2 An Introduction to L-presented groups

Let S be an alphabet, Q and R be subsets of the free group F_S over this alphabet, and Φ be a set of free group endomorphisms φ: F_S-> F_S. An *L-presentation* is a quadruple (S,Q,Φ,R) and it is called *finite* if the sets S, Q, Φ, and R are finite. A (finite) L-presentation (S,Q,Φ,R) defines the (*finitely*) *L-presented group*

G=\left\langle S \left| Q\cup \bigcup_{\varphi\in\Phi^*}R^\varphi\right.\right\rangle

where Φ^* denotes the free monoid generated by Φ; that is, the closure of Φ∪{ id} under composition.

The elements in Q are the *fixed relators* and the elements in R are the *iterated relators* of the L-presentation (S,Q,Φ,R). An L-presentation of the form (S,∅,Φ,R) is an *ascending L-presentation* and it is an *invariant L-presentation* if the normal subgroup

K=\left\langle Q\cup \bigcup_{\varphi\in\Phi^*}R^\varphi\right\rangle^{F_S}

is φ-invariant for each φ∈Φ; that is, if K satisfies K^φ⊂ K for each φ∈Φ. Note that every ascending L-presentation is invariant and for each L-presentation (S,Q,Φ,R) there is a unique *underlying ascending L-presentation* (S,∅,Φ,R) which is invariant. In general it is not decidable whether or not a given L-presentation is invariant as this would require a solution to the word-problem.

In the remainder of this manual, an L-presented group is always finitely L-presented.

The construction of an L-presented group is similar to the construction of a finitely presented group (see Chapter Reference: Finitely Presented Groups of the **GAP** Reference manual for further details).

`‣ LPresentedGroup` ( F, frels, endos, irels ) | ( function ) |

returns the **GAP** object of an L-presented group with the underlying free group `F`, the fixed relators `frels`, the set of endomorphisms `endos`, and the iterated relators `irels`. The input variables `frels` and `irels` need to be finite subsets of the underlying free group `F` and `endos` needs to be a finite list of homomorphisms F-> F.

For example, the Grigorchuk group,

\Big\langle a,b,c,d \Big| a^2,b^2,c^2,d^2,bcd,[d,d^a]^{\sigma^n},[d,d^{acaca}]^{\sigma^n},(n\inℕ_0) \Big\rangle,

can be constructed as follows.

gap> F:=FreeGroup( "a", "b", "c", "d" ); <free group on the generators [ a, b, c, d ]> gap> AssignGeneratorVariables( F ); #I Assigned the global variables [ a, b, c, d ] gap> frels:=[a^2, b^2, c^2, d^2, b*c*d];; gap> endos:=[GroupHomomorphismByImagesNC( F, F, [a, b, c, d], [c^a, d, b, c])];; gap> irels:=[Comm( d, d^a ), Comm( d, d^(a*c*a*c*a) )];; gap> G:=LPresentedGroup( F, frels, endos, irels ); <L-presented group on the generators [ a, b, c, d ]>

There are various examples of finitely L-presented groups available in the library of the **lpres**-package.

`‣ ExamplesOfLPresentations` ( n ) | ( function ) |

returns some well-known examples of finitely L-presented groups. The input of this function needs to be a positive integer at most 10.

**n=1**The Grigorchuk group on 4 generators; cf. [Gri80], [Lys85], and [Bar03, Theorem 4.6],

**n=2**the Grigorchuk group on 3 generators; cf. [Gri80], [Lys85], and [Bar03, Theorem 4.6],

**n=3**the lamplighter group ℤ/2≀ℤ; cf. [Bar03, Theorem 4.1],

**n=4**the Brunner-Sidki-Vieira group; cf. [BSV99] and [Bar03, Theorem 4.4],

**n=5**the Grigorchuk supergroup; cf. [BG02] and [Bar03, Theorem 4.6],

**n=6****n=7****n=8**an index-3 subgroup of the Gupta-Sidki group,

**n=9****n=10**Baumslag's finitely generated, infinitely related group with a trivial multiplier; cf. [Bau71].

Furthermore, every free group in a variety of groups satisfying finitely many identities is finitely L-presented. Some of these groups are available from the **lpres**-package using the following operations; for further details we refer to the diploma thesis [Har08].

