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href="chap9_mj.html#X84DB7FA4846075A </div></div> </div> <h3>9 <span class="Heading">Examples</span></h3> <p><strong class="pkg">FR</strong> predefines a large collection of machines and groups. The grou <p><a id="X7AF5DEF08531AFA5" name="X7AF5DEF08531AFA5"></a></p> <h4>9.1 <span class="Heading">Examples of groups</span></h4> <p><a id="X7D774B847D81E6DE" name="X7D774B847D81E6DE"></a></p> <h5>9.1-1 FullBinaryGroup</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Fu <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Fi <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Fi <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Bo <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Po <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Fi <p>These are the finitary, bounded, polynomial-growth, finite-state, or unrestricted groups <p>They may be used to draw random elements, or to test membership.</p> <p><a id="X813F53C57F41F5F5" name="X813F53C57F41F5F5"></a></p> <h5>9.1-2 BinaryKneadingGroup</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Bi <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Bi <p>Returns: The [machine generating the] iterated monodromy group of a quadratic polynomial.</ <p>The first function constructs a Mealy machine whose state closure is the binary kneading grou <p>The second function constructs a new FR group <code class="code">G</code>, which is the iterated <p>If the argument is a (rational) angle, the attribute <code class="code">Correspondence(G)</c <p>If there is one argument, which is a list or a string, it is treated as the kneading sequence of a p <p>If there are two arguments, which are lists or strings, they are treated as the preperiod and pe <p>As particular examples, <code class="code">BinaryKneadingMachine()</code> is the adding mac <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">BinaryKneadingGroup()=AddingGro true <span class="GAPprompt">gap></span> <span class="GAPinput">BinaryKneadingGroup(1/3)=Basili true <span class="GAPprompt">gap></span> <span class="GAPinput">g := BinaryKneadingGroup([0,1],[0 BinaryKneadingGroup("01","0") <span class="GAPprompt">gap></span> <span class="GAPinput">ForAll(Correspondence(g)[1],IsF true <span class="GAPprompt">gap></span> <span class="GAPinput">ForAll(Correspondence(g)[2],IsF false <span class="GAPprompt">gap></span> <span class="GAPinput">ForAll(Correspondence(g)[2],IsB true </pre></div> <p><a id="X7B8D49D079D336E8" name="X7B8D49D079D336E8"></a></p> <h5>9.1-3 BasilicaGroup</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Ba <p>The <em>Basilica group</em>. This is a shortcut for <code class="code">BinaryKneadingGroup("1" <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">IsBoundedFRSemigroup(BasilicaGr true <span class="GAPprompt">gap></span> <span class="GAPinput">pi := EpimorphismFromFreeGroup(Ba [ x1, x2 ] -> [ a, b ] <span class="GAPprompt">gap></span> <span class="GAPinput">sigma := GroupHomomorphismByImage [ x1, x2 ] -> [ x2, x1^2 ] <span class="GAPprompt">gap></span> <span class="GAPinput">ForAll([0..10],i->IsOne(Comm(F.2 true </pre></div> <p><a id="X8449487686E00D22" name="X8449487686E00D22"></a></p> <h5>9.1-4 FornaessSibonyGroup</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Fo <p>The <em>Fornaess-Sibony group</em>. This group was studied by Nekrashevych in <a href="chapBib_ <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">Size(NucleusOfFRSemigroup(Forna 288 <span class="GAPprompt">gap></span> <span class="GAPinput">List(AdjacencyBasesWithOne(Forn [ 128, 128, 36, 36, 16, 16, 8 ] <span class="GAPprompt">gap></span> <span class="GAPinput">p := AdjacencyPoset(FornaessSibon <general mapping: <object> -> <object> > <span class="GAPprompt">gap></span> <span class="GAPinput">Draw(HasseDiagramBinaryRelation </pre></div> <p>This produces (in a new window) the following picture: <img alt="Hasse Diagram" src="fornae <p><a id="X85F4FDF787173863" name="X85F4FDF787173863"></a></p> <h5>9.1-5 AddingGroup</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Ad <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Ad <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Ad <p>The second function constructs the adding machine on the alphabet <code class="code">[1..n]</ <p>The third function constructs the Mealy element on the adding machine, with initial state 2.