Inglés (pdf)
Articulo en XML
Referencias del artículo
Traducción automática
Enviar articulo por email
Citado por SciELO 
Citado por Google 
Similares en
SciELO 
Similares en Google 
Permalink
1. Introduction
Baumslag-Solitar groups constitute an important family of examples or counterexamples in the theory of combinatorial groups, see [2], [7], [5] and [4]. These groups were first introduced by G. Baumslag and D. Solitar in [2] in order to get non-Hopfian one-relator group presentations and they are given by the following short presentation
where m and n are non zero integer numbers.
Although the principal focus of the study of non-abelian representations has been on the family of classical knot groups, in particular on the collection of 2-bridge classical knot groups, we want to start a classification of the PSL(2, 
) non abelian representations on the family of Baumslag-Solitar groups. Recently, in a joint work with O. Salazar and J. Mira, see [8], we proved that BS(n, n +1) are the only Baumslag-Solitar groups that correspond to groups of 2-bridge non-classical virtual knots. From this we consider that the study of some properties of the Baumslag-Solitar groups becomes an important aim, in particular the classification of their non-abelian representations into the group PSL(2, ). It is well known that there exists an isomorphism between the orientation preserving isometries of 
and PSL(2, ).
In this paper we present a collection of algebraic varieties that encode information about the set of equivalence classes of non-abelian representations of BS(n, m). The definition of these algebraic varieties uses a classification of the conjugacy classes of elements of the Möbius group via diagonal matrices and Jordan forms which is not the standard classification of elements of the Möbius group given in [3]. It is worth noting that for |m| > 1, BS(1,m) is a linear group. It would seem that this result forms part of the folklore in the scope of Baumslag-Solitar group theory. As a consequence, there is no article in the literature which contains a formal proof of such fact. However, we can get one from mathoverflow. From the results displayed in this paper and some restrictions given in [7] and [1] about the residually finiteness of the Baumslag-Solitar groups, we prove that BS(n, m) only admits a faithful representation into SL(2, ) when n = 1 and |m| > 1. This result overlaps with what was mentioned in the previous paragraph, but we propose a different method that might be of interest for those working in Kleinian and Fuchsian groups. Moreover, if rm denotes the image of BS(1,m), for |m| > 1, into PSL(2, ), then we show that Tm is an elementary and non-discrete subgroup of Möbius transformations.
This paper is organized as follows. In section 2 we give a short list of preliminaries concerning Möbius transformations. We also recall the definition of SL(2, )-representations, residually finite groups and linear groups. Then, in Section 3 we show the existence of non-abelian representations of the Baumslag-Solitar groups and provide a classification of them into two families, pseudo-parabolic and pseudo-elliptic representations. We give functions of certain algebraic affine varieties into each of them. Finally, in Section 4 we show that BS(1, m) is a linear group, for |m| > 1. Moreover, we prove that its representation image in the Möbius transformations group is an elementary and non discrete subgroup.
2. Notation and preliminary results
In this section we introduce a short list of definitions, notations and some results necessary to understand this article.
2.1. Möbius transformations
The set of Möbius transformations
, with a,b,c,d Є and ad ≠ bc = 0, acting on
, is denoted by 
. This set has some additional structure; it is a group under composition of transformations. Besides, the homomorphism
, where
has kernel {±I}, and it provides a natural identification between and PSL(2, ).
We use tr(A) and At to denote the trace and the transpose of a matrix A.
The following definition come from the correspondence between SL(2, ) and the group of Möbius transformations PSL(2, ).
Definition 2.1. [3] Let A (A ≠ I) be a matrix in SL(2, ), then A is a parabolic matrix if and only if tr2(A) =4, A is an elliptic matrix if and only if tr2(A) Є [0,4), A is a hyperbolic matrix if and only if tr2(A) Є (4, +∞), and A is a strictly loxodromic matrix if and only of tr2(A) 
.
The following notation will be useful in order to simplify the proof and certain definitions given in the rest of this paper.
Notation 1. For 
and p Є,
We say that A Є SL(2, ) is a scalar matrix if A = D(δ), with δ2 = 1, and I = D(1).
The purpose of the next theorem is to classify the conjugacy classes of the matrices in SL(2, ). This classification involves diagonal matrices and matrices in Jordan form. Another classification using the theory of fixed points of Moobius transformations is given in [3].
Theorem 2.2. Every non scalar matrix A in SL(2, ) is a conjugate of one of the following matrices: D(1, 1), D(1, -1) or D(6).
Moreover, if A is taken in PSL(2, ), then it is conjugate to D(1, 1) or D(δ).
Proof. Let A Є SL(2, ) be a non scalar matrix with characteristic polynomial 
and consider the following cases.