`‣ FreeEngelGroup` ( n, num ) | ( operation ) |

returns an L-presentation for the free `n`-Engel group on `num` generators; that is, the free group in the variety of `num`-generated groups satisfying the `n`-Engel identity.

`‣ FreeBurnsideGroup` ( exp, num ) | ( operation ) |

returns an L-presentation for the free Burnside group on `num` generators with exponent `exp`; that is, the free group in the variety of `num`-generated groups with exponent `exp`.

`‣ FreeNilpotentGroup` ( c, num ) | ( operation ) |

returns an L-presentation for the free nilpotent group of class `c` on `num` generators; that is, the free group in the variety of `num`-generated, nilpotent groups with nilpotency class `c`.

`‣ GeneralizedFabrykowskiGuptaLpGroup` ( n ) | ( operation ) |

returns an L-presentation for the `n`-th generalized Fabrykowski-Gupta group as constructed in [BEH08].

`‣ LamplighterGroup` ( filter, int ) | ( operation ) |

`‣ LamplighterGroup` ( filter, pcgroup ) | ( operation ) |

returns a finite L-presentation for the lamplighter group on `int` lamp states in the first case, if `filter` is the filter `IsLpGroup`

. In the second case, the group `pcgroup` must be a finite cyclic group. Then the method returns a finite L-presentation for the lamplighter group on `Size(pcgroup)`

lamp states; for details on the L-presentation see [Bar03].

gap> LamplighterGroup( IsLpGroup, 2 ); <L-presented group on the generators [ a, t, u ]> gap> LamplighterGroup( IsLpGroup, CyclicGroup(3) ); <L-presented group on the generators [ a, t, u ]>

`‣ EmbeddingOfIASubgroup` ( a ) | ( operation ) |

computes an L-presentation for the IA-automorphism group of a free group. This is the subgroup of automorphisms of a free group f that act trivially on the abelianization of f.

The L-presentation is taken from [DP].

gap> f := FreeGroup(3); <free group on the generators [ f1, f2, f3 ]> gap> a := AutomorphismGroup(f); <group of size infinity with 3 generators> gap> ia := Source(EmbeddingOfIASubgroup(a)); <invariant LpGroup on the generators [ C(1,2), C(1,3), C(2,1), C(2,3), C(3,1), C(3,2), M(1,[2,3]), M(2,[1,3]), M(3,[1,2]) ]> gap> rank := 3; 3 gap> q := NilpotentQuotient(ia,rank);; gap> lcs := LowerCentralSeries(q);; gap> for i in [1..Length(lcs)-1] do > r := AbelianInvariants(lcs[i]/lcs[i+1]); > Print(i); if i>3 then Print("th"); else Print(ELM_LIST(["st","nd","rd"],i)); fi; > Print(" quotient: abelian invariants ",r," (collected ",Collected(r),")\n"); > od; 1st quotient: abelian invariants [ 0, 0, 0, 0, 0, 0, 0, 0, 0 ] (collected [ [ 0, 9 ] ]) 2nd quotient: abelian invariants [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ] (collected [ [ 0, 18 ] ]) 3rd quotient: abelian invariants [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3 ] (collected [ [ 0, 43 ], [ 2, 14 ], [ 3, 9 ] ])

An L-presented group is defined as an image of its underlying free group. Note that these are two different **GAP** objects. The elements of the L-presented group are represented by words in the underlying free group.

`‣ FreeGroupOfLpGroup` ( lpgroup ) | ( attribute ) |

Returns: the underlying free group of the L-presented group `lpgroup`

`‣ FreeGeneratorsOfLpGroup` ( lpgroup ) | ( attribute ) |

Returns: the generators of the free group which underlies the L-presented group `lpgroup`

`‣ GeneratorsOfGroup` ( lpgroup ) | ( attribute ) |

Returns: the generators of the L-presented group `lpgroup`. These are the images of the generators of the underlying free group under the natural homomorphism.

`‣ UnderlyingElement` ( elm ) | ( operation ) |

returns the preimage of an L-presented group element `elm` in the underlying free group. More precisely, each element of an L-presented group is represented by an element in the free group. This method returns the corresponding element in the free group.

