9 Examples

9.1 Examples of groups

9.1-1 FullBinaryGroup

9.1-2 BinaryKneadingGroup

9.1-3 BasilicaGroup

9.1-4 FornaessSibonyGroup

9.1-5 AddingGroup

9.1-6 BinaryAddingGroup

9.1-7 MixerGroup

9.1-8 SunicGroup

9.1-9 GrigorchukMachines

9.1-10 GrigorchukMachine

9.1-11 GrigorchukOverGroup

9.1-12 GrigorchukTwistedTwin

9.1-13 BrunnerSidkiVieiraGroup

9.1-14 AleshinGroups

9.1-15 AleshinGroup

9.1-16 BabyAleshinGroup

9.1-17 SidkiFreeGroup

9.1-18 GuptaSidkiGroups

9.1-19 GuptaSidkiGroup

9.1-20 NeumannGroup

9.1-21 FabrykowskiGuptaGroup

9.1-22 ZugadiSpinalGroup

9.1-23 GammaPQMachine

9.1-24 RattaggiGroup

9.1-25 HanoiGroup

9.1-26 DahmaniGroup

9.1-27 MamaghaniGroup

9.1-28 WeierstrassGroup

9.1-29 StrichartzGroup

9.1-30 FRAffineGroup

9.1-31 CayleyGroup

9.1-1 FullBinaryGroup

9.1-2 BinaryKneadingGroup

9.1-3 BasilicaGroup

9.1-4 FornaessSibonyGroup

9.1-5 AddingGroup

9.1-6 BinaryAddingGroup

9.1-7 MixerGroup

9.1-8 SunicGroup

9.1-9 GrigorchukMachines

9.1-10 GrigorchukMachine

9.1-11 GrigorchukOverGroup

9.1-12 GrigorchukTwistedTwin

9.1-13 BrunnerSidkiVieiraGroup

9.1-14 AleshinGroups

9.1-15 AleshinGroup

9.1-16 BabyAleshinGroup

9.1-17 SidkiFreeGroup

9.1-18 GuptaSidkiGroups

9.1-19 GuptaSidkiGroup

9.1-20 NeumannGroup

9.1-21 FabrykowskiGuptaGroup

9.1-22 ZugadiSpinalGroup

9.1-23 GammaPQMachine

9.1-24 RattaggiGroup

9.1-25 HanoiGroup

9.1-26 DahmaniGroup

9.1-27 MamaghaniGroup

9.1-28 WeierstrassGroup

9.1-29 StrichartzGroup

9.1-30 FRAffineGroup

9.1-31 CayleyGroup

**FR** predefines a large collection of machines and groups. The groups are, whenever possible, defined as state closures of corresponding Mealy machines.

`‣ FullBinaryGroup` | ( global variable ) |

`‣ FiniteDepthBinaryGroup` ( l ) | ( function ) |

`‣ FinitaryBinaryGroup` | ( global variable ) |

`‣ BoundedBinaryGroup` | ( global variable ) |

`‣ PolynomialGrowthBinaryGroup` | ( global variable ) |

`‣ FiniteStateBinaryGroup` | ( global variable ) |

These are the finitary, bounded, polynomial-growth, finite-state, or unrestricted groups acting on the binary tree. They are respectively shortcuts for `FullSCGroup([1..2])`

, `FullSCGroup([1..2],l)`

, `FullSCGroup([1..2],IsFinitaryFRSemigroup)`

, `FullSCGroup([1..2],IsBoundedFRSemigroup)`

, `FullSCGroup([1..2],IsPolynomialGrowthFRSemigroup)`

, and `FullSCGroup([1..2],IsFiniteStateFRSemigroup)`

.

They may be used to draw random elements, or to test membership.

`‣ BinaryKneadingGroup` ( angle/ks ) | ( function ) |

`‣ BinaryKneadingMachine` ( angle/ks ) | ( function ) |

Returns: The [machine generating the] iterated monodromy group of a quadratic polynomial.

The first function constructs a Mealy machine whose state closure is the binary kneading group.

The second function constructs a new FR group `G`

, which is the iterated monodromy group of a quadratic polynomial, either specified by its angle or by its kneading sequence(s).

If the argument is a (rational) angle, the attribute `Correspondence(G)`

is a function returning, for any rational, the corresponding generator of `G`

.

If there is one argument, which is a list or a string, it is treated as the kneading sequence of a periodic (superattracting) polynomial. The sequence is implicity assumed to end by '*'. The attribute `Correspondence(G)`

is a list of the generators of `G`

.

If there are two arguments, which are lists or strings, they are treated as the preperiod and period of the kneading sequence of a preperiodic polynomial. The last symbol of the arguments must differ. The attribute `Correspondence(G)`

is a pair of lists of generators; `Correspondence(G)[1]`

is the preperiod, and `Correspondence(G)[2]`

is the period. The attribute `KneadingSequence(G)`

returns the kneading sequence, as a pair of strings representing preperiod and period respectively.

As particular examples, `BinaryKneadingMachine()`

is the adding machine; `BinaryKneadingGroup()`

is the adding machine; and `BinaryKneadingGroup("1")`

is `BasilicaGroup`

(9.1-3).

gap> BinaryKneadingGroup()=AddingGroup(2); true gap> BinaryKneadingGroup(1/3)=BasilicaGroup; true gap> g := BinaryKneadingGroup([0,1],[0]); BinaryKneadingGroup("01","0") gap> ForAll(Correspondence(g)[1],IsFinitaryFRElement); true gap> ForAll(Correspondence(g)[2],IsFinitaryFRElement); false gap> ForAll(Correspondence(g)[2],IsBoundedFRElement); true

`‣ BasilicaGroup` | ( global variable ) |

The *Basilica group*. This is a shortcut for `BinaryKneadingGroup("1")`

. It is the first-discovered amenable group that is not subexponentially amenable, see [BV05] and [G{\.Z}02].

gap> IsBoundedFRSemigroup(BasilicaGroup); true gap> pi := EpimorphismFromFreeGroup(BasilicaGroup); F := Source(pi);; [ x1, x2 ] -> [ a, b ] gap> sigma := GroupHomomorphismByImages(F,F,[F.1,F.2],[F.2,F.1^2]); [ x1, x2 ] -> [ x2, x1^2 ] gap> ForAll([0..10],i->IsOne(Comm(F.2,F.2^F.1)^(sigma^i*pi))); true

`‣ FornaessSibonyGroup` | ( global variable ) |

The *Fornaess-Sibony group*. This group was studied by Nekrashevych in [Nek08b]. It is the iterated monodromy group of the endomorphism of CP^2 defined by (z,p)↦((1-2z/p)^2,(1-2/p)^2).

gap> Size(NucleusOfFRSemigroup(FornaessSibonyGroup)); 288 gap> List(AdjacencyBasesWithOne(FornaessSibonyGroup),Length); [ 128, 128, 36, 36, 16, 16, 8 ] gap> p := AdjacencyPoset(FornaessSibonyGroup); <general mapping: <object> -> <object> > gap> Draw(HasseDiagramBinaryRelation(p));

This produces (in a new window) the following picture:

`‣ AddingGroup` ( n ) | ( function ) |

`‣ AddingMachine` ( n ) | ( function ) |

`‣ AddingElement` ( n ) | ( function ) |

The second function constructs the adding machine on the alphabet `[1..n]`

. This machine has a trivial state 1, and a non-trivial state 2. It implements the operation "add 1 with carry" on sequences.

