<html><head><title>[automgrp] 1 Introduction</title></head> <body text="#000000" bgcolor="#ffffff"> [<a href = "chapters.htm">Up</a>] [<a href ="CHAP002.htm">Next</a>] [<a href = "theindex.htm">Index</a>] <h1>1 Introduction</h1><p> <P> <H3>Sections</H3> <oL> <li> <A HREF="CHAP001.htm#SECT001">Short math background</a> <li> <A HREF="CHAP001.htm#SECT002">Installation instructions</a> <li> <A HREF="CHAP001.htm#SECT003">Quick example</a> </ol><p> <p> The <code>AutomGrp</code> 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. <p> 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 <a href="biblio.htm#BGK06"><cite>BGK06</cite></a>, <a href="biblio.htm#BGK07"><cite>BGK07</cite></a>). <p> 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. <p> 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. <p> Both authors were partially supported by NSF grants DMS-0600975, DMS-0456185 and DMS-0308985. <p> <p> <h2><a name="SECT001">1.1 Short math background</a></h2> <p><p> 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 <a href="biblio.htm#GNS00"><cite>GNS00</cite></a>. <p> The infinite connected tree with selected vertex, called the <var>root</var>, in which the degree of every vertex except the root is <i>d</i>+1 and the degree of the root is <i>d</i> is called the <var>regular homogeneous rooted tree of degree <i>d</i> (or <i>d</i>-ary tree)</var>. The rooted tree of degree 2 is called the <var>binary tree</var>. <p> The <i>n</i>-th <var>level</var> of the tree consists of all vertices located at distance <i>n</i> from the root (here we mean combinatorial distance in the graph). <p> Similarly one defines <var>spherically homogeneous</var> (or <var>spherically-transitive</var>) rooted trees as rooted trees, such that the degrees of all vertices on the same level coincide. <p> Given a finite alphabet <i>X</i>={1,2,…,<i>d</i>} the set <i>X</i><sup>*</sup> of all finite words over <i>X</i> may be endowed with the structure of <i>d</i>-ary tree in which the empty word ∅ is the <var>root</var>, the <var>level</var> <i>n</i> in <i>X</i><sup>*</sup> consists of the words of length <i>n</i> over <i>X</i> and every vertex <i>v</i> has <i>d</i> children, labeled by <i>vx</i>, for <i>x</i> ∈ <i>X</i>. <p> Any automorphism <i>f</i> of the rooted tree <i>T</i> fixes the root and the levels. For any vertex <i>v</i> of the tree the automorphism <i>f</i> induces the automorphism <i>f</i>|<sub><i>v</i></sub> of the subtree hanging down from the vertex <i>v</i> by <i>f</i>|<sub><i>v</i></sub>(<i>u</i>)=<i>w</i> if <i>f</i>(<i>vu</i>)=<i>v</i>′<i>w</i> for some <i>v</i>′ ∈ <i>X</i><sup>|<i>v</i>|</sup> from the same level as <i>v</i> (here |<i>v</i>| denotes the combinatorial distance from <i>v</i> to the root of the tree). This automorphism is called <var>the section</var> of <i>f</i> at <i>v</i>. <p> If the tree <i>T</i> is regular, then the subtrees hanging down from vertices of <i>T</i> are canonically isomorphic to <i>T</i> and, thus, the sections of any automorphism <i>f</i> of <i>T</i> can be considered as automorphisms of <i>T</i> again. <p> A group <i>G</i> of automorphisms of the regular rooted tree <i>T</i> is called <var>self-similar</var> if all sections of every element of <i>G</i> belong to <i>G</i>. <p> A self-similar group <i>G</i> is called <var>contracting</var> if there is a finite set <i>N</i> of elements of <i>G</i>, such that for any <i>g</i> in <i>G</i> there is a level <i>n</i> such that all sections of <i>g</i> at vertices of levels bigger than <i>n</i> belong to <i>N</i>. The smallest set with such a property is called the <var>nucleus</var> of <i>G</i>. <p> Any automorphism <i>f</i> of a rooted tree can be decomposed as <br clear="all" /><table border="0" width="100%"><tr><td><table align="center" cellspacing="0" cellpadding="2"><tr><td nowrap="nowrap" align="center"><i>f</i>=(<i>f</i><sub>1</sub>,<i>f</i><sub>2</sub>,…,<i>f</i><sub><i>d</i></sub>)σ,</td></tr></table></td></tr></table> <p> where <i>f</i><sub>1</sub>,…,<i>f</i><sub><i>d</i></sub> are the sections of <i>f</i> at the vertices of the first level and σ is the permutation which permutes the subtrees hanging down from these vertices. <p> This notation is very convenient for performing multiplication of elements. If <i>f</i>=(<i>f</i><sub>1</sub>,<i>f</i><sub>2</sub>,…,<i>f</i><sub><i>d</i></sub>)σ and <i>g</i>=(<i>g</i><sub>1</sub>,<i>g</i><sub>2</sub>,…,<i>g</i><sub><i>d</i></sub>)π, then <p> <br clear="all" /><table border="0" width="100%"><tr><td><table align="center" cellspacing="0" cellpadding="2"><tr><td nowrap="nowrap" align="center"><i>f</i>·<i>g</i>=(<i>f</i><sub>1</sub>·<i>g</i><sub>σ(1)</sub>,…,<i>f</i><sub><i>d</i></sub>·<i>g</i><sub>σ(<i>d</i>)</sub>)σπ,</td></tr></table></td></tr></table> <p> <br clear="all" /><table border="0" width="100%"><tr><td><table align="center" cellspacing="0" cellpadding="2"><tr><td nowrap="nowrap" align="center"><i>f</i><sup>−1</sup>=(<i>f</i><sub>σ<sup>−1</sup>(1)</sub><sup>−1</sup>,…,<i>f</i><sub>σ<sup>−1</sup>(<i>d</i>)</sub><sup>−1</sup>)σ<sup>−1</sup>.