[automgrp] 5 Miscellaneous

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# 5 Miscellaneous

### Sections

1. Converters to and from FR package
2. Trees
3. Some predefined groups

In this chapter we present the functionality that does not quite fit in other chapters and the list of predefined groups and semigroups.

## 5.1 Converters to and from FR package

- `FR2AutomGrp(` G`) O`

  This operation is designed to convert data structures defined in FR package written by Laurent Bartholdi to corresponding structures in AutomGrp package. Currently it is implemented for functionally recursive groups, semigroups, and their sub(semi)groups and elements.

      gap> ZZ := FRGroup("t=<,t>[2,1]");
      <state-closed group over [ 1 .. 2 ] with 1 generator>
      gap> AZZ := FR2AutomGrp(ZZ);
      < t >
      gap> Display(AZZ);
      < t = (1, t)(1,2) >

      gap> i4 := FRMonoid("s=(1,2)","f=<s,f>[1,1]");
      <state-closed monoid over [ 1 .. 2 ] with 2 generators>
      gap> Ai4 := FR2AutomGrp(i4);
      < 1, s, f >
      gap> Display(Ai4);
      < 1 = (1, 1),
        s = (1, 1)(1,2),
        f = (s, f)[1,1] >

      gap> S := FRGroup("a=<a*b^-2,b^3>(1,2)","b=<b^-1*a,1>");
      <state-closed group over [ 1 .. 2 ] with 2 generators>
      gap> AS := FR2AutomGrp(S);
      < a, b >
      gap> Display(AS);
      < a = (a*b^-2, b^3)(1,2),
        b = (b^-1*a, 1) >
      gap> AssignGeneratorVariables(S);
      #I  Assigned the global variables [ "a", "b" ]
      gap> x := a^3*b*a^-2;
      <2|a^3*b*a^-2>
      gap> DecompositionOfFRElement(x);
      [ [ <2|a*b^-2>, <2|b^3*a^2*b^-1*a^-1> ], [ 2, 1 ] ]
      gap> y := FR2AutomGrp(x);
      a^3*b*a^-2
      gap> Decompose(y);
      (a*b^-2, b^3*a^2*b^-1*a^-1)(1,2)

- `AutomGrp2FR(` G`) O`

  This operation is designed to convert data structures defined in AutomGrp to corresponding structures in AutomGrp package written by Laurent Bartholdi. Currently it is implemented for automaton and self-similari (or, functionally recursive in L.Bartholdi's terminology) groups, semigroups, their sub(semi)groups and elements.

      gap> G:=AutomatonGroup("a=(b,a)(1,2),b=(a,b)");
      < a, b >
      gap> FG := AutomGrp2FR(G);
      <state-closed group over [ 1 .. 2 ] with 2 generators>
      gap> DecompositionOfFRElement(FG.1);
      [ [ <2|b>, <2|a> ], [ 2, 1 ] ]
      gap> DecompositionOfFRElement(FG.2);
      [ [ <2|a>, <2|b> ], [ 1, 2 ] ]

      gap> G := SelfSimilarGroup("a=(a*b^-2,b)(1,2),b=(b^2,a*b*a^-2)");
      < a, b >
      gap> F := AutomGrp2FR(G);
      <state-closed group over [ 1 .. 2 ] with 2 generators>
      gap> DecompositionOfFRElement(F.1);
      [ [ <2|a*b^-2>, <2|b> ], [ 2, 1 ] ]

      gap> G := AutomatonGroup("a=(b,a)(1,2),b=(a,b),c=(c,a)(1,2)");
      < a, b, c >
      gap> H := Group([a*b,b*c^-2,a]);
      < a*b, b*c^-2, a >
      gap> FH := AutomGrp2FR(H);
      <recursive group over [ 1 .. 2 ] with 3 generators>
      gap> DecompositionOfFRElement(FH.1);
      [ [ <2|b^2>, <2|a^2> ], [ 2, 1 ] ]

      gap> G := SelfSimilarSemigroup("a=(a*b^2,b*a)[1,1],b=(b,a*b*a)(1,2)");
      < a, b >
      gap> S := AutomGrp2FR(G);
      <state-closed semigroup over [ 1 .. 2 ] with 2 generators>
      gap> DecompositionOfFRElement(S.1);
      [ [ <2|a*b^2>, <2|b*a> ], [ 1, 1 ] ]

      gap> G := AutomatonGroup("a=(b,a)(1,2),b=(a,b),c=(c,a)(1,2)");
      < a, b, c >
      gap> Decompose(a*b^-2);
      (b^-1, a^-1)(1,2)
      gap> x := AutomGrp2FR(a*b^-2);
      <2|a*b^-2>
      gap> DecompositionOfFRElement(x);
      [ [ <2|b^-1>, <2|a^-1> ], [ 2, 1 ] ]