The third function constructs the Mealy element on the adding machine, with initial state 2.

The first function constructs the state-closed group generated by the adding machine on `[1..n]`

. This group is isomorphic to the `Integers`

.

gap> Display(AddingElement(3)); | 1 2 3 ---+-----+-----+-----+ a | a,1 a,2 a,3 b | a,2 a,3 b,1 ---+-----+-----+-----+ Initial state: b gap> ActivityPerm(FRElement(AddingMachine(3),2),2); (1,4,7,2,5,8,3,6,9) gap> G := AddingGroup(3); <self-similar group over [ 1 .. 3 ] with 1 generator> gap> Size(G); infinity gap> IsRecurrentFRSemigroup(G); true gap> IsLevelTransitiveFRGroup(G); true

`‣ BinaryAddingGroup` | ( global variable ) |

`‣ BinaryAddingMachine` | ( global variable ) |

`‣ BinaryAddingElement` | ( global variable ) |

These are respectively the same as `AddingGroup(2)`

, `AddingMachine(2)`

and `AddingElement(2)`

.

`‣ MixerGroup` ( A, B, f[, g] ) | ( function ) |

`‣ MixerMachine` ( A, B, f[, g] ) | ( function ) |

Returns: A Mealy "mixer" machine/group.

The second function constructs a Mealy "mixer" machine `m`

. This is a machine determined by a permutation group `A`, a finitely generated group `B`, and a matrix of homomorphisms from `B` to `A`. If `A` acts on `[1..d]`

, then each row of `f` contains at most d-1 homomorphisms. The optional last argument is an endomorphism of `B`. If absent, it is treated as the identity map on `B`.

The states of the machine are 1, followed by some elements `a`

of `A`, followed by as many copies of some elements `b`

of `B` as there are rows in `f`. The elements in `B` that are taken is the smallest sublist of `B` containing its generators and closed under `g`. The elements in `A` that are taken are the generators of `A` and all images of all taken elements of `B` under maps in `f`.

The transitions from `a`

are towards 1 and the outputs are the permutations in `A`. The transitions from `b`

are periodically given by `f`, completed by trivial elements, and finally by b^g; the output of `b`

is trivial.

This construction is described in more detail in [B{Š}01] and [BG{Š}03].

`Correspondence(m)`

is a list, with in first position the subset of the states that correspond to `A`, in second position the states that correspond to the first copy of `B`, etc.

The first function constructs the group generated by the mixer machine. For examples see `GrigorchukGroups`

(9.1-9), `NeumannGroup`

(9.1-20), `GuptaSidkiGroups`

(9.1-18), and `ZugadiSpinalGroup`

(9.1-22).

gap> grigorchukgroup := MixerGroup(Group((1,2)),Group((1,2)), [[IdentityMapping(Group((1,2)))],[IdentityMapping(Group((1,2)))],[]])); <self-similar group over [ 1 .. 2 ] with 4 generators> gap> IdGroup(Group(grigorchukgroup.1,grigorchukgroup.2)); [ 32, 18 ]

`‣ SunicGroup` ( phi ) | ( function ) |

`‣ SunicMachine` ( phi ) | ( function ) |

Returns: The Sunic machine associated with the polynomial `phi`.

A "Sunic machine" is a special kind of `MixerMachine`

(9.1-7), in which the group A is a finite field GF(q), the group B is an extension A[x]/(ϕ), where ϕ is a monic polynomial; there is a map f:B-> A, given say by evaluation; and there is an endomorphism of g:B-> B given by multiplication by ϕ.

These groups were described in [{Š}un07]. In particular, the case q=2 and ϕ=x^2+x+1 gives the original `GrigorchukGroup`

(9.1-10).

gap> x := Indeterminate(GF(2));; gap> g := SunicGroup(x^2+x+1); SunicGroup(x^2+x+Z(2)^0) gap> g = GrigorchukGroup; true

`‣ GrigorchukMachines` ( omega ) | ( function ) |

`‣ GrigorchukGroups` ( omega ) | ( function ) |

Returns: One of the Grigorchuk groups.

This function constructs the Grigorchuk machine or group associated with the infinite sequence `omega`, which is assumed (pre)periodic; `omega` is either a periodic list (see `PeriodicList`

(11.2-2)) or a plain list describing the period. Entries in the list are integers in `[1..3]`

.

These groups are `MixerGroup`

(9.1-7)s. The most famous example is `GrigorchukGroups([1,2,3])`

, which is also called `GrigorchukGroup`

(9.1-10).

These groups are all 4-generated and infinite. They are described in [Gri84]. `GrigorchukGroups([1])`

is infinite dihedral. If `omega` contains at least 2 different digits, `GrigorchukGroups([1])`

has intermediate word growth. If `omega` contains all 3 digits, `GrigorchukGroups([1])`

is torsion.

The growth of `GrigorchukGroups([1,2])`

has been studied in [Ers04].

gap> G := GrigorchukGroups([1]); GrigorchukGroups([ 1 ]) gap> Index(G,DerivedSubgroup(G)); IsAbelian(DerivedSubgroup(G)); 4 true gap> H := GrigorchukGroups([1,2]); GrigorchukGroups([ 1, 2 ]) gap> Order(H.1*H.2); 8 gap> Order(H.1*H.4); infinity gap> IsSubgroup(H,G); true

`‣ GrigorchukMachine` | ( global variable ) |

`‣ GrigorchukGroup` | ( global variable ) |

This is Grigorchuk's first group, introduced in [Gri80]. It is a 4-generated infinite torsion group, and has intermediate word growth. It could have been defined as `FRGroup("a=(1,2)","b=<a,c>","c=<a,d>","d=<,b>")`

, but is rather defined using Mealy elements.

The command `EpimorphismFromFpGroup(GrigorchukGroup,n)`

constructs an approximating presentation for the Grigorchuk group, as proven in [Lys85]. Adding the relations `Image(sigma^(n-2),(a*d)^2)`

, `Image(sigma^(n-1),(a*b)^2)`

and `Image(sigma^(n-2),(a*c)^4)`

yields the quotient acting on 2^n points, as a finitely presented group.