</td></tr></table></td></tr></table> <p> The group of automorphisms of a rooted tree is said to be <var>level-transitive</var> if it acts transitively on each level of the tree. <p> 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. <p> 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. <p> A <var>Mealy automaton</var> (<var>transducer</var>, <var>synchronous automaton</var>, or, simply, <var>automaton</var>) is a tuple <i>A</i>=(<i>Q</i>,<i>X</i>,ρ,τ), where <i>Q</i> is a set of <var>states</var>, <i>X</i> is a finite <var>alphabet</var> of cardinality <i>d</i> ≥ 2, ρ:<i>Q</i> ×<i>X</i> → <i>X</i> is a map, called <var>output map</var>, τ:<i>Q</i> ×<i>X</i> → <i>Q</i> is a map, called <var>transition map</var>. <p> If for each state <i>q</i> in <i>Q</i>, the restriction ρ<sub><i>q</i></sub>: <i>X</i> → <i>X</i> given by ρ<sub><i>q</i></sub>(<i>x</i>)=ρ(<i>q</i>,<i>x</i>) is a permutation, the automaton is called <var>invertible</var>. <p> If the set <i>Q</i> of states is finite, the automaton is called <var>finite</var>. <p> If some state <i>q</i> in <i>Q</i> of the automaton <i>A</i> is selected to be initial, the automaton is called <var>initial</var> and denoted <i>A</i><sub><i>q</i></sub>. If an initial state is not specified, the automaton is called <var>noninitial</var>. <p> An initial automaton naturally acts on <i>X</i><sup>*</sup> by homomorphisms (automorphisms in case of an invertible automation). Given a word <i>x</i><sub>1</sub><i>x</i><sub>2</sub>…<i>x</i><sub><i>n</i></sub> the automaton starts at the initial state <i>q</i>, reads the first input letter <i>x</i><sub>1</sub>, outputs the letter ρ<sub><i>q</i></sub>(<i>x</i><sub>1</sub>) and changes its state to <i>q</i><sub>1</sub>=τ(<i>q</i>,<i>x</i><sub>1</sub>). The rest of the input word is handled by the new state <i>q</i><sub>1</sub> in the same way. Formally speaking, the functions ρ and τ can be extended to ρ:<i>Q</i> ×<i>X</i><sup>*</sup> → <i>X</i><sup>*</sup> and τ:<i>Q</i> ×<i>X</i><sup>*</sup> → <i>Q</i>. <p> Given an automaton <i>A</i> the group <i>G</i>(<i>A</i>) of automorphisms of <i>X</i><sup>*</sup> generated by the states of <i>A</i> (as initial automata) is called the <var>automaton group</var> defined by <i>A</i>. <p> Every automaton group is self-similar, because the section of <i>A</i><sub><i>q</i></sub> at vertex <i>v</i> is just <i>A</i><sub>τ(<i>q</i>,<i>v</i>)</sub>. <p> 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. <p> Finite automata are often described by <var>recursive relations</var> of the form <p> <br clear="all" /><table border="0" width="100%"><tr><td><table align="center" cellspacing="0" cellpadding="2"><tr><td nowrap="nowrap" align="center"><i>q</i>=(τ(<i>q</i>,1),…,τ(<i>q</i>,<i>d</i>)) ρ<sub><i>q</i></sub></td></tr></table></td></tr></table> <p> for every state <i>q</i>. For example, the line <i>a</i>=(<i>a</i>,<i>b</i>)(1,2), <i>b</i>=(<i>a</i>,<i>b</i>) describes the automaton with 2 states <i>a</i> and <i>b</i>, such that <i>a</i> permutes the letters 1 and 2 and <i>b</i> does not; and independently of current state the automaton changes its initial state to <i>a</i> if it reads 1 and to <i>b</i> if it reads 2. This particular automaton generates the, so-called, Lamplighter group. <p> One may also consider semigroups generated by noninvertible automata. <p> <p> <h2><a name="SECT002">1.2 Installation instructions</a></h2> <p><p> <code>AutomGrp</code> package requires <font face="Gill Sans,Helvetica,Arial">GAP</font> version at least 4.4.6 and <code>FGA</code> (Free Group Algorithms) package available at <a href="http://www.gap-system.org/Packages/fga.html">http://www.gap-system.org/Packages/fga.html</a> <p> The installation of the <code>AutomGrp</code> package follows the standard <font face="Gill Sans,Helvetica,Arial">GAP</font> rules, i.e. to