`‣ ElementOfLpGroup` ( fam, elm ) | ( operation ) |

returns the element in the L-presented group represented by the word `elm` on the generators of the underlying free group, if `fam` is the family of L-presented group elements.

gap> F:=FreeGroup( 2 );; gap> G:=LPresentedGroup( F, [ F.1^2 ], [ IdentityMapping( F ) ], [ F.2 ] );; gap> FreeGroupOfLpGroup( G ) = F; true gap> GeneratorsOfGroup( G ); [ f1, f2 ] gap> FreeGeneratorsOfLpGroup( G ); [ f1, f2 ] gap> last = last2; false gap> UnderlyingElement( G.1 ); f1 gap> last in F; true gap> ElementOfLpGroup( ElementsFamily( FamilyObj( G ) ), last2 ) in G; true

The fixed relators, the iterated relators, and the endomorphisms of an L-presented group are accessible with the following methods.

`‣ FixedRelatorsOfLpGroup` ( lpgroup ) | ( attribute ) |

Returns: the fixed relators of the L-presented group `lpgroup` as elements of the underlying free group.

`‣ IteratedRelatorsOfLpGroup` ( lpgroup ) | ( attribute ) |

Returns: the iterated relators of the L-presented group `lpgroup` as elements of the underlying free group.

`‣ EndomorphismsOfLpGroup` ( lpgroup ) | ( attribute ) |

Returns: the endomorphisms of the L-presented group `lpgroup` as endomorphisms of the underlying free group.

gap> F:=FreeGroup( 2 );; gap> G:=LPresentedGroup( F, [ F.1^2 ], [ IdentityMapping( F ) ], [ F.2 ] ); <L-presented group on the generators [ f1, f2 ]> gap> FixedRelatorsOfLpGroup( G ); [ f1^2 ] gap> IteratedRelatorsOfLpGroup( G ); [ f2 ] gap> EndomorphismsOfLpGroup( G ); [ IdentityMapping( <free group on the generators [ f1, f2 ]> ) ]

For the method-selection of the nilpotent quotient algorithm, an L-presented group may have the following attributes and properties.

`‣ UnderlyingAscendingLPresentation` ( lpgroup ) | ( attribute ) |

returns the underlying ascending L-presentation of `lpgroup`; that is, if `lpgroup` is finitely L-presented by (S,Q,Φ,R), the underlying ascending L-presentation is (S,∅,Φ,R).

`‣ UnderlyingInvariantLPresentation` ( lpgroup ) | ( attribute ) |

attempts to compute a ``good'' underlying invariant L-presentation for `lpgroup`; that is, if `lpgroup` is finitely L-presented by (S,Q,Φ,R), then this method seeks to find a subset Q'⊆ Q such that (S,Q',Φ,R) is an invariant L-presentation. Note that there is always the underlying ascending L-presentation (S,∅,Φ,R). However, for the efficiency of the nilpotent quotient algorithm it is important that the subset Q' is as big as possible.

Since it is undecidable, in general, whether or not a given L-presentation is invariant, there is no algorithm which can determine the best possible underlying invariant L-presentation. The method implemented for this attribute tries to compute a ``good'' invariant L-presentation and will return the underlying ascending L-presentation in the worst case.

This attribute can be set manually using `SetUnderlyingInvariantLPresentation`

. For instance, the Grigorchuk group

\Big\langle a,b,c,d \Big| a^2,b^2,c^2,d^2,bcd,[d,d^a]^{\sigma^n}, [d,d^{acaca}]^{\sigma^n},(n\inℕ_0) \Big\rangle,

is invariantly L-presented and therefore, it should be constructed as follows:

gap> F:=FreeGroup( "a", "b", "c", "d" );; gap> AssignGeneratorVariables( F ); #I Assigned the global variables [ a, b, c, d ] gap> frels:=[ a^2, b^2, c^2, d^2, b*c*d ];; gap> endos:=[ GroupHomomorphismByImagesNC( F, F, [ a, b, c, d ], [ c^a, d, b, c ]) ];; gap> irels:=[ Comm( d, d^a ), Comm( d, d^(a*c*a*c*a) ) ];; gap> G:=LPresentedGroup( F, frels, endos, irels ); <L-presented group on the generators [ a, b, c, d ]> gap> SetUnderlyingInvariantLPresentation( G, G );;

`‣ IsAscendingLPresentation` ( lpgroup ) | ( property ) |

checks whether the L-presentation of `lpgroup` is ascending; that is, if the set of fixed relators is empty. This property is set automatically when creating an L-presented group with no fixed relators using the function `LPresentedGroup`

(2.2-1).