Case 1: If 
, then, because A is not a scalar matrix, its minimal polynomial must be equal to 
, hence, from the canonical Jordan form, the matrix A is a conjugate of some matrix of the form D(1 , δ ), where δ2 = 1.
Case 2 If 
, then A is a diagonalizable matrix, therefore the matrix A is a conjugate of some matrix of the form D(δ), where δ2 =1
For the last part, note that, in PSL(2, ), D(1 , - 1) = D( -1 , 1). Because D(1, i)D(1,1)D-1, = D( -1,1), then D(1, -1) and D(1,1) are in the same conjugacy class. 
Let A Є SL(2, ), we define the norm of A, denoted ||A||, as
The topology on SL(2, ) induced by this norm is denoted by
.
The homomorphism Π induces the quotient topology 
on 
with respect to which Π is continuous. Besides, has the topology 
of uniform convergence with respect to the chordal metric on 
These topologies are the same. (see [3].
Definition 2.3. A subgroup G of SL(2, ) is discrete if and only if the relative topology on G is the discrete topology.
The proof of the following lemma can be found in [3].
Lemma 2.4. A subgroup G of SL(2, ) is discrete if and only if for each positive k, the set 
is finite.
Definition 2.5. A subgroup G of 
is said to be elementary if and only if any two elements of infinite order have a common fixed point.
A subgroup G of acts discontinuously on a G-invariant disc Δ (or on the half-plane), if for every compact 
except for a finite number of g Є G.
Definition 2.6. A non-elementary subgroup G of 
is a Kleinian group if it is discrete. If, moreover, G has a G-invariant disc Δ on which G acts discontinuously, we say that G is a Fuchsian group.
The proof of the following theorem can be found in [3].
Theorem 2.7.
Let
with
and not of order two. Let θ : 
, where 
. If for some n, 
= f, then (f, g) is elementary.
Therefore, any elementary Fuchsian group is either cyclic or it is conjugate to some group (f, g), where g(z) = kz (k > 1) and f (z) = -1/z.
2.2. SL(2, C )-representations of Baumslag-Solitar groups
In this section we will give some background material concerning the representation and character of the Baumslag-Solitar groups.
A representation of BS(n, m) into SL(2, ) is understood as a homomor-phism p : 
. The set of all representations is denoted by R(BS(n, m)). It is not hard to verify that R (BS(n, m)) can be endowed with the structure of an affine algebraic set.
Recall that two representations p and p' are equivalent (conjugate), denoted 
, if there exists 
such that 
for every g 
.
An immediate consequence of the definition above is, two representations p and p' of BS(n, m) are equivalent if and only if there exists 
such that 
and
.
The character of a representation p is the function 
defined by
. Since, equivalent representations have the same character, the map
where 
induce a well-defined function
.
Definition 2.8. Let 
. Then p is called irreducible if the only subspaces of 
which are invariant under p(BS(n, m)) are {0} and . In other case, we say that p is reducible.
From the previous definition, a representation p Є R(BS(n, m)) is reducible if and only if all p(g), with g Є BS(n, m), have a common one-dimensional eigenspace. Therefore, a representation p Є R(BS(n, m)) is reducible if and only if p(x) and p(y) have a common eigenvector.
The proof of the following theorem can be found in [9].
Theorem 2.9. Let p Є R(BS(n, m)), then p is reducible if and only if 
= 2, for each element c of the commutator subgroup of BS(n, m).
In this case, if 
denotes the image of p, then every element of the commutator subgroup 
of
is parabolic.
A representation p is abelian if its image is an abelian subgroup of SL(2, ), and nonabelian otherwise. We denote the set of nonabelian representations of BS(n, m) by nab-rep(BS(n, m)).
Definition 2.10. Let p Є R(BS(n,m)). Then p is called parabolic if p(y) is conjugate to D(1,1) or D(1,-1). Besides, if p(y) is conjugate to D(δ), then p is elliptic, hyperbolic or strictly loxodromic depending of the subset of C containing
.
A representation 
is said to be faithful if the kernel of p is trivial.
We recall the definition of residually finite groups.
Definition 2.11. A group G is called residually finite if for every g and h in G there exist a finite group F and a homomorphism y : 
such that
.
The following theorem gives us a complete classification of the Baumslag-Solitar groups in terms of the previous definition. Its proof can be found in [7].
Theorem 2.12. The group BS(m,n) is residually finite if and only if |m| = |n| or |m| = 1 or | n| = 1 .
Definition 2.13. A group G is said to be a linear group if there exists a faithful representation 
The proof of the next theorem can be found in [6].
Theorem 2.14. Let R be a field, and let M be a finite set of n by n matrices with elements in R and with non-vanishing determinant. Then the set of matrices in M generate a residually finite group.
Corollary 2.15. Linear groups are residually finite.
Therefore, BS(n, m) does not have faithful representations if |m| ≠ |n| and |m| ≠ 1 and |n| ≠ 1.
