP</strong> objects. The elements of the L-presented group are represented by words in the underlying free group.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ FreeGroupOfLpGroup</code>( <var class="Arg">lpgroup</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: the underlying free group of the L-presented group <var class="Arg">lpgroup</var></p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ FreeGeneratorsOfLpGroup</code>( <var class="Arg">lpgroup</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: the generators of the free group which underlies the L-presented group <var class="Arg">lpgroup</var></p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ GeneratorsOfGroup</code>( <var class="Arg">lpgroup</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: the generators of the L-presented group <var class="Arg">lpgroup</var>. These are the images of the generators of the underlying free group under the natural homomorphism.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ UnderlyingElement</code>( <var class="Arg">elm</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>returns the preimage of an L-presented group element <var class="Arg">elm</var> in the underlying free group. More precisely, each element of an L-presented group is represented by an element in the free group. This method returns the corresponding element in the free group.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ElementOfLpGroup</code>( <var class="Arg">fam</var>, <var class="Arg">elm</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>returns the element in the L-presented group represented by the word <var class="Arg">elm</var> on the generators of the underlying free group, if <var class="Arg">fam</var> is the family of L-presented group elements.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ FixedRelatorsOfLpGroup</code>( <var class="Arg">lpgroup</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: the fixed relators of the L-presented group <var class="Arg">lpgroup</var> as elements of the underlying free group.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IteratedRelatorsOfLpGroup</code>( <var class="Arg">lpgroup</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: the iterated relators of the L-presented group <var class="Arg">lpgroup</var> as elements of the underlying free group.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ EndomorphismsOfLpGroup</code>( <var class="Arg">lpgroup</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: the endomorphisms of the L-presented group <var class="Arg">lpgroup</var> as endomorphisms of the underlying free group.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">F:=FreeGroup( 2 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">G:=LPresentedGroup( F, [ F.1^2 ], [ IdentityMapping( F ) ], [ F.2 ] );</span>
<L-presented group on the generators [ f1, f2 ]>
<span class="GAPprompt">gap></span> <span class="GAPinput">FixedRelatorsOfLpGroup( G );</span>
[ f1^2 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">IteratedRelatorsOfLpGroup( G );</span>
[ f2 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">EndomorphismsOfLpGroup( G );</span>
[ IdentityMapping( <free group on the generators [ f1, f2 ]> ) ]
</pre></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ UnderlyingAscendingLPresentation</code>( <var class="Arg">lpgroup</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>returns the underlying ascending L-presentation of <var class="Arg">lpgroup</var>; that is, if <var class="Arg">lpgroup</var> is finitely L-presented by <span class="SimpleMath">\((S,Q,\Phi,R)\)</span>, the underlying ascending L-presentation is <span class="SimpleMath">\((S,\emptyset,\Phi,R)\)</span>.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ UnderlyingInvariantLPresentation</code>( <var class="Arg">lpgroup</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>attempts to compute a ``good'' underlying invariant L-presentation for <var class="Arg">lpgroup</var>; that is, if <var class="Arg">lpgroup</var> is finitely L-presented by <span class="SimpleMath">\((S,Q,\Phi,R)\)</span>, then this method seeks to find a subset <span class="SimpleMath">\(Q'\subseteq Q\) such that \((S,Q',\Phi,R)\)</span> is an invariant L-presentation. Note that there is always the underlying ascending L-presentation <span class="SimpleMath">\((S,\emptyset,\Phi,R)\)</span>. However, for the efficiency of the nilpotent quotient algorithm it is important that the subset <span class="SimpleMath">\(Q'\) is as big as possible.
<p>Since it is undecidable, in general, whether or not a given L-presentation is invariant, there is no algorithm which can determine the best possible underlying invariant L-presentation. The method implemented for this attribute tries to compute a ``good'' invariant L-presentation and will return the underlying ascending L-presentation in the worst case.</p>
<p>This attribute can be set manually using <code class="code">SetUnderlyingInvariantLPresentation</code>. For instance, the Grigorchuk group</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsAscendingLPresentation</code>( <var class="Arg">lpgroup</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>checks whether the L-presentation of <var class="Arg">lpgroup</var> is ascending; that is, if the set of fixed relators is empty. This property is set automatically when creating an L-presented group with no fixed relators using the function <code class="func">LPresentedGroup</code> (<a href="chap2_mj.html#X7BBBE4C082AE4D5A"><span class="RefLink">2.2-1</span></a>).</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsInvariantLPresentation</code>( <var class="Arg">lpgroup</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>attempts to check whether the L-presentation of <var class="Arg">lpgroup</var> is invariant. In general, one cannot decide whether or not a given L-presentation is invariant. There are mainly two methods implemented for this property. The first method seeks to find a ``good'' underlying invariant L-presentation using the operation <code class="func">UnderlyingInvariantLPresentation</code> (<a href="chap2_mj.html#X86F017E085082624"><span class="RefLink">2.5-2</span></a>). If this latter L-presentation coincides with the L-presentation of <var class="Arg">lpgroup</var>, then <var class="Arg">lpgroup</var> is invariantly L-presented. If this method fails, then the second method uses the nilpotent quotient algorithm for L-presented groups which yields a necessary condition for an L-presented group to be invariantly L-presented. Note that the latter method may not terminate. For instance, both methods fail on Baumslag's finitely generated, infinitely related group with trivial multiplier returned by ExamplesOfLPresentations (2.2-2).