  ## 5.2 Trees

- `NumberOfVertex(` ver`,` deg`) F`

  One can naturally enumerate all the vertices of the n-th level of the tree by the numbers 1,…,deg n . This function returns the number that corresponds to the vertex ver of the deg-ary tree. The vertex can be defined either as a list or as a string.

      gap> NumberOfVertex([1,2,1,2], 2);
      6
      gap> NumberOfVertex("333", 3);
      27

- `VertexNumber(` num`,` lev`,` deg`) F`

  One can naturally enumerate all the vertices of the lev-th level of the deg-ary tree by the numbers 1,…,deg n . This function returns the vertex of this level that has number num.

      gap> VertexNumber(1, 3, 2);
      [ 1, 1, 1 ]
      gap> VertexNumber(4, 4, 3);
      [ 1, 1, 2, 1 ]

  ## 5.3 Some predefined groups

  Several groups are predefined as fields in the global variable `AG_Groups`. Here is how to access, for example, Grigorchuk group

      gap> G:=AG_Groups.GrigorchukGroup;
      < a, b, c, d >

  To perform operations with elements of `G` one can use `AssignGeneratorVariables` function.

      gap> AssignGeneratorVariables(G);
      #I  Global variable `a' is already defined and will be overwritten
      #I  Global variable `b' is already defined and will be overwritten
      #I  Global variable `c' is already defined and will be overwritten
      #I  Global variable `d' is already defined and will be overwritten
      #I  Assigned the global variables [ a, b, c, d ]
      gap> Decompose(a*b);
      (c, a)(1,2)

  Below is a list of all predefined groups with short description and references.

- `GrigorchukGroup V`

  is the first Grigorchuk group, an infinite 2-group of intermediate growth constructed in Gri80 (see Gri05 for a survey on this group). It is defined as the group generated by the automaton

  a=(1,1)(1,2), b=(a,c), c=(a,d), d=(1,b).

  The group is stored in the global variable `AG_Groups.GrigorchukGroup`

- `UniversalGrigorchukGroup V`

  is the universal group for the family of groups Gω (see Gri84). It is defined as a group acting on the 6-ary tree, generated by the automaton

  a=(1,1,1,1,1,1)(1,2), b=(a,a,1,b,b,b), c=(a,1,a,c,c,c), d=(1,a,a,d,d,d).

  The group is stored in the global variable `AG_Groups.UniversalGrigorchukGroup`

- `Basilica V`

  is the Basilica group. It was first studied in GZ02a and GZ02b. Later it became the first example of amenable, but not subexponentially amenable group (see BV05). It is the iterated monodromy group of the map f(z)=z2−1. It is generated by the automaton

  u=(v,1)(1,2), v=(u,1).

  The group is stored in the global variable `AG_Groups.Basilica`

- `Lamplighter V`

  is the lamplighter group. This group was the key ingredient in the counterexample to the strong Atiyah conjecture (see GLSZ00). It is generated by the automaton

  a=(a,b)(1,2), b=(a,b).

  The group is stored in the global variable `AG_Groups.Lamplighter`

- `AddingMachine V`

  is the free abelian group of rank 1 (see GNS00) generated by the automaton

  a=(1,a)(1,2).

  The group is stored in the global variable `AG_Groups.AddingMachine`

- `InfiniteDihedral V`

  is the infinite dihedral group (see GNS00) generated by the automaton

  a=(a,a)(1,2), b=(b,a).

  The group is stored in the global variable `AG_Groups.InfiniteDihedral`

- `AleshinGroup V`

  is a group generated by the Aleshin automaton (see Ale83) defined by the following wreath recursion:

  a=(b,c)(1,2), b=(c,b)(1,2), c=(a,a).

  It is isomorphic to the free group of rank 3 as was proved by M.Vorobets and Y.Vorobets (see VV05). The group is stored in the global variable `AG_Groups.AleshinGroup`

- `Bellaterra V`

  is a group generated by the Aleshin automaton (see Ale83) defined by the following wreath recursion:

  a=(c,c)(1,2), b=(a,b), c=(b,a).

  It is isomorphic to the free product of 3 cyclic groups of order 2 (see BGK07) The group is stored in the global variable `AG_Groups.Bellaterra`

- `SushchanskyGroup V`

  is the self-similar closure of the faithful level-transitive action of the Sushchansky group on the ternary tree. The original groups constructed in Sus79 are infinite p-groups of intermediate growth acting on the p-ary tree. In BS07 the action of these groups on the tree was simplified, so that, in particular, the self-similar closure of one of the 3-groups is generated by the automaton

  A=(1,1,1)(1,2,3), A2=(1,1,1)(1,3,2), B=(r1,q1,A),

  r1=(r2,A,1), r2=(r3,1,1), r3=(r4,1,1),

  r4=(r5,A,1), r5=(r6,A2,1), r6=(r7,A,1),

  r7=(r8,A,1), r8=(r9,A,1), r9=(r1,A2,1),

  q1=(q2,1,1), q2=(q3,A,1), q3=(q1,A,1).