</ <p>The first function constructs the state-closed group generated by the adding machine on <code <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">Display(AddingElement(3));</span> | 1 2 3 ---+-----+-----+-----+ a | a,1 a,2 a,3 b | a,2 a,3 b,1 ---+-----+-----+-----+ Initial state: b <span class="GAPprompt">gap></span> <span class="GAPinput">ActivityPerm(FRElement(AddingMa (1,4,7,2,5,8,3,6,9) <span class="GAPprompt">gap></span> <span class="GAPinput">G := AddingGroup(3);</span> <self-similar group over [ 1 .. 3 ] with 1 generator> <span class="GAPprompt">gap></span> <span class="GAPinput">Size(G);</span> infinity <span class="GAPprompt">gap></span> <span class="GAPinput">IsRecurrentFRSemigroup(G);</span> true <span class="GAPprompt">gap></span> <span class="GAPinput">IsLevelTransitiveFRGroup(G);</sp true </pre></div> <p><a id="X7A4BB24A805CDF63" name="X7A4BB24A805CDF63"></a></p> <h5>9.1-6 BinaryAddingGroup</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Bi <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Bi <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Bi <p>These are respectively the same as <code class="code">AddingGroup(2)</code>, <code class="cod <p><a id="X78AFA63B86C94227" name="X78AFA63B86C94227"></a></p> <h5>9.1-7 MixerGroup</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Mi <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Mi <p>Returns: A Mealy "mixer" machine/group.</p> <p>The second function constructs a Mealy "mixer" machine <code class="code">m</code>. This is a ma <p>The states of the machine are 1, followed by some elements <code class="code">a</code> of <var clas <p>The transitions from <code class="code">a</code> are towards 1 and the outputs are the permutati <p>This construction is described in more detail in <a href="chapBib_mj.html#biBMR1856923">[B <p><code class="code">Correspondence(m)</code> is a list, with in first position the subset of the <p>The first function constructs the group generated by the mixer machine. For examples see <code <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">grigorchukgroup := MixerGroup(Gro [[IdentityMapping(Group((1,2)))],[IdentityMapping(Group((1,2)))],[]])); <self-similar group over [ 1 .. 2 ] with 4 generators> <span class="GAPprompt">gap></span> <span class="GAPinput">IdGroup(Group(grigorchukgroup.1 [ 32, 18 ] </pre></div> <p><a id="X84C97E0687F119C0" name="X84C97E0687F119C0"></a></p> <h5>9.1-8 SunicGroup</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Su <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Su <p>Returns: The Sunic machine associated with the polynomial <var class="Arg">phi</var>.</p> <p>A "Sunic machine" is a special kind of <code class="func">MixerMachine</code> (<a href="chap9_ <p>These groups were described in <a href="chapBib_mj.html#biBMR2318546">[{Š}un07]</a>. In part <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">x := Indeterminate(GF(2));;</span> <span class="GAPprompt">gap></span> <span class="GAPinput">g := SunicGroup(x^2+x+1);</span> SunicGroup(x^2+x+Z(2)^0) <span class="GAPprompt">gap></span> <span class="GAPinput">g = GrigorchukGroup;</span> true </pre></div> <p><a id="X79E3F3BE80F34590" name="X79E3F3BE80F34590"></a></p> <h5>9.1-9 GrigorchukMachines</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Gr <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Gr <p>Returns: One of the Grigorchuk groups.