[automgrp] 5 Miscellaneous

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# 5 Miscellaneous

### Sections

1. Trees
2. Some predefined groups

## 5.1 Trees

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

  Let ver belong to the n-th level of the deg-ary tree. One can naturally enumerate all the vertices of this level by the numbers 1,…,deg n . This function returns the number that corresponds to the vertex ver.

      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 ]

  ## 5.2 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`

  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).

- `UniversalGrigorchukGroup`

  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).

- `Basilica`

  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).

- `Lamplighter`

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

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

- `AddingMachine`

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

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

- `InfiniteDihedral`

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

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

- `AleshinGroup`

  is the free group of rank 3 generated by the Aleshin automaton (see Ale83)

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

  It was proved just recently by M.Vorobets and Ya.Vorobets (see VV05) that the group is indeed free of rank 3.

- `Bellaterra`

  is the free product of 3 cyclic groups of oreder 2 (see BGK07)

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

- `SushchanskyGroup`

  is the self-similar closure of the faithful level-transitive action of 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.

- `Hanoi3`
- `Hanoi4`

  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).

- `GuptaSidki3Group`

  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).

- `GuptaFabrikowskiGroup`

  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).

- `BartholdiGrigorchukGroup`

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

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

- `GrigorchukErschlerGroup`

  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).

- `BartholdiNonunifExponGroup`

  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).

- `IMG_z2plusI`

  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).

- `Airplane`
- `Rabbit`

  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).

- `TwoStateSemigroupOfIntermediateGrowth`

  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]·

- `UniversalD_omega`

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

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

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