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# 1 Introduction

### Sections

1. Short math background
2. Installation instructions
3. Quick example

The `AutomGrp` package provides methods for computations with groups and semigroups generated by finite automata or given by wreath recursions, as well as with their finitely generated subgroups, subsemigroups and elements.

The project originally started in 2000 mostly for personal use. It was gradually expanding during consequent years, including both addition of new algorithms and simplification of user interface. It was used in the process of classification of groups generated by 3-state automata over a 2-letter alphabet (see BGK06, BGK07).

First author thanks Sveta and Max Muntyan for their infinite patience and understanding. Second author thanks Olga and Anna Savchuk for their help and understanding. This project would be impossible without them.

We would like to express our warm gratitude to Rostislav Grigorchuk, Zoran Sunic, Volodymyr Nekrashevych, Ievgen Bondarenko, Rostyslav Kravchenko, Yaroslav and Maria Vorobets and Ben Steinberg for their support, valuable comments, feature requests and constant interest in the project.

Both authors were partially supported by NSF grants DMS-0600975, DMS-0456185 and DMS-0308985.

## 1.1 Short math background

This package deals mostly with groups acting on rooted trees. In this section we recall necessary definitions and notation that will be used throughout the manual. For more detailed introduction in the theory of groups generated by automata we refer the reader to GNS00.

The infinite connected tree with selected vertex, called the root, in which the degree of every vertex except the root is d+1 and the degree of the root is d is called the regular homogeneous rooted tree of degree d (or d-ary tree). The rooted tree of degree 2 is called the binary tree.

The n-th level of the tree consists of all vertices located at distance n from the root (here we mean combinatorial distance in the graph).

Similarly one defines spherically homogeneous (or spherically-transitive) rooted trees as rooted trees, such that the degrees of all vertices on the same level coincide.

Given a finite alphabet X={1,2,…,d} the set X* of all finite words over X may be endowed with the structure of d-ary tree in which the empty word ∅ is the root, the level n in X* consists of the words of length n over X and every vertex v has d children, labeled by vx, for x ∈ X.

Any automorphism f of the rooted tree T fixes the root and the levels. For any vertex v of the tree the automorphism f induces the automorphism f|v of the subtree hanging down from the vertex v by f|v(u)=w if f(vu)=v′w for some v′ ∈ X|v| from the same level as v (here |v| denotes the combinatorial distance from v to the root of the tree). This automorphism is called the section of f at v.

If the tree T is regular, then the subtrees hanging down from vertices of T are canonically isomorphic to T and, thus, the sections of any automorphism f of T can be considered as automorphisms of T again.

A group G of automorphisms of the regular rooted tree T is called self-similar if all sections of every element of G belong to G.

A self-similar group G is called contracting if there is a finite set N of elements of G, such that for any g in G there is a level n such that all sections of g at vertices of levels bigger than n belong to N. The smallest set with such a property is called the nucleus of G.

Any automorphism f of a rooted tree can be decomposed as

f=(f1,f2,…,fd)σ,

where f1,…,fd are the sections of f at the vertices of the first level and σ is the permutation which permutes the subtrees hanging down from these vertices.

This notation is very convenient for performing multiplication of elements. If f=(f1,f2,…,fd)σ and g=(g1,g2,…,gd)π, then

f·g=(f1·gσ(1),…,fd·gσ(d))σπ,

f−1=(fσ−1(1)−1,…,fσ−1(d)−1)σ−1.

The group of automorphisms of a rooted tree is said to be level-transitive if it acts transitively on each level of the tree.

Everything above applies also for homomorphisms of rooted trees (maps preserving the root and incidence relation of the vertices). The only difference is that in this case we get semigroups and monoids of tree homomorphisms.

A special class of self-similar groups is the class of groups generated by finite automata. This class is especially nice from algorithmic point of view. Recall basic definitions.

A Mealy automaton (transducer, synchronous automaton, or, simply, automaton) is a tuple A=(Q,X,ρ,τ), where Q is a set of states, X is a finite alphabet of cardinality d ≥ 2, ρ:Q ×X → X is a map, called output map, τ:Q ×X → Q is a map, called transition map.

If for each state q in Q, the restriction ρq: X → X given by ρq(x)=ρ(q,x) is a permutation, the automaton is called invertible.

If the set Q of states is finite, the automaton is called finite.

If some state q in Q of the automaton A is selected to be initial, the automaton is called initial and denoted Aq. If an initial state is not specified, the automaton is called noninitial.

An initial automaton naturally acts on X* by homomorphisms (automorphisms in case of an invertible automation). Given a word x1x2…xn the automaton starts at the initial state q, reads the first input letter x1, outputs the letter ρq(x1) and changes its state to q1=τ(q,x1). The rest of the input word is handled by the new state q1 in the same way. Formally speaking, the functions ρ and τ can be extended to ρ:Q ×X* → X* and τ:Q ×X* → Q.