`‣ GrigorchukOverGroup` | ( global variable ) |

This is a group strictly containing the Grigorchuk group (see `GrigorchukGroup`

(9.1-10)). It also has intermediate growth (see [BG02], but it contains elements of infinite order. It could have been defined as `FRGroup("a=(1,2)","b=<a,c>","c=<,d>","d=<,b>")`

, but is rather defined using Mealy elements.

gap> IsSubgroup(GrigorchukOverGroup,GrigorchukGroup); true gap> Order(Product(GeneratorsOfGroup(GrigorchukOverGroup))); infinity gap> GrigorchukGroup.2=GrigorchukSuperGroup.2*GrigorchukSuperGroup.3; true

The command `EpimorphismFromFpGroup(GrigorchukOverGroup,n)`

will will construct an approximating presentation for the Grigorchuk overgroup, as proven in [Bar03a].

`‣ GrigorchukTwistedTwin` | ( global variable ) |

This is a group with same closure as the Grigorchuk group (see `GrigorchukGroup`

(9.1-10)), but not isomorphic to it. It could have been defined as `FRGroup("a=(1,2)","x=<y,a>","y=<a,z>","z=<,x>")`

, but is rather defined using Mealy elements.

gap> AbelianInvariants(GrigorchukTwistedTwin); [ 2, 2, 2, 2 ] gap> AbelianInvariants(GrigorchukGroup); [ 2, 2, 2 ] gap> PermGroup(GrigorchukGroup,8)=PermGroup(GrigorchukTwistedTwin,8); true

`‣ BrunnerSidkiVieiraGroup` | ( global variable ) |

`‣ BrunnerSidkiVieiraMachine` | ( global variable ) |

This machine is the sum of two adding machines, one in standard form and one conjugated by the element `d`

described below. The group that it generates is described in [BSV99]. It could have been defined as `FRGroup("tau=<,tau>(1,2)","mu=<,mu^-1>(1,2)")`

, but is rather defined using Mealy elements.

gap> V := FRGroup("d=<e,e>(1,2)","e=<d,d>"); <self-similar group over [ 1 .. 2 ] with 2 generators> gap> Elements(V); [ <2|identity ...>, <2|e>, <2|d>, <2|e*d> ] gap> AssignGeneratorVariables(BrunnerSidkiVieiraGroup); #I Assigned the global variables [ "tau", "mu", "" ] gap> List(V,x->tau^x)=[tau,mu,mu^-1,tau^-1]; true

`‣ AleshinGroups` ( l ) | ( function ) |

`‣ AleshinMachines` ( l ) | ( function ) |

Returns: The Aleshin machine with `Length(l)`

states.

This function constructs the bireversible machines introduced by Aleshin in [Ale83]. The argument `l` is a list of permutations, either `()`

or `(1,2)`

. The groups that they generate are contructed as `AleshinGroups`

.

If `l=[(1,2),(1,2),()]`

, this is `AleshinGroup`

(9.1-15). More generally, if `l=[(1,2,(1,2),(),...,()]`

has odd length, this is a free group of rank `Length(l)`

, see [VV07] and [VV06].

If `l=[(1,2),(1,2),...]`

has even length and contains an even number of `()`

's, then this is also a free group of rank `Length(l)`

, see [SVV06].

If `l=[(),(),(1,2)]`

, this is `BabyAleshinGroup`

(9.1-16). more generally, if `l=[(),(),...]`

has even length and contains an even number of `(1,2)`

's, then this is the free product of `Length(l)`

copies of the order-2 group.

`‣ AleshinGroup` | ( global variable ) |

`‣ AleshinMachine` | ( global variable ) |

This is the first example of non-abelian free group. It is the group generated by `AleshinMachine([(1,2),(1,2),()])`

. It could have been defined as `FRGroup("a=<b,c>(1,2)","b=<c,b>(1,2)","c=<a,a>")`

, but is rather defined using Mealy elements.

`‣ BabyAleshinGroup` | ( global variable ) |

`‣ BabyAleshinMachine` | ( global variable ) |

There are only two connected, transitive bireversible machines on a 2-letter alphabet, with 3 generators: `AleshinMachine(3)`

and the baby Aleshin machine.

The group generated by this machine is abstractly the free product of three C_2's, see [Nek05, 1.10.3]. It could have been defined as `FRGroup("a=<b,c>","b=<c,b>","c=<a,a>(1,2)")`

, but is rather defined here using Mealy elements.

gap> K := Kernel(GroupHomomorphismByImagesNC(BabyAleshinGroup,Group((1,2)), GeneratorsOfGroup(BabyAleshinGroup),[(1,2),(1,2),(1,2)])); <self-similar group over [ 1 .. 2 ] with 4 generators> gap> Index(BabyAleshinGroup,K); 2 gap> IsSubgroup(AleshinGroup,K); true

`‣ SidkiFreeGroup` | ( global variable ) |

This is a group suggested by Sidki as an example of a non-abelian state-closed free group. Nothing is known about that group: whether it is free as conjectured; whether generator `a` is state-closed, etc. It is defined as `FRGroup("a=<a^2,a^t>","t=<,t>(1,2)")]]>`

.

`‣ GuptaSidkiGroups` ( n ) | ( function ) |

`‣ GeneralizedGuptaSidkiGroups` ( n ) | ( function ) |

`‣ GuptaSidkiMachines` ( n ) | ( function ) |

Returns: The Gupta-Sidki group/machine on an `n`-letter alphabet.

This function constructs the machines introduced by Gupta and Sidki in [GS83]. They generate finitely generated infinite torsion groups: the exponent of every element divides some power of `n`. The special case n=3 is defined as `GuptaSidkiGroup`

(9.1-19) and `GuptaSidkiMachine`

(9.1-19).

For n>3, there are (at least) two natural ways to generalize the Gupta-Sidki construction: the original one involves the recursion `"t=<a,a^-1,1,...,1,t>"`

, while the second (called here `generalized') involves the recursion `"t=<a,a^2,...,a^-1,t>"`

. A finite L-presentation for the latter is implemented. It is not as computationally efficient as the L-presentation of the normal closure of `t`

(a subgroup of index p), which is an ascending L-presented group. The inclusion of that subgroup may be recoverd via `EmbeddingOfAscendingSubgroup(GuptaSidkiGroup)`

.

`‣ GuptaSidkiGroup` | ( global variable ) |

`‣ GuptaSidkiMachine` | ( global variable ) |

This is an infinite, 2-generated, torsion 3-group. It could have been defined as `FRGroup("a=(1,2,3)","t=<a,a^-1,t>")`

, but is rather defined using Mealy elements.

`‣ NeumannGroup` ( P ) | ( function ) |

`‣ NeumannMachine` ( P ) | ( function ) |

Returns: The Neumann group/machine on the permutation group `P`.

The first function constructs the Neumann group associated with the permutation group `P`. These groups were first considered in [Neu86]. In particular, `NeumannGroup(PSL(3,2))`

is a group of non-uniform exponential growth (see [Bar03b]), and `NeumannGroup(Group((1,2,3)))`

is `FabrykowskiGuptaGroup`

(9.1-21).