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ EmbeddingOfAscendingSubgroup</code>( <var class="Arg">lpgroup</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>stores an embedding of an ascendingly L-presented subgroup of the L-presented group <var class="Arg">lpgroup</var>. This attribute is set for ascending L-presentations only. In this case, the identity map of <var class="Arg">lpgroup</var> is returned. This attribute is used in the <strongclass="pkg">FR</strong>-package which can construct various finitely L-presented groups. The embedding is useful for a nilpotent quotient algorithm of a non-invariantly L-presented group.</p>
<h4>2.6 <span class="Heading">Methods for L-presented groups</span></h4>
<p>Some operations are natural extensions of the operations for finitely generated groups. For example, <code class="code">MappedWord(x,gens,imgs)</code>, when applied to a word <code class="code">x</code> in an L-presented group, returns the group element obtained by replacing each occurrence of a generator in <code class="code">gens</code> by the corresponding element in the list <code class="code">imgs</code>. The lists <code class="code">gens</code> and <code class="code">imgs</code> need to have the same length.</p>
<p>Equality test of elements of L-presented groups is implemented using the operation <code class="func">NqEpimorphismNilpotentQuotient</code> (<a href="../../../pkg/nq-2.5.11/doc/chap3_mj.html#X8758F663782AE655"><span class="RefLink">nq: NqEpimorphismNilpotentQuotient</span></a>) to compare the images in a nilpotent quotient of the group. The implemented method successively increases the class of the considered quotient until the images differ. Hence, this method may not terminate and it will only determine whether the elements are different.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ EpimorphismFromFpGroup</code>( <var class="Arg">lpgroup</var>, <var class="Arg">n</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>returns an epimorphism from a finitely presented group onto <var class="Arg">lpgroup</var>. The finitely presented group is obtained from <var class="Arg">lpgroup</var> by applying only words of length at most <var class="Arg">n</var> in the endomorphisms of <var class="Arg">lpgroup</var> to the iterated relators of <var class="Arg">lpgroup</var>.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SplitExtensionByAutomorphismsLpGroup</code>( <var class="Arg">lpgroup</var>, <var class="Arg">H</var>, <var class="Arg">auts</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>returns an L-presentation for the split extension of <var class="Arg">lpgroup</var> by an L-presented or by a finitely presented group <var class="Arg">H</var>. The action of a generator of <var class="Arg">H</var> on <var class="Arg">lpgroup</var> is given by an automorphism in the list <var class="Arg">auts</var>. Thus for each generator of <var class="Arg">H</var> there must be an automorphism in the list <var class="Arg">auts</var>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">F := FreeGroup( "a" );</span>
<free group on the generators [ a ]>
<span class="GAPprompt">gap></span> <span class="GAPinput">H := F / [ F.1^3 ];</span>
<fp group on the generators [ a ]>
<span class="GAPprompt">gap></span> <span class="GAPinput">U := ExamplesOfLPresentations( 8 );</span>
<L-presented group on the generators [ t, u, v ]>
<span class="GAPprompt">gap></span> <span class="GAPinput">aut:=GroupHomomorphismByImagesNC( U, U, [ U.1, U.2, U.3 ], [ U.2, U.3, U.1 ] );</span>
[ t, u, v ] -> [ u, v, t ]
<span class="GAPprompt">gap></span> <span class="GAPinput">SplitExtensionByAutomorphismsLpGroup( U, H, [ aut ] );</span>
<L-presented group on the generators [ t, u, v, a ]>
</pre></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AsLpGroup</code>( <var class="Arg">G</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>returns an ascending L-presentation for a finitely presented group <var class="Arg">G</var> or for a free group <var class="Arg">G</var>.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsomorphismLpGroup</code>( <var class="Arg">G</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>returns an isomorphism from a finitely presented group <var class="Arg">G</var> or from a free group <var class="Arg">G</var> to the L-presented group obtained from the method <code class="func">AsLpGroup</code> (<a href="chap2_mj.html#X84F112247DA4037C"><span class="RefLink">2.6-3</span></a>).</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">F:=FreeGroup( 2 );</span>
<free group on the generators [ f1, f2 ]>
<span class="GAPprompt">gap></span> <span class="GAPinput">G:=F/[ F.1^2, F.2^2, Comm( F.1, F.2 ) ];</span>
<fp group on the generators [ f1, f2 ]>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsomorphismLpGroup( G );</span>
[ f1, f2 ] -> [ f1, f2 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">Range(last);</span>
<L-presented group on the generators [ f1, f2 ]>
<span class="GAPprompt">gap></span> <span class="GAPinput">Display(last);</span>
generators = [ f1, f2 ]
fixed relators = [ ]
endomorphism = [
IdentityMapping( <free group on the generators [ f1, f2 ]> ) ]
iterated relators = [
f1^2,
f2^2,
f1^-1*f2^-1*f1*f2 ]
</pre></div>
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