install it unpack the archive into the <code>pkg</code> directory of your GAP distribution. This will create <code>automgrp</code> subdirectory. <p> To load package issue the command <pre> 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 </pre> <p> To test the installation, issue the command <pre> gap> Read( Filename( DirectoriesLibrary( "pkg/automgrp/tst"), "testall.g")); </pre> in the <font face="Gill Sans,Helvetica,Arial">GAP</font> command line. <p> <p> <h2><a name="SECT003">1.3 Quick example</a></h2> <p><p> Here is how to define Grigorchuk group and Basilica group. <p> <pre> 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 > </pre> <p> 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 (<a href="biblio.htm#BRS06"><cite>BRS06</cite></a>) <p> <pre> gap> SG := AutomatonSemigroup( "f0=(f0,f0)(1,2), f1=(f1,f0)[2,2]" ); < f0, f1 > </pre> <p> Another type of groups (semigroups) implemented in the package is the class of groups (semigroups) defined by wreath recursion (finitely generated self-similar groups). <p> <pre> gap> WRG := SelfSimilarGroup("x=(1,y)(1,2),y=(z^-1,1)(1,2),z=(1,x*y)"); < x, y, z > </pre> <p> Now we can compute several properties of <code>GrigorchukGroup</code>, <code>Basilica</code> and <code>SG</code> <p> <pre> gap> IsFinite(GrigorchukGroup); false gap> IsSphericallyTransitive(GrigorchukGroup); true gap> IsFractal(GrigorchukGroup); true gap> IsAbelian(GrigorchukGroup); false gap> IsTransitiveOnLevel(GrigorchukGroup, 4); true </pre> <p> We can also check that <code>Basilica</code> and <code>WRG</code> are contracting and compute their nuclei <pre> 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 ] </pre> <p> The group <code>GrigorchukGroup</code> is generated by a bounded automaton and, thus, is amenable (see <a href="biblio.htm#BKNV05"><cite>BKNV05</cite></a>) <pre> gap> IsGeneratedByBoundedAutomaton(GrigorchukGroup); true gap> IsAmenable(GrigorchukGroup); true </pre> <p> We can compute the stabilizers of levels and vertices <pre> 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 > </pre> <p> In case of a finite group we can produce an isomorphism into a permutational group <pre> 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 </pre> <p> Here is how to find relations in <code>Basilica</code> between elements of length not greater than 5. <pre> 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 ] </pre> <p> Or relations in the subgroup 〈<i>p</i>=<i>uv</i><sup>−1</sup>, <i>q</i>=<i>vu</i>〉 <pre> 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 ] </pre> <p> Or relations in the semigroup <code>SG</code> <p> <pre> 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 ] ] </pre> <p> Some basic operations with elements are the following: <p> The function <code>IsOne</code> computes whether an element represents the trivial automorphism of the tree <pre> gap> IsOne( (a*b)^16 ); true </pre> <p> Here is how to compute the order (this function might not stop in some cases) <pre> gap> Order(a*b); 16 gap> Order(u^22*v^-15*u^2*v*u^10); infinity </pre> <p> One can check if a particular element acts spherically transitively on the tree (this function might not stop in some cases) <pre> gap> IsSphericallyTransitive(a*b); false gap> IsSphericallyTransitive(u*v); true </pre> <p> The sections of an element can be obtained as follows <pre> 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) </pre> <p> One can try to compute whether the elements of group <code>WRG</code> defined by wreath recursion are finite-state and calculate corresponding automaton <p> <pre> 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 </pre> <p> To get the action of an element on a vertex or on a particular level of the tree use the following commands <pre> 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) </pre> <p> The action of the whole group <code>GrigorchukGroup</code> on some level can be computed via <code>PermGroupOnLevel</code> (see <a href="CHAP002.htm#SSEC003.1">PermGroupOnLevel</a>). <pre> 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 </pre> <p> The next example shows how to find all elements of Grigorchuk group of length at most 5, which have order 16. <pre> 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 ] </pre> <p> <p> [<a href = "chapters.htm">Up</a>] [<a href ="CHAP002.htm">Next</a>] [<a href = "theindex.htm">Index</a>] <P> <address>automgrp manual<br>September 2008 </address></body></html>