Given an automaton A the group G(A) of automorphisms of X* generated by the states of A (as initial automata) is called the automaton group defined by A.

Every automaton group is self-similar, because the section of Aq at vertex v is just Aτ(q,v).

A special case is the case of groups generated by finite automata and their subgroups. In this class we can solve the word problem, which makes it much nicer from computational point of view.

Finite automata are often described by recursive relations of the form

q=(τ(q,1),…,τ(q,d)) ρq

for every state q. For example, the line a=(a,b)(1,2), b=(a,b) describes the automaton with 2 states a and b, such that a permutes the letters 1 and 2 and b does not; and independently of current state the automaton changes its initial state to a if it reads 1 and to b if it reads 2. This particular automaton generates the, so-called, Lamplighter group.

One may also consider semigroups generated by noninvertible automata.

## 1.2 Installation instructions

`AutomGrp` package requires GAP version at least 4.4.6 and `FGA` (Free Group Algorithms) package available at http://www.gap-system.org/Packages/fga.html

The installation of the `AutomGrp` package follows the standard GAP rules, i.e. to install it unpack the archive into the `pkg` directory of your GAP distribution. This will create `automgrp` subdirectory.

To load package issue the command

    gap> LoadPackage("automgrp");
    -----------------------------------------------------------------------------
    Loading  AutomGrp 1.1.1 (Automata Groups)
    by Yevgen Muntyan (http://www.math.tamu.edu/~muntyan/) and
       Dmytro Savchuk (http://www.math.tamu.edu/~savchuk/).
    -----------------------------------------------------------------------------
    true

To test the installation, issue the command

    gap> Read( Filename( DirectoriesLibrary( "pkg/automgrp/tst"), "testall.g"));

in the GAP command line.

## 1.3 Quick example

Here is how to define Grigorchuk group and Basilica group.

    gap> GrigorchukGroup := AutomatonGroup("a=(1,1)(1,2),b=(a,c),c=(a,d),d=(1,b)");
    < a, b, c, d >
    gap> Basilica := AutomatonGroup( "u=(v,1)(1,2), v=(u,1)" );
    < u, v >

Similarly one can define a group (or semigroup) generated by a noninvertible automaton. As an example we consider the semigroup of intermediate growth generated by the two state automaton (BRS06)

    gap> SG := AutomatonSemigroup( "f0=(f0,f0)(1,2), f1=(f1,f0)[2,2]" );
    < f0, f1 >

Another type of groups (semigroups) implemented in the package is the class of groups (semigroups) defined by wreath recursion (finitely generated self-similar groups).

    gap> WRG := SelfSimilarGroup("x=(1,y)(1,2),y=(z^-1,1)(1,2),z=(1,x*y)");
    < x, y, z >

Now we can compute several properties of `GrigorchukGroup`, `Basilica` and `SG`

    gap> IsFinite(GrigorchukGroup);
    false
    gap> IsSphericallyTransitive(GrigorchukGroup);
    true
    gap> IsFractal(GrigorchukGroup);
    true
    gap> IsAbelian(GrigorchukGroup);
    false
    gap> IsTransitiveOnLevel(GrigorchukGroup, 4);
    true

We can also check that `Basilica` and `WRG` are contracting and compute their nuclei

    gap> IsContracting(Basilica);
    true
    gap> GroupNucleus(Basilica);
    [ 1, u, v, u^-1, v^-1, u^-1*v, v^-1*u ]
    gap> IsContracting( WRG );
    true
    gap> GroupNucleus( WRG );
    [ 1, y*z^-1*x*y, z^-1*y^-1*x^-1*y*z^-1, z^-1*y^-1*x^-1, y^-1*x^-1*z*y^-1,
      z*y^-1*x*y*z, x*y*z ]

The group `GrigorchukGroup` is generated by a bounded automaton and, thus, is amenable (see BKNV05)

    gap> IsGeneratedByBoundedAutomaton(GrigorchukGroup);
    true
    gap> IsAmenable(GrigorchukGroup);
    true

We can compute the stabilizers of levels and vertices

    gap> StabilizerOfLevel(GrigorchukGroup, 2);
    < a*b*a*d*a^-1*b^-1*a^-1, d, b*a*d*a^-1*b^-1, a*b*c*a^-1, b*a*b*a*b^-1*a^-1*b^
    -1*a^-1, a*b*a*b*a*b^-1*a^-1*b^-1 >
    gap> StabilizerOfVertex(GrigorchukGroup, [2, 1]);
    < a*b*a*d*a^-1*b^-1*a^-1, d, a*c*b^-1*a^-1, c, b, a*b*a*c*a^-1*b^-1*a^
    -1, a*b*a*b*a^-1*b^-1*a^-1 >