`‣ IsInvariantLPresentation` ( lpgroup ) | ( property ) |

attempts to check whether the L-presentation of `lpgroup` is invariant. In general, one cannot decide whether or not a given L-presentation is invariant. There are mainly two methods implemented for this property. The first method seeks to find a ``good'' underlying invariant L-presentation using the operation `UnderlyingInvariantLPresentation`

(2.5-2). If this latter L-presentation coincides with the L-presentation of `lpgroup`, then `lpgroup` is invariantly L-presented. If this method fails, then the second method uses the nilpotent quotient algorithm for L-presented groups which yields a necessary condition for an L-presented group to be invariantly L-presented. Note that the latter method may not terminate. For instance, both methods fail on Baumslag's finitely generated, infinitely related group with trivial multiplier returned by `ExamplesOfLPresentations`

(2.2-2).

`‣ EmbeddingOfAscendingSubgroup` ( lpgroup ) | ( attribute ) |

stores an embedding of an ascendingly L-presented subgroup of the L-presented group `lpgroup`. This attribute is set for ascending L-presentations only. In this case, the identity map of `lpgroup` is returned. This attribute is used in the **FR**-package which can construct various finitely L-presented groups. The embedding is useful for a nilpotent quotient algorithm of a non-invariantly L-presented group.

Some operations are natural extensions of the operations for finitely generated groups. For example, `MappedWord(x,gens,imgs)`

, when applied to a word `x`

in an L-presented group, returns the group element obtained by replacing each occurrence of a generator in `gens`

by the corresponding element in the list `imgs`

. The lists `gens`

and `imgs`

need to have the same length.

Equality test of elements of L-presented groups is implemented using the operation `NqEpimorphismNilpotentQuotient`

(nq: NqEpimorphismNilpotentQuotient) to compare the images in a nilpotent quotient of the group. The implemented method successively increases the class of the considered quotient until the images differ. Hence, this method may not terminate and it will only determine whether the elements are different.

`‣ EpimorphismFromFpGroup` ( lpgroup, n ) | ( operation ) |

returns an epimorphism from a finitely presented group onto `lpgroup`. The finitely presented group is obtained from `lpgroup` by applying only words of length at most `n` in the endomorphisms of `lpgroup` to the iterated relators of `lpgroup`.

`‣ SplitExtensionByAutomorphismsLpGroup` ( lpgroup, H, auts ) | ( operation ) |

returns an L-presentation for the split extension of `lpgroup` by an L-presented or by a finitely presented group `H`. The action of a generator of `H` on `lpgroup` is given by an automorphism in the list `auts`. Thus for each generator of `H` there must be an automorphism in the list `auts`.

gap> F := FreeGroup( "a" ); <free group on the generators [ a ]> gap> H := F / [ F.1^3 ]; <fp group on the generators [ a ]> gap> U := ExamplesOfLPresentations( 8 ); <L-presented group on the generators [ t, u, v ]> gap> aut:=GroupHomomorphismByImagesNC( U, U, [ U.1, U.2, U.3 ], [ U.2, U.3, U.1 ] ); [ t, u, v ] -> [ u, v, t ] gap> SplitExtensionByAutomorphismsLpGroup( U, H, [ aut ] ); <L-presented group on the generators [ t, u, v, a ]>

`‣ AsLpGroup` ( G ) | ( operation ) |

returns an ascending L-presentation for a finitely presented group `G` or for a free group `G`.

`‣ IsomorphismLpGroup` ( G ) | ( operation ) |

returns an isomorphism from a finitely presented group `G` or from a free group `G` to the L-presented group obtained from the method `AsLpGroup`

(2.6-3).

gap> F:=FreeGroup( 2 ); <free group on the generators [ f1, f2 ]> gap> G:=F/[ F.1^2, F.2^2, Comm( F.1, F.2 ) ]; <fp group on the generators [ f1, f2 ]> gap> IsomorphismLpGroup( G ); [ f1, f2 ] -> [ f1, f2 ] gap> Range(last); <L-presented group on the generators [ f1, f2 ]> gap> Display(last); generators = [ f1, f2 ] fixed relators = [ ] endomorphism = [ IdentityMapping( <free group on the generators [ f1, f2 ]> ) ] iterated relators = [ f1^2, f2^2, f1^-1*f2^-1*f1*f2 ]

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