3. Non-abelian representations of BS(n, m)
In this section we will construct a collection of algebraic varieties that encode information about the set of equivalence classes of non-abelian representations of BS(n, m).
For an ideal J of
, we denote by 
(J) the algebraic variety
.
3.1. The case BS(n, n)
Let us start this section with the following theorem.
Theorem 3.1. Let A Є SL(2, ) be a non scalar matrix and λ = tr( A) ≠ 0. Consider the infinite sequence of polynomials in
, where
. Then
for every n Є
.
Proof. From the Cayley-Hamilton theorem
. Therefore
Now suppose that
. From the equation (1),
and so
The proof of the following lemma is a direct consequence of the Cayley-Hamilton theorem, so we will omit it.
Lemma 3.2. Let A in SL(2, ) be a non scalar matrix such that tr(A) = 0, then
.
Corollary 3.3. Let A, B in SL(2, ) be non scalar matrices such that 
Let A be a non scalar matrix with λ ≠ tr(A) ≠ 0 and
. Then
. Therefore A is a diagonalizable matrix, so there exists 
and 
such that
, with, 
and 
.
Now, let B Є SL(2, ) be a non-scalar matrix. It is well known that AB = BA implies that B = WD(y)W-1, with y2 - 1 ≠ 0. We have proven the following theorem.
Theorem 3.4. Let φ be a representation of BS(n, n), with φ(x) = B and φ(y) = A non-scalar matrices, and let λ = tr(A) ≠ 0, then:
(a) If 
, then A is a diagonalizable matrix. Moreover, AB = BA if and only if A and B are simultaneously diagonalizable.
(b) If 
then AB = BA.
Therefore, if φ is a non abelian representation, then A must be elliptic and
The following lemma will give us a easy way to compute the polynomial 
for the case of diagonalizable matrices.
Lemma 3.5.
Let
be a non-scalar matrix, with
and
. Then,
(a) If n is an even number, then
and
(b) ifn is a odd number, then
Proof. From Theorem 3.1,
Then
Therefore 
and so
If we multiply both sides by an, we obtain 
1. This implies that
The rest of the proof follows by factorizing polynomials of the form
Theorem 3.6. With the notation above.
(1) If n is a even number, then there exists an injective function from the algebraic set 
(J) into nab-rep(BS(n,n))/ ≈, where J is the ideal of 
spanned by
and
(2) If n is an odd number, then there exists an injective function from the algebraic set 
(J) into nab-rep(BS(n,n))/ ≈, where J is the ideal of
spanned by
and 
Proof. (1) Let 
and let 
the canonical homomorphism such that H(x) = D(y, β) and H(y) = D(α). Suppose that n = 2t, t > 0.
Now, consider the following two cases for the trace, A, of D(a).
Case 1: If λ = 0, then 
and
Case 2: if λ ≠ 0, then 
Therefore 
From the previous cases, and the fact that H(x) and H(y) are not simultaneously diagonalizable, H extends to a nonabelian representation 
. So, we can define the function
where 
is the equivalence class of the representation 
.
Let 
be such that 
then there exists 
and 
By expanding the matrix equality 
we obtain two possibilities, either 
If we suppose that 
then C11 = C22 = 0 and C21 = 
If we replace them in the equation CD(x
3, x
2)C
-1 = D(y
3, y
2), we get that C
12 = 0, but this is a contradiction, and so x
1 = y
1, and therefore C = I. Hence, ç is an injective function.
(2) Let (α, β,Υ ,σ) Є 
(J) and let, again, H : 
be the canonical homomorphism given by H(y) = D(α) and H(x) = D(Υ, β).
Because g(α, β,Υ ,σ) = 0, then 
therefore 
and, from Lemma 3.5, 
Therefore H extends to a non-abelian representation 
. So, we can define the function
where 
is the equivalence class of the representation 
The proof that ç is an injective function is quite similar to what we did in the proof of (1). 
We conclude this section with the fact that for every 
all the representations in the equivalence class 
are reducible.
Theorem 3.7.
Every representation in
is reducible.
Proof.
Let 
be the representative element of 
where 
and 
The eigenspace 
corresponds to the eigenvalue α-1, of the matrix 
(y), and has the property that it is an eigenspace of the matrix (x). Therefore 
is an (BS(n, n))-invariant subspace of 
3.2. The case BS(n,m), with n ≠ m
We start this section with the following theorem.
Theorem 3.8. There are no non-abelian representations 
of BS(n,m) such that(x) and (y) are both parabolic matrices.
Proof. Suppose that there exists a non-abelian representations 
of BS(n,m) in which (x) and (y) are both parabolic matrices. Then is a conjugate of the non-abelian representations 
such that 
Since 
then 
if and only if
Then we prove that (2) is true if and only if w12 = 0 and 
. Since 
then 
and so 
Hence, 
where 
and 
, that is a contradiction.