In case of a finite group we can produce an isomorphism into a permutational group

    gap> f := IsomorphismPermGroup(Group(a,b));
    [ a, b ] ->
    [ (1,2)(3,5)(4,6)(7,9)(8,10)(11,13)(12,14)(15,17)(16,18)(19,21)(20,22)(23,
        25)(24,26)(27,29)(28,30)(31,32), (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,
        15)(14,16)(17,19)(18,20)(21,23)(22,24)(25,27)(26,28)(29,31)(30,32) ]
    gap> Size(Image(f));
    32

Here is how to find relations in `Basilica` between elements of length not greater than 5.

    gap> FindGroupRelations(Basilica, 6);
    v*u*v*u^-1*v^-1*u*v^-1*u^-1
    v*u^2*v^-1*u^2*v*u^-2*v^-1*u^-2
    v^2*u*v^2*u^-1*v^-2*u*v^-2*u^-1
    [ v*u*v*u^-1*v^-1*u*v^-1*u^-1, v*u^2*v^-1*u^2*v*u^-2*v^-1*u^-2,
      v^2*u*v^2*u^-1*v^-2*u*v^-2*u^-1 ]

Or relations in the subgroup 〈p=uv−1, q=vu〉

    gap> FindGroupRelations([u*v^-1,v*u], ["p", "q"], 5);
    q*p^2*q*p^-1*q^-2*p^-1
    [ q*p^2*q*p^-1*q^-2*p^-1 ]

Or relations in the semigroup `SG`

    gap> FindSemigroupRelations(SG, 4);
    f0^3 = f0
    f0^2*f1 = f1
    f1*f0^2 = f1
    f1^3 = f1
    [ [ f0^3, f0 ], [ f0^2*f1, f1 ], [ f1*f0^2, f1 ], [ f1^3, f1 ] ]

Some basic operations with elements are the following:

The function `IsOne` computes whether an element represents the trivial automorphism of the tree

    gap> IsOne( (a*b)^16 );
    true

Here is how to compute the order (this function might not stop in some cases)

    gap> Order(a*b);
    16
    gap> Order(u^22*v^-15*u^2*v*u^10);
    infinity

One can check if a particular element acts spherically transitively on the tree (this function might not stop in some cases)

    gap> IsSphericallyTransitive(a*b);
    false
    gap> IsSphericallyTransitive(u*v);
    true

The sections of an element can be obtained as follows

    gap> Section(u*v^2*u, 2);
    u^2*v
    gap> Decompose(u*v^2*u);
    (v, u^2*v)
    gap> Decompose(u*v^2*u, 3);
    (v, 1, 1, 1, u*v, 1, u, 1)(1,2)(5,6)

One can try to compute whether the elements of group `WRG` defined by wreath recursion are finite-state and calculate corresponding automaton

    gap> IsFiniteState(x*y^-1);
    true
    gap> AllSections(x*y^-1);
    [ x*y^-1, z, 1, x*y, y*z^-1, z^-1*y^-1*x^-1, y^-1*x^-1*z*y^-1, z*y^-1*x*y*z,
      y*z^-1*x*y, z^-1*y^-1*x^-1*y*z^-1, x*y*z, y, z^-1, y^-1*x^-1, z*y^-1 ]
    gap> A := MealyAutomaton(x*y^-1);
    <automaton>
    gap> NumberOfStates(A);
    15

To get the action of an element on a vertex or on a particular level of the tree use the following commands

    gap> [1,2,1,1]^(a*b);
    [ 2, 2, 1, 1 ]
    gap> PermOnLevel(u*v^2*v, 3);
    (1,6,4,8,2,5,3,7)

The action of the whole group `GrigorchukGroup` on some level can be computed via `PermGroupOnLevel` (see PermGroupOnLevel).

    gap> PermGroupOnLevel(GrigorchukGroup, 3);
    Group([ (1,5)(2,6)(3,7)(4,8), (1,3)(2,4)(5,6), (1,3)(2,4), (5,6) ])
    gap> Size(last);
    128

The next example shows how to find all elements of Grigorchuk group of length at most 5, which have order 16.

    gap> FindElements(GrigorchukGroup, Order, 16, 5);
    [ a*b, b*a, c*a*d, d*a*c, a*b*a*d, a*c*a*d, a*d*a*b, a*d*a*c, b*a*d*a,
      c*a*d*a, d*a*b*a, d*a*c*a, a*c*a*d*a, a*d*a*c*a, b*a*b*a*c, b*a*c*a*c,
      c*a*b*a*b, c*a*c*a*b ]

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