The second function constructs the Neumann machine associated to the permutation group `P`. It is a shortcut for `MixerMachine(P,P,[[IdentityMapping(P)]])`

.

The attribute `Correspondence(G)`

is set to a list of homomorphisms from `P` to the `A`

and `B`

copies of `P`

.

`‣ FabrykowskiGuptaGroup` | ( global variable ) |

`‣ FabrykowskiGuptaGroups` ( p ) | ( function ) |

This is an infinite, 2-generated group of intermediate word growth. It was studied in [FG85] and [FG91]. It could have been defined as `FRGroup("a=(1,2,3)","t=<a,,t>")`

, but is rather defined using Mealy elements. It has a torsion-free subgroup of index 3 and is branched.

The natural generalization, replacing 3 by p, is constructed by the second form. It is a specific case of Neumann group (see `NeumannGroup`

(9.1-20)), for which an ascending L-presentation is known.

`‣ ZugadiSpinalGroup` | ( global variable ) |

This is an infinite, 2-generated group, which was studied in [BG02]. It has a torsion-free subgroup of index 3, and is virtually branched but not branched. It could have been defined as `FRGroup("a=(1,2,3)","t=<a,a,t>")`

, but is rather defined using Mealy elements.

Amaia Zugadi computed its Hausdorff dimension to be 1/2.

`‣ GammaPQMachine` ( p, q ) | ( function ) |

`‣ GammaPQGroup` ( p, q ) | ( function ) |

Returns: The quaternion-based machine / SC group.

The first function constructs a bireversible (see `IsBireversible`

(5.2-7)) Mealy machine based on quaternions, see [BM00a] and [BM00b]. This machine has p+1 states indexed by integer quaternions of norm `p`, and an alphabet of size q+1 indexed by quaternions of norm `q`. These quaternions are congruent to 1mod 2 or imod 2 depending on whether the odd prime p or q is 1 or 3mod 4.

The structure of the machine is such that there is a transition from (q,a) to (q',a') precisely when qa'=± q'a. In particular, the relations of the `StructuralGroup`

(3.5-1) hold up to a sign, when the alphabet/state letters are substituted for the appropriate quaternions.

The quaternions themselves can be recovered through `Correspondence`

(3.5-12), which is a list with in first position the quaternions of norm p and in second those of norm q.

The second function constructs the quaternion lattice that is the `StructuralGroup`

(3.5-1) of that machine. It has actions on two trees, given by `VerticalAction`

(9.5-2) and `HorizontalAction`

(9.5-2); the ranges of these actions are groups generated by automata, which are infinitely-transitive (see `IsInfinitelyTransitive`

(7.2-13)) and free by [GM05, Proposition 3.3]; see also [Ale83].

`‣ RattaggiGroup` | ( global variable ) |

This record contains interesting examples of VH groups, that were studied by Rattaggi in his PhD thesis [Rat04]. His Example 2.x appears as `RattaggiGroup.2_x`

.

The following summary of the examples' properties are copied from Rattaggi's thesis. RF means "residually finite"; JI means "just infinite"; VS means "virtually simple".

Example | P_h | P_v | Irred? | Linear? | RF? | JI? | VS? |

2.2 | 2tr | 2tr | Y | N | N? | Y | Y? |

2.15 | |||||||

2.18 | |||||||

2.21 | |||||||

2.26 | 2tr | q-prim | Y | N | N | N | N |

2.30 | 2tr | 2tr | Y | N | N | Y | Y? |

2.36 | |||||||

2.39 | |||||||

2.43 | 2tr | 2tr | Y | N | N | Y | Y |

2.46 | |||||||

2.48 | |||||||

2.50 | |||||||

2.52 | tr | 2tr | Y | N | N | N | N |

2.56 | |||||||

2.58 | 2tr | prim | Y | N | N? | N | N |

2.70 | |||||||

3.26 | 2tr | 2tr | Y | Y | Y | Y | N |

3.28 | |||||||

3.31 | |||||||

3.33 | |||||||

3.36 | |||||||

3.38 | |||||||

3.40 | |||||||

3.44 | |||||||

3.46 | |||||||

3.72 |

`‣ HanoiGroup` ( n ) | ( function ) |

Returns: The Hanoi group on an `n` pegs.

This function constructs the Hanoi group on `n` pegs. Generators of the group correspond to moving a peg, and tree vertices correspond to peg configurations. This group is studied in [G{Š}06].

`‣ DahmaniGroup` | ( global variable ) |

This is an example of a non-contracting weakly branched group. It was first studied in [Dah05]. It could have been defined as `FRGroup("a=<c,a>(1,2)","b=<b,a>(1,2)","c=<b,c>")`

, but is rather defined using Mealy elements.

It has relators abc, [a^2c,[a,c]], [cab,a^-1c^-1ab] and [ac^2,c^-1b^-1c^2] among others.

It admits an endomorphism on its derived subgroup. Indeed `FRElement(1,Comm(a,b))=Comm(c^-1,b/a)`

, `FRElement(1,Comm(a,c))=Comm(a/b,c)`

, `FRElement(1,Comm(b,c))=Comm(c,(a/b)^c)`

.

`‣ MamaghaniGroup` | ( global variable ) |

This group was studied in [Mam03]. It is fractal, but not contracting. It could have been defined as `FRGroup("a=<,b>(1,2)","b=<a,c>","c=<a,a^-1>(1,2)")]]>`

, but is rather defined using Mealy elements. It partially admits branching on its subgroup `Subgroup(G,[a^2,(a^2)^b,(a^2)^c])`

, and, setting `x=Comm(a^2,b)`

, on `Subgroup(G,[x,x^a,x^b,x^(b*a),x^(b/a)])`

. One has `FRElement(1,x)=(x^-1)^b/x`

.

`‣ WeierstrassGroup` | ( global variable ) |

This is the iterated monodromy group associated with the Weierstrass ℘-function.

Some relators in the group: (atbt)^4, ((atbt)(bt)^4n)^4, ((atbt)^2(bt)^4n)^2.

Set x=[a,t], y=[b,t], z=[c,t], and w=[x,y]. Then G'=⟨ x,y,z⟩ of index 8, and γ_3=⟨[{x,y,z},{a,b,c}]⟩ of index 32, and γ_4=G''=⟨ w⟩^G, of index 256, and G''>(G'')^4 since [[t^a,t],[t^b,t]]=(w,1,1,1).

The Schreier graph is obtained in the complex plane as the image of a 2^n× 2^n lattice in the torus, via Weierstrass's ℘-function.

The element at has infinite order.

[c,t,b][b,t,c][a,t,c][c,t,a] has order 2, and belongs to G''; so there exist elements of arbitrary large finite order in the group.