</p> <p>This function constructs the Grigorchuk machine or group associated with the infinite seque <p>These groups are <code class="func">MixerGroup</code> (<a href="chap9_mj.html#X78AFA63B86C <p>These groups are all 4-generated and infinite. They are described in <a href="chapBib_mj.htm <p>The growth of <code class="code">GrigorchukGroups([1,2])</code> has been studied in <a href="c <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">G := GrigorchukGroups([1]);</span> GrigorchukGroups([ 1 ]) <span class="GAPprompt">gap></span> <span class="GAPinput">Index(G,DerivedSubgroup(G)); IsA 4 true <span class="GAPprompt">gap></span> <span class="GAPinput">H := GrigorchukGroups([1,2]);</spa GrigorchukGroups([ 1, 2 ]) <span class="GAPprompt">gap></span> <span class="GAPinput">Order(H.1*H.2);</span> 8 <span class="GAPprompt">gap></span> <span class="GAPinput">Order(H.1*H.4);</span> infinity <span class="GAPprompt">gap></span> <span class="GAPinput">IsSubgroup(H,G);</span> true </pre></div> <p><a id="X85BAE48780E665A4" name="X85BAE48780E665A4"></a></p> <h5>9.1-10 GrigorchukMachine</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Gr <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Gr <p>This is Grigorchuk's first group, introduced in [Gri80]. It is a 4-generated infi FRGroup("a=(1,2)","b=<a,c>","c=<a,d>","d=<,b>"), but is rather defined using Mealy e<p>The command <code class="code">EpimorphismFromFpGroup(GrigorchukGroup,n)</code> constru <p><a id="X800640597E9C707D" name="X800640597E9C707D"></a></p> <h5>9.1-11 GrigorchukOverGroup</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Gr <p>This is a group strictly containing the Grigorchuk group (see <code class="func">GrigorchukG <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">IsSubgroup(GrigorchukOverGroup, true <span class="GAPprompt">gap></span> <span class="GAPinput">Order(Product(GeneratorsOfGroup infinity <span class="GAPprompt">gap></span> <span class="GAPinput">GrigorchukGroup.2=GrigorchukSup true </pre></div> <p>The command <code class="code">EpimorphismFromFpGroup(GrigorchukOverGroup,n)</code> wil <p><a id="X7E765AF77AAC21A6" name="X7E765AF77AAC21A6"></a></p> <h5>9.1-12 GrigorchukTwistedTwin</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Gr <p>This is a group with same closure as the Grigorchuk group (see <code class="func">GrigorchukGr <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">AbelianInvariants(GrigorchukTwi [ 2, 2, 2, 2 ] <span class="GAPprompt">gap></span> <span class="GAPinput">AbelianInvariants(GrigorchukGro [ 2, 2, 2 ] <span class="GAPprompt">gap></span> <span class="GAPinput">PermGroup(GrigorchukGroup,8)=Pe true </pre></div> <p><a id="X7F93EC437B5AE276" name="X7F93EC437B5AE276"></a></p> <h5>9.1-13 BrunnerSidkiVieiraGroup</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Br <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Br <p>This machine is the sum of two adding machines, one in standard form and one conjugated by the el <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">V := FRGroup("d=<e,e>(1,2)","e=<d,d>" <self-similar group over [ 1 .. 2 ] with 2 generators> <span class="GAPprompt">gap></span> <span class="GAPinput">Elements(V);</span> [ <2|identity ...>, <2|e>, <2|d>, <2|e*d> ] <span class="GAPprompt">gap></span> <span class="GAPinput">AssignGeneratorVariables(Brunne #I Assigned the global variables [ "tau", "mu", "" ] <span class="GAPprompt">gap></span> <span class="GAPinput">List(V,x->tau^x)=[tau,mu,mu^-1,t true </pre></div> <p><a id="X7F8A028B799946D3" name="X7F8A028B799946D3"></a></p> <h5>9.1-14 AleshinGroups</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Al <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Al <p>Returns: The Aleshin machine with <code class="code">Length(l)</code> states.</p> <p>This function constructs the bireversible machines introduced by Aleshin in <a href="chapBi <p>If <code class="code">l=[(1,2),(1,2),()]</code>, this is <code class="func">AleshinGroup</co <p>If <code class="code">l=[(1,2),(1,2),...]</code> has even length and contains an even number o Length(l), see<p>If <code class="code">l=[(),(),(1,2)]</code>, this is <code class="func">BabyAleshinGroup</c Length(l) copie<p><a id="X7C286D3A84790ECE" name="X7C286D3A84790ECE"></a></p> <h5>9.1-15 AleshinGroup</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Al <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Al <p>This is the first example of non-abelian free group. It is the group generated by <code class="c <p><a id="X7E024B4D7BA411B1" name="X7E024B4D7BA411B1"></a></p> <h5>9.1-16 BabyAleshinGroup</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Ba <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Ba <p>There are only two connected, transitive bireversible machines on a 2-letter alphabet, with <p>The group generated by this machine is abstractly the free product of three <span class="Simpl FRGroup(, but is rather defined here using Mealy elements.