Corollary 3.9. Let 
m and denote by L the ideal of
spanned by f (z, w). Then there exists a function from the algebraic variety
(L) into nab-rep(BS(n, m))/ ≠.
Proof. Let 
and consider the map
where 
is the representation given by 
=
It is not hard to prove that Ψ is a well-defined function. 
The function Ψ is not injective because 
and 
Theorem 3.10. With the above notation. The representation 
is reducible.
Proof.
Consider the eigenspace 
. Then, E is a K(BS(n, m))-invariant one dimensional subspace of 
Theorem 3.11. If 
nab-rep(BS(n, m)) and
(y) is not a parabolic matrix, then (y) has finite order.
Proof. Suppose that 
nab-rep(BS(n, m)) and 
(y) is not a parabolic matrix. Then must be equivalent to a representation
, such that 
The matrix equality 
is true if and only if
Now, 
therefore D (δ) is an elliptic matrix of finite order. Moreover, since 
is a non abelian representation, then we have to add one of the following inequalities 
…………………………………………….
Theorem 3.12. We suppose that m > n. Let g1(α, β, Υ, σ) 
,
and let I the ideal of
[α, β, Υ, σ] spanned by {g
1
, g
2
}. Then, there exist a injective function from the algebraic variety
(I) into nab-rep(BS(n, m)) .
Proof. Consider the map
nab-rep(BS(n,m)), 
where 
is the representation given by 
and 
.
Due to the fact that 
and from Theorem 3.11, we get that 
is a non-abelian representation, therefore we have shown that 
is well defined.
The proof that is an injective function, is straightforward. 
Theorem 3.13. The representation 
given by
is reducible.
4. On non-abelian faithful representations of BS(n,m)
From the Theorem 2.12, the group BS(n, m) could have a non-abelian faithful representation in SL(2, 
) only for the cases in which n = m or n = 1 or m = 1. But, we know that BS(n,n) does not have non-abelian faithful representations into SL(2,
). (see Theorem 3.6). Therefore there only remains the case n = 1, because for m =1 we have that BS(1, m) 
BS(m, 1).
Before the proof that BS(1,m) is a linear group, consider the following theorem.
Theorem 4.1.
Each element
Є BS(1, m) is uniquely represented by a word of the form
where p, k, q are integers and p, q 0.
Proof.
From 
we obtain the infinite family of relators { 
}. So, we have the following four kinds of equations:
and
Without loss of generality, we may suppose that the words in BS(1,m) have the form 
where 
We will complete the proof by induction on the length n.
When n = 1, then 
So, if t1 > 0, then from (a) and (d), 
Now assume that 
. then 
. By the induction hypothesis 
where a, c ≥ 0, so w = 
If c + tn ≥ 0, from (a) and (d), 
, but if c + tn ≤ 0, from (b) and (c), 
Corollary 4.2. Let m £ Z such that |m| > 1, then BS(1, m) is a linear group, and it is residually finite.
Proof. We give the proof for the case m > 1. From Theorem 3.8, there exists a representation 
: BS(1,m) 
SL(2, 
), such that 
(x) = D(m-1/2) and 
(y) = D(1 , 1).
Let w be a word in BS(1 ,m) such that (w) = I. Because w is uniquely represented by a word of the form 
where 
and p, q ≥ 0, then
hence, 
(w) = I if and only if p = q and k = 0. Therefore w =1 and so is injective. 
If m = -1, then we have that D2(δ, i) = D(-1), therefore D(δ, i) is a matrix of order 4, as a consequence, neither of the representations given in the proof of Theorem 3.8 are faithful.
Let 
and 
where A = D(1,1) and B = D(m-1/2). Let 
, with m > 1, be the subgroup of 
generated by 
and 
Proposition 4.3. The subgroup 
is an elementary subgroup of 
. Moreover, 
is not a Fuchsian group.
Proof.
Since 
then
therefore, from Theorem 2.7 we have that 
is an elementary subgroup of 
. Besides, 
is not abelian, and hence it is not cyclic. Since A is not a diagonalizable matrix, is not conjugate to some group of the form 
where g(z) = kz (k > 1) and f (z) =-1/z -then is not a Fuchsian group. 
Theorem 4.4.
The subgroup
is not discrete.
Proof.
Let 
then there exists p, q, k in 
with p, q ≥ 0 such that 
Due to the fact that
Then
Let t ≥ 0 such that 
. Then
If we take t = 3 and p = q, then the previous inequality becomes 2m2p + k2 ≤ 3m2p, and so k2 ≤ m2p. It is not hard to prove that there are infinitely many pairs (k,p) G 
such that k2 ≤ m2p. 