`‣ StrichartzGroup` | ( global variable ) |

This group generates the graph of the Strichartz hexacarpet.

The Strichartz hexacarpet is the dual graph to the infinitely iterated barycentric subdivision of the triangle. The Strichartz group acts on {1,dots,6}^n for all n, and the Schreier graph with 6^n vertices is the nth Strichartz graph.

Conjecturally, that graph's radius is 1/18(2^(n+1)(13+3n)+(-1)^n-9) and its diameter is 1/9(2^(n-1)(31+12n)+2(-1)^(n-1)-18).

See [BKN+12] for details.

`‣ FRAffineGroup` ( d, R, u[, transversal] ) | ( operation ) |

Returns: The `d`-dimensional affine group over `R`.

This function constructs a new FR group `G`

, which is finite-index subgroup of the `d`-dimensional affine group over R_u, the local ring over `R` in which all non-multiples of `u` are invertible. Since no generators of `G`

are known, `G`

is in fact returned as a full SC group; only its attribute `Correspondence(G)`

, which is a homomorphism from GL_d+1(R_u) to `G`

, is relevant.

The affine group can also be described as a subgroup of GL_d+1(R_u), consisting of those matrices M with M_i,d+1=0 and M_d+1,d+1=1. The finite-index subgroup is defined by the conditions u|M_i,j for all j<i.

The only valid arguments are `R=Integers`

and `R=PolynomialRing(S)`

for a finite ring `S`

. The alphabet of the affine group is R/RuR; an explicit transversal of RuR be specified as last argument.

The following examples construct the "Baumslag-Solitar group" Z[frac12]⋊_2 Z first introduced in [BS62], the "lamplighter group" ( Z/2)≀ Z, and a 2-dimensional affine group. Note that the lamplighter group may also be defined via `CayleyGroup`

(9.1-31).

gap> A := FRAffineGroup(1,Integers,3); <self-similar group over [ 1 .. 3 ]> gap> f := Correspondence(A); MappingByFunction( ( Integers^ [ 2, 2 ] ), <self-similar group over [ 1 .. 3 ]>, function( mat ) ... end ) gap> BaumslagSolitar := Group([[2,0],[0,1]]^f,[[1,0],[1,1]]^f); <self-similar group over [ 1 .. 3 ] with 2 generators> gap> BaumslagSolitar.2^BaumslagSolitar.1=BaumslagSolitar.2^2; true gap> R := PolynomialRing(GF(2));; gap> A := FRAffineGroup(1,R,R.1);; gap> f := Correspondence(A);; gap> Lamplighter := Group(([[1+R.1,0],[0,1]]*One(R))^f,([[1,0],[1,1]]*One(R))^f); <self-similar group over [ 1 .. 2 ] with 2 generators> gap> Lamplighter = CayleyGroup(Group((1,2))); true gap> StructureDescription(Group(Lamplighter.2,Lamplighter.2^Lamplighter.1)); "C2 x C2" gap> ForAll([1..10],i->IsOne(Comm(Lamplighter.2,Lamplighter.2^(Lamplighter.1^i)))); true gap> A := FRAffineGroup(2,Integers,2);; gap> f := Correspondence(A);; gap> a := [[1,4,0],[2,3,0],[1,0,1]]; [ [ 1, 4, 0 ], [ 2, 3, 0 ], [ 1, 0, 1 ] ] gap> b := [[1,2,0],[4,3,0],[0,1,1]]; [ [ 1, 2, 0 ], [ 4, 3, 0 ], [ 0, 1, 1 ] ] gap> Display(b^f); | 1 2 ----+------+------+ a | b,1 c,2 b | d,2 e,1 c | a,2 f,1 ... bh | cb,1 be,2 ca | bd,1 bf,2 cb | ae,2 bh,1 ----+------+------+ Initial state: a gap> a^f*b^f=(a*b)^f; true

`‣ CayleyGroup` ( G ) | ( function ) |

`‣ CayleyMachine` ( G ) | ( function ) |

`‣ LamplighterGroup` ( IsFRGroup, G ) | ( method ) |

Returns: The Cayley machine/group of the group `G`.

The *Cayley machine* of a group `G` is a machine with alphabet and stateset equal to `G`, and with output and transition functions given by multiplication in the group, in the order `state*letter`

.

The second function constructs a new FR group `CG`

, which acts on `[1..Size(G)]`

. Its generators are in bijection with the elements of `G`, and have as output the corresponding permutation of `G` induced by right multiplication, and as transitions all elements of `G`; see `CayleyMachine`

. This construction was introduced in [SS05].

If `G` is an abelian group, then `CG`

is the wreath product G≀ Z; it is created by the constructor `LamplighterGroup(IsFRGroup,G)`

.

The attribute `Correspondence(CG)`

is a list. Its first entry is a homomorphism from `G` into `CG`

. Its second entry is the generator of `CG`

that has trivial output. `CG`

is generated `Correspondence(CG)[2]`

and the image of `Correspondence(CG)[1]`

.

In the example below, recall the definition of `Lamplighter`

in the example of `FRAffineGroup`

(9.1-30).

gap> L := CayleyGroup(Group((1,2))); CayleyGroup(Group( [ (1,2) ] )) gap> L=LamplighterGroup(IsFRGroup,CyclicGroup(2)); true gap> (1,2)^Correspondence(L)[1]; <Mealy element on alphabet [ 1, 2 ] with 2 states, initial state 1> gap> IsFinitaryFRElement(last); Display(last2); true | 1 2 ---+-----+-----+ a | b,2 b,1 b | b,1 b,2 ---+-----+-----+ Initial state: a

`‣ I2Machine` | ( global variable ) |

`‣ I2Monoid` | ( global variable ) |

The Mealy machine I_2, and the monoid that it generates. This is the smallest Mealy machine generating a monoid of intermediate word growth; see [BRS06].

For sample calculations in this monoid see `SCSemigroup`

(7.1-3).

`‣ I4Machine` | ( global variable ) |

`‣ I4Monoid` | ( global variable ) |

The Mealy machine generating I_4, and the monoid that it generates. This is a very small Mealy machine generating a monoid of intermediate word growth; see [BR08].

For sample calculations in this monoid see `SCMonoid`

(7.1-3).

`‣ PSZAlgebra` ( k[, m] ) | ( function ) |

This function creates an associative algebra `A`

, over the field `k` of positive characteristic, generated by `m` derivations `d0,...,d(m-1),v`

. If the argument `m` is absent, it is taken to be `2`

.

This algebra has polynomial growth, and is not nilpotent. Petrogradsky showed in [Pet06] that the Lie subalgebra of `PSZAlgebra(GF(2))`

generated by v,[u,v] is nil; this result was generalized by Shestakov and Zelmanov in [SZ08] to arbitrary `k` and m=2.