<div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">K := Kernel(GroupHomomorphismByIm GeneratorsOfGroup(BabyAleshinGroup),[(1,2),(1,2),(1,2)])); <self-similar group over [ 1 .. 2 ] with 4 generators> <span class="GAPprompt">gap></span> <span class="GAPinput">Index(BabyAleshinGroup,K);</span> 2 <span class="GAPprompt">gap></span> <span class="GAPinput">IsSubgroup(AleshinGroup,K);</spa true </pre></div> <p><a id="X8108E3A8872A6FFE" name="X8108E3A8872A6FFE"></a></p> <h5>9.1-17 SidkiFreeGroup</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Si <p>This is a group suggested by Sidki as an example of a non-abelian state-closed free group. Noth <p><a id="X82D3CB6A7C189C78" name="X82D3CB6A7C189C78"></a></p> <h5>9.1-18 GuptaSidkiGroups</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Gu <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Ge <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Gu <p>Returns: The Gupta-Sidki group/machine on an <var class="Arg">n</var>-letter alphabet.</p> <p>This function constructs the machines introduced by Gupta and Sidki in <a href="chapBib_mj.h <p>For <span class="SimpleMath">\(n>3\)</span>, there are (at least) two natural ways to generaliz "t=<a,a^2,...,a^-1,t>". A finite L-presentation for the latter is implemented.t (a subgroup of index \(p\)), which is an ascending L-presented group. TheEmbeddingOfAscendingSubgroup(Gupt.<p><a id="X83E59288860EF661" name="X83E59288860EF661"></a></p> <h5>9.1-19 GuptaSidkiGroup</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Gu <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Gu <p>This is an infinite, 2-generated, torsion 3-group. It could have been defined as <code class=" <p><a id="X8521B4FF7BA189B2" name="X8521B4FF7BA189B2"></a></p> <h5>9.1-20 NeumannGroup</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Ne <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Ne <p>Returns: The Neumann group/machine on the permutation group <var class="Arg">P</var>.</p> <p>The first function constructs the Neumann group associated with the permutation group <var cl <p>The second function constructs the Neumann machine associated to the permutation group <var c <p>The attribute <code class="code">Correspondence(G)</code> is set to a list of homomorphisms fr <p><a id="X878D1C7080EA9797" name="X878D1C7080EA9797"></a></p> <h5>9.1-21 FabrykowskiGuptaGroup</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Fa <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Fa <p>This is an infinite, 2-generated group of intermediate word growth. It was studied in <a href=" <p>The natural generalization, replacing 3 by <span class="SimpleMath">\(p\)</span>, is constru <p><a id="X7C5ADAE77EA3876D" name="X7C5ADAE77EA3876D"></a></p> <h5>9.1-22 ZugadiSpinalGroup</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Zu <p>This is an infinite, 2-generated group, which was studied in <a href="chapBib_mj.html#biBMR <p>Amaia Zugadi computed its Hausdorff dimension to be <span class="SimpleMath">\(1/2\)</span>.< <p><a id="X7A0319827CB51ED0" name="X7A0319827CB51ED0"></a></p> <h5>9.1-23 GammaPQMachine</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Ga <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Ga <p>Returns: The quaternion-based machine / SC group.</p> <p>The first function constructs a bireversible (see <code class="func">IsBireversible</code> (< <p>The structure of the machine is such that there is a transition from <span class="SimpleMath">\ <p>The quaternions themselves can be recovered through <code class="func">Correspondence</cod <p>The second function constructs the quaternion lattice that is the <code class="func">Structu <p><a id="X80B617717C2887D4" name="X80B617717C2887D4"></a></p> <h5>9.1-24 RattaggiGroup</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Ra <p>This record contains interesting examples of VH groups, that were studied by Rattaggi in his P <p>The following summary of the examples' properties are copied from Rattaggi's thesis. R <div class="pcenter"><table class="GAPDocTable"> <tr> <td class="tdright">Example</td> <td class="tdcenter"><span class="SimpleMath">\(P_h\)</span></td> <td class="tdcenter"><span class="SimpleMath">\(P_v\)</span></td> <td class="tdcenter">Irred?