  The group 〈A,B〉 is isomorphic to the original Sushchansky 3-group. The group is stored in the global variable `AG_Groups.SushchanskyGroup`

- `Hanoi3 V`
- `Hanoi4 V`

  Groups related to the Hanoi towers game on 3 and 4 pegs correspondingly (see GS06a and GS06b). For 3 pegs `Hanoi3` is generated by the automaton

  a23=(a23,1,1)(2,3), a13=(1,a13,1)(1,3), a12=(1,1,a12)(1,2),

  while the automaton generating `Hanoi4` is

  a12=(1,1,a12,a12)(1,2), a13=(1,a13,1,a13)(1,3), a14=(1,a14,a14,1)(1,4),

  a23=(a23,1,1,a23)(2,3), a24=(a24,1,a24,1)(2,4), a34=(a34,a34,1,1)(3,4).

  The groups are stored in the global variables `AG_Groups.Hanoi3` and `AG_Groups.Hanoi4`

- `GuptaSidki3Group V`

  is the Gupta-Sidki infinite 3-group constructed in GS83 and generated by the automaton

  a=(1,1,1)(1,2,3), b=(a,a−1,b).

  The group is stored in the global variable `AG_Groups.GuptaSidki3Group`

- `GuptaFabrikowskiGroup V`

  is the Gupta-Fabrykowski group of intermediate growth constructed in FG85 and generated by the automaton

  a=(1,1,1)(1,2,3), b=(a,1,b).

  The group is stored in the global variable `AG_Groups.GuptaFabrikowskiGroup`

- `BartholdiGrigorchukGroup V`

  is the Bartholdi-Grigorchuk group studied in BG02 and generated by the automaton

  a=(1,1,1)(1,2,3), b=(a,a,b).

  The group is stored in the global variable `AG_Groups.BartholdiGrigorchukGroup`

- `GrigorchukErschlerGroup V`

  is the group of subexponential growth studied by Erschler in Ers04. It grows faster than exp(nα) for any α < 1. It belongs to the class of groups constructed by Grigorchuk in Gri84 and corresponds to the sequence 01010101…. It is generated by the automaton

  a=(1,1)(1,2), b=(a,b), c=(a,d), d=(1,c).

  The group is stored in the global variable `AG_Groups.GrigorchukErschlerGroup`

- `BartholdiNonunifExponGroup V`

  is the group of nonuniformly exponential growth constructed by Bartholdi in Bar03. Similar examples were constructed earlier in Wil04 by Wilson. It is generated by the automaton

  x=(1,1,1,1,1,1,1)(1,5)(3,7), y=(1,1,1,1,1,1,1)(2,3)(6,7), z=(1,1,1,1,1,1,1)(4,6)(5,7),

  x1=(x1,x,1,1,1,1,1), y1=(y1,y,1,1,1,1,1), z1=(z1,z,1,1,1,1,1).

  The group is stored in the global variable `AG_Groups.BartholdiNonunifExponGroup`

- `IMG_z2plusI V`

  The iterated monodromy group of the map f(z)=z2+i. It has intermediate growth (see BP06) and was studied in GSS07.

  a=(1,1)(1,2), b=(a,c), c=(b,1).

  The group is stored in the global variable `AG_Groups.IMG_z2plusI`

- `Airplane V`
- `Rabbit V`

  These are iterated monodromy groups of certain quadratic polynomials studied in BN06. It was proved there that they are not isomorphic. The automata generating them are the following

  a=(b,1)(1,2), b=(c,1), c=(a,1);

  a=(b,1)(1,2), b=(1,c), c=(a,1).

  The groups are stored in the global variables `AG_Groups.Airplane` and `AG_Groups.Rabbit`

- `TwoStateSemigroupOfIntermediateGrowth V`

  is the semigroup of intermediate growth studied in BRS06. It is generated by the automaton

  f0=(f0,f0)(1,2), f1=(f1,f0)[2,2]·

  The group is stored in the global variable `AG_Groups.TwoStateSemigroupOfIntermediateGrowth`

- `UniversalD_omega V`

  is the group constructed in Nek07 as the universal group which covers an uncountable family of groups parameterized by infinite binary sequences. It is contracting with nucleus consisting of 35 elements. Its generating automaton is as follows (it acts on the 4-ary tree):

  a=(1,2)(3,4), b=(a,c,a,c), c=(b,1,1,b).

  The group is stored in the global variable `AG_Groups.UniversalD_omega`

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