This ring is Z^m-graded; the attribute `Grading`

is set. It is graded nil [PSZ].

gap> a := PSZAlgebra(2); PSZAlgebra(GF(2)) gap> Nillity(a.1); Nillity(a.2); 2 4 gap> IsNilElement(LieBracket(a.1,a.2)); true gap> DecompositionOfFRElement(LieBracket(a.1,a.2))=DiagonalMat([a.2,a.2]); true

`‣ GrigorchukThinnedAlgebra` ( k ) | ( function ) |

This function creates the associative envelope `A`

, over the field `k`, of Grigorchuk's group `GrigorchukGroup`

(9.1-10).

`k` may be a field or an prime representing the prime field `GF(k)`

. In characteristic 2, this ring is graded, and the attribute `Grading`

is set.

For more information on the structure of this thinned algebra, see [Bar06].

gap> R := GrigorchukThinnedAlgebra(2); <self-similar algebra-with-one on alphabet GF(2)^2 with 4 generators, of dimension infinity> gap> GrigorchukGroup.1^Embedding(GrigorchukGroup,R)=R.1; true gap> Nillity(R.2+R.1); 16 gap> x := 1+R.1+R.2+(R.1-1)*(R.4-1); # x has infinite order <Linear element on alphabet GF(2)^2 with 5-dimensional stateset> gap> Inverse(x); <Linear element on alphabet GF(2)^2 with 9-dimensional stateset> gap> Grading(R).hom_components(4); <vector space of dimension 6 over GF(2)> gap> Random(last); <Linear element on alphabet GF(2)^2 with 6-dimensional stateset> gap> Nillity(last); 4

`‣ GuptaSidkiThinnedAlgebra` ( k ) | ( function ) |

This function creates the associative envelope `A`

, over the field `k`, of Gupta-Sidki's group `GuptaSidkiGroup`

(9.1-19).

`k` may be a field or an prime representing the prime field `GF(k)`

.

For more information on the structure of this thinned algebra, see [Sid97].

gap> R := GuptaSidkiThinnedAlgebra(3); <self-similar algebra-with-one on alphabet GF(3)^3 with 4 generators> gap> Order(R.1); 3 gap> R.1*R.3; <Identity linear element on alphabet GF(3)^3> gap> IsOne(R.2*R.4); true gap> x := 1+R.2*(1+R.1+R.3); # a non-invertible element <Linear element on alphabet GF(3)^3 with 5-dimensional stateset> gap> Inverse(x); #I InverseOp: extending to depth 3 #I InverseOp: extending to depth 4 #I InverseOp: extending to depth 5 #I InverseOp: extending to depth 6 Error, user interrupt in

`‣ GrigorchukLieAlgebra` ( k ) | ( function ) |

`‣ GuptaSidkiLieAlgebra` ( k ) | ( function ) |

Two more examples of self-similar Lie algebras; see [Bar10].

`‣ SidkiFreeAlgebra` ( k ) | ( function ) |

This is an example of a free 2-generated associative ring over the Z, defined by self-similar matrices. It is due to S. Sidki. The argument is a field on which to construct the algebra. The recursion is `s=[[1,0],[0,2*s]]`

and `t=[[0,2*s],[0,2*t]]`

.

gap> R := SidkiFreeAlgebra(Rationals); <self-similar algebra-with-one on alphabet Rationals^2 with 2 generators> gap> V := VectorSpace(Rationals,[R.1,R.2]); <vector space over Rationals, with 2 generators> gap> P := [V];; for i in [1..3] do Add(P,ProductSpace(P[i],V)); od; gap> List(P,Dimension); [ 2, 4, 8, 16 ] gap> R := SidkiFreeAlgebra(GF(3)); <self-similar algebra-with-one on alphabet GF(3)^2 with 2 generators> gap> V := VectorSpace(GF(3),[R.1,R.2]);; gap> P := [V];; for i in [1..3] do Add(P,ProductSpace(P[i],V)); od; gap> List(P,Dimension); [ 2, 4, 7, 12 ]

`‣ SidkiMonomialAlgebra` ( k ) | ( function ) |

This is an example of a self-similar algebra that does not come from a semigroup; it is due to S. Sidki. The argument is a field on which to construct the algebra. The recursion is `s=[[0,0],[1,0]]`

and `t=[[0,t],[0,s]]`

. Sidki shows that this algebra, like the Grigorchuk thinned algebra in characteristic 2 (see `GrigorchukThinnedAlgebra`

(9.3-2)), admits a monomial presentation, and in particular is a graded ring.

gap> R := SidkiMonomialAlgebra(Rationals); <self-similar algebra-with-one on alphabet Rationals^2 with 2 generators> gap> m := FreeSemigroup("s","t");; gap> lambda := MagmaEndomorphismByImagesNC(m,[m.2,m.1*m.2]);; gap> u := [m.1^2];; for i in [1..3] do u[2*i] := m.2*u[2*i-1]^lambda; u[2*i+1] := u[2*i]^lambda; od; gap> u; # first relations [ s^2, t^3, s*t*s*t*s*t, t^2*s*t^2*s*t^2*s*t, s*t*s*t^2*s*t*s*t^2*s*t*s*t^2*s*t, t^2*s*t^2*s*t*s*t^2*s*t^2*s*t*s*t^2*s*t^2*s*t*s*t^2*s*t, s*t*s*t^2*s*t*s*t^2*s*t^2*s*t*s*t^2*s*t*s*t^2*s*t^2*s*t*s*t^2*s*t*s*t^2*s*t^2*s*t*s*t^2*s*t ] gap> pi := MagmaHomomorphismByImagesNC(m,R,[R.1,R.2]);; gap> Image(pi,u); [ <Zero linear element on alphabet Rationals^2> ] gap> # growth given by fibonacci numbers gap> List([0..6],Grading(R).hom_components); [ <vector space over Rationals, with 1 generators>, <vector space over Rationals, with 2 generators>, <vector space of dimension 3 over Rationals>, <vector space of dimension 4 over Rationals>, <vector space of dimension 5 over Rationals>, <vector space of dimension 7 over Rationals>, <vector space of dimension 8 over Rationals> ]

In his paper [Bac08], Roland Bacher exhibits striking formulas for determinants of matrices obtained from binomial coefficients.

The general construction is as follows: let P be an infinite matrix, and let P(n) be its upper n× n corner. To evaluate det P(n), decompose P=LDR where L,D,R are respectively lower triangular, diagonal, and upper triangular, with 1's on the diagonals of L and R. Then that determinant is the product of the first n entries of D.

Bacher considers some natural examples of matrices arising from binomial coefficients, and notes that the matrix P is the limit of a convergent vector element (see `IsConvergent`

(6.1-9)). He also notes that the decomposition P=LDR may be achieved within vector elements.