</td> <td class="tdcenter">Linear?</td> <td class="tdcenter">RF?</td> <td class="tdcenter">JI?</td> <td class="tdcenter">VS?</td> </tr> <tr> <td class="tdright">2.2</td> <td class="tdcenter">2tr</td> <td class="tdcenter">2tr</td> <td class="tdcenter">Y</td> <td class="tdcenter">N</td> <td class="tdcenter">N?</td> <td class="tdcenter">Y</td> <td class="tdcenter">Y?</td> </tr> <tr> <td class="tdright">2.15</td> <td> </td> <td> </td> <td> </td> <td> </td> <td> </td> <td> </td> <td> </td> </tr> <tr> <td class="tdright">2.18</td> <td> </td> <td> </td> <td> </td> <td> </td> <td> </td> <td> </td> <td> </td> </tr> <tr> <td class="tdright">2.21</td> <td> </td> <td> </td> <td> </td> <td> </td> <td> </td> <td> </td> <td> </td> </tr> <tr> <td class="tdright">2.26</td> <td class="tdcenter">2tr</td> <td class="tdcenter">q-prim</td> <td class="tdcenter">Y</td> <td class="tdcenter">N</td> <td class="tdcenter">N</td> <td class="tdcenter">N</td> <td class="tdcenter">N</td> </tr> <tr> <td class="tdright">2.30</td> <td class="tdcenter">2tr</td> <td class="tdcenter">2tr</td> <td class="tdcenter">Y</td> <td class="tdcenter">N</td> <td class="tdcenter">N</td> <td class="tdcenter">Y</td> <td class="tdcenter">Y?</td> </tr> <tr> <td class="tdright">2.36</td> <td> </td> <td> </td> <td> </td> <td> </td> <td> </td> <td> </td> <td> </td> </tr> <tr> <td class="tdright">2.39</td> <td> </td> <td> </td> <td> </td> <td> </td> <td> </td> <td> </td> <td> </td> </tr> <tr> <td class="tdright">2.43</td> <td class="tdcenter">2tr</td> <td class="tdcenter">2tr</td> <td class="tdcenter">Y</td> <td class="tdcenter">N</td> <td class="tdcenter">N</td> <td class="tdcenter">Y</td> <td class="tdcenter">Y</td> </tr> <tr> <td class="tdright">2.46</td> <td> </td> <td> </td> <td> </td> <td> </td> <td> </td> <td> </td> <td> </td> </tr> <tr> <td class="tdright">2.48</td> <td> </td> <td> </td> <td> </td> <td> </td> <td> </td> <td> </td> <td> </td> </tr> <tr> <td class="tdright">2.50</td> <td> </td> <td> </td> <td> </td> <td> </td> <td> </td> <td> </td> <td> </td> </tr> <tr> <td class="tdright">2.52</td> <td class="tdcenter">tr</td> <td class="tdcenter">2tr</td> <td class="tdcenter">Y</td> <td class="tdcenter">N</td> <td class="tdcenter">N</td> <td class="tdcenter">N</td> <td class="tdcenter">N</td> </tr> <tr> <td class="tdright">2.56</td> <td> </td> <td> </td> <td> </td> <td> </td> <td> </td> <td> </td> <td> </td> </tr> <tr> <td class="tdright">2.58</td> <td class="tdcenter">2tr</td> <td class="tdcenter">prim</td> <td class="tdcenter">Y</td> <td class="tdcenter">N</td> <td class="tdcenter">N?</td> <td class="tdcenter">N</td> <td class="tdcenter">N</td> </tr> <tr> <td class="tdright">2.70</td> <td> </td> <td> </td> <td> </td> <td> </td> <td> </td> <td> </td> <td> </td> </tr> <tr> <td class="tdright">3.26</td> <td class="tdcenter">2tr</td> <td class="tdcenter">2tr</td> <td class="tdcenter">Y</td> <td class="tdcenter">Y</td> <td class="tdcenter">Y</td> <td class="tdcenter">Y</td> <td class="tdcenter">N</td> </tr> <tr> <td class="tdright">3.28</td> <td> </td> <td> </td> <td> </td> <td> </td> <td> </td> <td> </td> <td> </td> </tr> <tr> <td class="tdright">3.31</td> <td> </td> <td> </td> <td> </td> <td> </td> <td> </td> <td> </td> <td> </td> </tr> <tr> <td class="tdright">3.33</td> <td> </td> <td> </td> <td> </td> <td> </td> <td> </td> <td> </td> <td> </td> </tr> <tr> <td class="tdright">3.36</td> <td> </td> <td> </td> <td> </td> <td> </td> <td> </td> <td> </td> <td> </td> </tr> <tr> <td class="tdright">3.38</td> <td> </td> <td> </td> <td> </td> <td> </td> <td> </td> <td> </td> <td> </td> </tr> <tr> <td class="tdright">3.40</td> <td> </td> <td> </td> <td> </td> <td> </td> <td> </td> <td> </td> <td> </td> </tr> <tr> <td class="tdright">3.44</td> <td> </td> <td> </td> <td> </td> <td> </td> <td> </td> <td> </td> <td> </td> </tr> <tr> <td class="tdright">3.46</td> <td> </td> <td> </td> <td> </td> <td> </td> <td> </td> <td> </td> <td> </td> </tr> <tr> <td class="tdright">3.72</td> <td> </td> <td> </td> <td> </td> <td> </td> <td> </td> <td> </td> <td> </td> </tr> </table><br /> </div> <p><a id="X7A0BE9F57B401C5C" name="X7A0BE9F57B401C5C"></a></p> <h5>9.1-25 HanoiGroup</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Ha <p>Returns: The Hanoi group on an <var class="Arg">n</var> pegs.