As a first example, consider the n× n matrix P(n) with coefficients P_s,t=s+t choose smod 2. Here and below, indices start at 0. Let also ds(n) denote the digit-sum of the integer n. Then

\det(P(n))=\cases{ (-1)^{n/2} & if $n$ is even,\cr (-1)^{(n-1)/2+ds((n-1)/2)} & if $n$ is odd.}

For the proof, note that P is a convergent infinite matrix; it may be presented as a self-similar linear element by `FRAlgebra("P=[[P,P],[P,0]]")`

. It then suffices to construct an LR decomposition of P within FR vector elements, following Bacher:

gap> AssignGeneratorVariables(FRAlgebra(Rationals, "P=[[P,P],[P,0]]","L=[[L,0],[L,L]]","D=[[D,0],[0,-D]]")); gap> L*D*TransposedFRElement(L)=P; true

and to analyse the elements of the diagonal matrix D.

For a more complicated example, let v_2 denote 2-valuation of a rational, and construct the n× n matrix V(n) with coefficients V_s,t=i^v_2(s+t choose s). Then

\det(V(n))=\det(P(n))\prod_{k=1}^{n-1}(1-f(k)i),

where f(k) is the regular paper-folding sequence defined by f(2^n)=1 and f(2^n+a)=-f(2^n-a) for 1le a<2^n.

This is again proved by noticing that V is a corner in a self-similar element, namely

gap> AssignGeneratorVariables(FRAlgebra(GaussianRationals, "V1=[[V1,V2],[V2,E(4)*V1]]", "V2=[[V1,-E(4)*V1+(1+E(4))*V2],[-E(4)*V1+(1+E(4))*V2,-V1]]")); gap> Activity(V1,3)= List([0..7],s->List([0..7],t->E(4)^ValuationInt(Binomial(s+t,s),2))); true

The LR decomposition of V=V1 can be checked as follows:

gap> AssignGeneratorVariables(FRAlgebra(GaussianRationals, "L1=[[L1,0],[L3,L4]]", "L2=[[0,-E(4)*L2],[-L1+L3,-E(4)*L2-E(4)*L4]]:0", "L3=[[L1,L2],[-E(4)*L1+(1+E(4))*L3,L2+(1+E(4))*L4]]", "L4=[[L1,0],[(1-E(4))*L1+E(4)*L3,L4]]", "D1=[[D1,0],[0,D2]]", "D2=[[D3,0],[0,2*D1-D2+2*D3]]:-1+E(4)", "D3=[[D3,0],[0,-D2]]:-1+E(4)")); gap> L1*D1*TransposedFRElement(L1)=V1; true

The LR decomposition can also, in favourable situations, be discovered by **FR** through the command `LDUDecompositionFRElement`

(6.1-11). This approach will be followed below.

For the next example, consider "Beeblebrox reduction" β(4k±1)=±1,β(2k)=0, and construct the n× n matrix Z(n) (named after Zaphod Beeblebrox) with coefficients Z_s,t=β(s+t choose s). Then

\det(Z(n))=\prod_{k=1}^{n-1}g(k),

where g(∑ a_i2^i)=(-1)^a_03^#{i:a_i=a_i+1=1}-#{i:a_i≠ a_i+1=1} with all a_i∈{0,1}.

This is again proved by noticing that Z is a corner in a self-similar element, namely

gap> beta := n->Jacobi(-1,n)*(n mod 2);; gap> Zaphod := GuessVectorElement(List([0..7],i->List([0..7],j->beta(Binomial(i+j,j))))); <Linear element on alphabet Rationals^2 with 3-dimensional stateset> gap> Display(Zaphod); Rationals | 1 | 2 | -----------+----------+----------+ 1 | 1 0 0 | 0 1 0 | | 1 0 0 | 0 1 0 | | 1 0 0 | 0 -1 0 | -----------+----------+----------+ 2 | 0 0 1 | 0 0 0 | | 0 0 -1 | 0 0 0 | | 0 0 1 | 0 0 0 | -----------+----------+----------+ Output: 1 1 1 Initial state: 1 0 0 gap> LDUDecompositionFRElement(guessZ); [ <Linear element on alphabet Rationals^2 with 4-dimensional stateset>, <Linear element on alphabet Rationals^2 with 2-dimensional stateset>, <Linear element on alphabet Rationals^2 with 4-dimensional stateset> ] gap> Display(last[2]); Rationals | 1 | 2 | -----------+---------+---------+ 1 | 1 0 | 0 0 | | 3 0 | 0 0 | -----------+---------+---------+ 2 | 0 0 | 0 1 | | 0 0 | 0 1/3 | -----------+---------+---------+ Output: 1 -1 Initial state: 1 0

and now the recursion read on this diagonal self-similar matrix gives immediately Bacher's recursion for det(Z(n)).

Bacher notes that the group generated by a=L_1,b=L_2/2,c=L_3,d=L_4 in the last example may be of interest. A quick check produces the following relations (slightly rewritten):

gap> AssignGeneratorVariables(FRAlgebra(Rationals, "a=[[a,0],[c,d]]","b=[[-1/3*a,2*b],[1/3*c,d]]", "c=[[a,2*b],[c,d]]","d=[[a,0],[1/3*c,d]]")); gap> g := Group(List([a,b,c,d], x->Activity(x,3))); <matrix group with 4 generators> gap> FindShortGroupRelations(g,10); [ b*d^-1*c*a^-1, c*a^-1*c*a^-1, c*a*d^-1*a^-1*d^2*a^-1*b^-1, c*a*d^-1*c^-1*b*d*a^-1*b^-1, c*d*a^-2*d*a*d^-1*b^-1, c*a^2*d^-1*a^-2*d*a*d*a^-2*b^-1, d^2*a*d^-2*b^-1*c*a*d*a^-3, c*d*a*d^-2*a^-1*d*a*d*a^-2*b^-1 ]

Consider next the "triangular Beeblebrox matrix" with entries L_s,t=β(s choose t). The recurrence is now given by

gap> A := FRAlgebra(Rationals, "L1=[[L1,0],[L2,L3]]", "L2=[[L1,0],[L2,-L3]]", "L3=[[L1,0],[-L2,L3]]"); <self-similar algebra on alphabet Rationals^2 with 3 generators>

and it is striking that A is a graded algebra, with L_1,L_2,L_3 homogeneous of degree 1, and each homogeneous component is 3-dimensional; all of L_1,L_2,L_3 are invertible (with inverses have degree -1), and generate a group that admits a faithful 3× 3 linear representation. As a final example, Bacher considers the "Jacobi character" χ(8ℤ±1)=1,χ(8ℤ±3)=-1,χ(2ℤ)=0, and the associated matrix J_s,t=χ(s+t choose s). He gives an easily-computed, but complicated formula for det(J(n)). We can recover this formula, as before, by "guessing" an LR decomposition for J, which is self-similar and convergent:

gap> chi := function(x) if x mod 8 in [1,7] then return 1; elif x mod 8 in [3,5] then return -1; else return 0; fi; end;; gap> m := List([0..63],i->List([0..63],j->chi(Binomial(i+j,j))));; gap> J := GuessVectorElement(m,2); <Linear element on alphabet Rationals^2 with 9-dimensional stateset> gap> LDUDecompositionFRElement(J); [ <Linear element on alphabet Rationals^2 with 20-dimensional stateset>, <Linear element on alphabet Rationals^2 with 4-dimensional stateset>, <Linear element on alphabet Rationals^2 with 20-dimensional stateset> ] gap> time; 26869 gap> Display(last2[2]); Rationals | 1 | 2 | -----------+-----------------+-----------------+ 1 | 1 0 0 0 | 0 0 0 0 | | 0 0 1 0 | 0 0 0 0 | | 3 0 0 0 | 0 0 0 0 | | 0 0 3 0 | 0 0 0 0 | -----------+-----------------+-----------------+ 2 | 0 0 0 0 | 0 1 0 0 | | 0 0 0 0 | 0 0 0 1 | | 0 0 0 0 | 0 1/3 0 0 | | 0 0 0 0 | 0 0 0 1/3 | -----------+-----------------+-----------------+ Output: 1 -1 3 -1/3 Initial state: 1 0 0 0

**FR** understands a special kind of finitely presented groups, called *VH groups*. These are groups with two distinguished sets of generators, V and H, and such that for every choice of v∈ V,h∈ H there are unique v'∈ V,h'∈ H such that vh=h'v' and conversely. In other words, these are finitely presented groups whose Cayley complex is a product of two trees.

These groups are of particular interest thanks to the work of Burger and Mozes, see [BM00a] and [BM00b], who constructed the first examples of finitely presented simple groups in this manner.

VH groups are connected to groups generated by automata as follows. Given a VH group, consider the automaton with stateset V, acting on alphabet H; its output and transition are determined by Φ(v,h)=(h',v') where v',h' are determined by the equation vh=h'v'.

Conversely, any bireversible automaton gives rise to a VH group by the inverse construction.

**FR** contains commands that automatize the verification that a VH group is non-residually finite, or virtually simple. Inspiration came from Diego Rattaggi's PhD thesis [Rat04].

`‣ VHStructure` ( g ) | ( operation ) |

`‣ IsVHGroup` ( g ) | ( filter ) |

Returns: A VH-structure for the group `g`.

A *VH-structure* on a group `g` is a partition of the generators in two sets V,H such that every relator of `g` is of the form vhv'h', and such that for all v∈ V,h∈ H there exist unique v'∈ V,h'∈ H such that vhv'h'=1.

The VH structure is stored as a record with fields `v,h`

containing lists of generators, and integer matrices `transitions,output`

such that `transitions[v][h']=v'`

and `output[v][h']=h`

.

The filter recognizes groups with a VH structure.

`‣ VerticalAction` ( g ) | ( attribute ) |

`‣ HorizontalAction` ( g ) | ( attribute ) |

Returns: A homomorphism to an FR group.

A group with VH structure admits a *vertical action* of its subgroup ⟨ V⟩; this is the group generated by the automaton `MealyMachine(trans,out)`

. The function returns the group homomorphism from the subgroup ⟨ V⟩ to that FR group.

The horizontal action is that of the dual automaton (see `DualMachine`

(5.2-3)).

gap> v := VerticalAction(RattaggiGroup.2_21); [ a1, a2, a3 ] -> [ <Mealy element on alphabet [ 1 .. 8 ] with 6 states>, <Mealy element on alphabet [ 1 .. 8 ] with 6 states>, <Mealy element on alphabet [ 1 .. 8 ] with 6 states> ] gap> RattaggiGroup.2_21.1^v; <Mealy element on alphabet [ 1 .. 8 ] with 6 states> gap> Range(v); <state-closed group over [ 1, 2, 3, 4, 5, 6, 7, 8 ] with 3 generators> gap> PermGroup(last,1); Group([ (3,4)(5,6), (1,7,8,2)(3,4,6,5), (1,7,5,3)(2,8,6,4) ]) gap> DisplayCompositionSeries(last); G (3 gens, size 1344) | A(1,7) = L(2,7) ~ B(1,7) = O(3,7) ~ C(1,7) = S(2,7) ~ 2A(1,7) = U(2,7) ~ A(2,2) = L(3,2) S (3 gens, size 8) | Z(2) S (2 gens, size 4) | Z(2) S (1 gens, size 2) | Z(2) 1 (0 gens, size 1)

`‣ VHGroup` ( l1, l2, ... ) | ( function ) |

Returns: A new VH group.

This function constructs the VH group specified by the squares `l1, l2, ...`. Each `li` is a list of length 4, of the form `[v,h,v',h']`

. Here the entries are indices of vertical, respectively horizontal generators, if positive; and their inverses if negative.

gap> # the Baby-Aleshin group gap> g := VHGroup([[1,1,-2,-1],[1,2,-3,-2],[2,1,-3,-1], [2,2,-2,-2],[3,1,-1,-2],[3,2,-1,-1]]); <VH group on the generators [ a1, a2, a3 | b1, b2 ]> gap> Display(g); generators = [ a1, a2, a3, b1, b2 ] ## relators = [ a1*b1*a2^-1*b1^-1, a1*b2*a3^-1*b2^-1, a2*b1*a3^-1*b1^-1, a2*b2*a2^-1*b2^-1, a3*b1*a1^-1*b2^-1, a3*b2*a1^-1*b1^-1 ]

`‣ IsIrreducibleVHGroup` ( g ) | ( property ) |

Returns: Whether `g` is an irreducible lattice.

A VH group is *irreducible* if its projections on both trees is dense.

gap> Display(RattaggiGroup.2_21); generators = [ a1, a2, a3, b1, b2, b3, b4 ] relators = [ a1*b1*a1^-1*b1^-1, a1*b2*a1^-1*b2^-1, a1*b3*a1^-1*b4^-1, a1*b4*a2^-1*b3^-1, a1*b4^-1*a2^-1*b3, a2*b1*a2^-1*b2^-1, a2*b2*a3^-1*b1, a2*b3*a2^-1*b4, a2*b2^-1*a3*b1^-1, a3*b1*a3*b3^-1, a3*b2*a3*b4^-1, a3*b3*a3*b4 ] gap> IsIrreducibleVHGroup(RattaggiGroup.2_21); true

`‣ MaximalSimpleSubgroup` ( g ) | ( attribute ) |

Returns: A maximal simple subgroup of `g`, if possible.

A VH group is never simple, but in favourable cases it admits a finite-index simple subgroup, see [BM97]. This function attempts to construct such a subgroup. It returns `fail`

if no such subgroup can be found.

The current implementation is not smart enough to work with the Rattaggi examples (see `IsVirtuallySimpleGroup`

(7.3-4)).

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