</p> <p>This function constructs the Hanoi group on <var class="Arg">n</var> pegs. Generators of the gro <p><a id="X7C7A0EEF7EFF8B99" name="X7C7A0EEF7EFF8B99"></a></p> <h5>9.1-26 DahmaniGroup</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Da <p>This is an example of a non-contracting weakly branched group. It was first studied in <a href=" <p>It has relators <span class="SimpleMath">\(abc\)</span>, <span class="SimpleMath">\([a^2c,[ <p>It admits an endomorphism on its derived subgroup. Indeed <code class="code">FRElement(1,Co <p><a id="X7C958AB78484E256" name="X7C958AB78484E256"></a></p> <h5>9.1-27 MamaghaniGroup</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Ma <p>This group was studied in <a href="chapBib_mj.html#biBMR2139928">[Mam03]</a>. It is fractal, <p><a id="X86D952E8784E4D97" name="X86D952E8784E4D97"></a></p> <h5>9.1-28 WeierstrassGroup</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ We <p>This is the iterated monodromy group associated with the Weierstrass <span class="SimpleMat <p>Some relators in the group: <span class="SimpleMath">\((atbt)^4\)</span>, <span class="Simpl <p>Set <span class="SimpleMath">\(x=[a,t]\)</span>, <span class="SimpleMath">\(y=[b,t]\)</spa <p>The Schreier graph is obtained in the complex plane as the image of a <span class="SimpleMath">\ <p>The element <span class="SimpleMath">\(at\)</span> has infinite order.</p> <p><span class="SimpleMath">\([c,t,b][b,t,c][a,t,c][c,t,a]\)</span> has order 2, and belongs <p><a id="X80D59AFF7E7D3B8B" name="X80D59AFF7E7D3B8B"></a></p> <h5>9.1-29 StrichartzGroup</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ St <p>This group generates the graph of the Strichartz hexacarpet.</p> <p>The Strichartz hexacarpet is the dual graph to the infinitely iterated barycentric subdivis <p>Conjecturally, that graph's radius is \(1/18(2^(n+1)(13+3n)+(-1)^n-9)\) and its d <p>See <a href="chapBib_mj.html#biBMR3004256">[BKN+12]</a> for details.</p> <p><a id="X86B124758135DFBD" name="X86B124758135DFBD"></a></p> <h5>9.1-30 FRAffineGroup</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ FR <p>Returns: The <var class="Arg">d</var>-dimensional affine group over <var class="Arg">R</var>.</p> <p>This function constructs a new FR group <code class="code">G</code>, which is finite-index subg <p>The affine group can also be described as a subgroup of <span class="SimpleMath">\(GL_{d+1}(R <p>The only valid arguments are <code class="code">R=Integers</code> and <code class="code">R=Pol <p>The following examples construct the "Baumslag-Solitar group" <span class="SimpleMath">\ <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">A := FRAffineGroup(1,Integers,3);< <self-similar group over [ 1 .. 3 ]> <span class="GAPprompt">gap></span> <span class="GAPinput">f := Correspondence(A);</span> MappingByFunction( ( Integers^ [ 2, 2 ] ), <self-similar group over [ 1 .. 3 ]>, function( mat ) ... end ) <span class="GAPprompt">gap></span> <span class="GAPinput">BaumslagSolitar := Group([[2,0],[ <self-similar group over [ 1 .. 3 ] with 2 generators> <span class="GAPprompt">gap></span> <span class="GAPinput">BaumslagSolitar.2^BaumslagSolit true <span class="GAPprompt">gap></span> <span class="GAPinput">R := PolynomialRing(GF(2));;</span> <span class="GAPprompt">gap></span> <span class="GAPinput">A := FRAffineGroup(1,R,R.1);;</spa <span class="GAPprompt">gap></span> <span class="GAPinput">f := Correspondence(A);;</span> <span class="GAPprompt">gap></span> <span class="GAPinput">Lamplighter := Group(([[1+R.1,0], <self-similar group over [ 1 .. 2 ] with 2 generators> <span class="GAPprompt">gap></span> <span class="GAPinput">Lamplighter = CayleyGroup(Group(( |
