On a family of groups generated by parabolic matrices

We study various aspects of the family of groups generated by the parabolic matrices A(t1ζ), . . . , A(tmζ) where A(z) = ( 1 z 0 1 ) and by the elliptic matrix ( 0 −1 1 0 ). The elements of the matrices W in such groups can be computed by a recursion formula. These groups are special cases of the generalized parametrized modular groups introduced in [16]. We study the sets {z : tr W (z) ∈ [−2,+2]} [13] and their critical points and geometry, furthermore some finite index subgroups and the discretness of subgroups.


Introduction
A parabolic matrix is determined by one parameter. In this paper we study a family of groups generated by a finite number of parabolic matrices, where the parameter lies in a polynomial ring of one variable over the complex numbers. The groups that we consider have one additional generator, an elliptic element of order four. More specifically, we consider the parabolic matrix A(ξ) = 1 ξ 0 1 and the elliptic matrix B = 0 −1 1 0 . For ξ = 1, the group generated by A(1) = ( 1 1 0 1 ) and B is the classical modular group. In [13] we studied a more general case when ξ runs through all complex numbers, so we introduce the parametrized modular group Π = A(ξ), B; B 4 = I ⊂ SL (2, C[ξ]), see also the paper of P.M.Cohn [3]. A free purely parabolic subgroup of Π with index 4 had been considered by various authors, for instance J.Gilman and P.Waterman [8] and in [14].
In [16] we had considered the more general case of m parabolic matrices A(p k ) where the p k are any polynomials in C[ξ]. All group elements can be written in a precise form, see (2.2) below.
The group defined in [16] is too general for many purposes. In the present paper however, we restrict ourselves to polynomials of the special form p k = t k ξ where t k are complex numbers, therefore we study the group Π = Π[t 1 ξ, . . . , t m ξ] := A(t 1 ξ), . . . , A(t m ξ), B; B 4 = I . Now the matrices in Π can be computed by a recursion formula, see Section 2. In a way this paper takes up more the ideas of our first paper [13].
In Section 3 we study the critical points, i.e, the points where our group has an additional relation. We consider two methods to find critical points, namely by the Riley operator and by using Chebyshev polynomials.
Section 4 is about the singular set. For a member W of our group we form the set of all complex numbers ζ for which the trace of W lies in the interval [−2, +2]. The singular set S is then the union of all the sets formed. Its closure is of particular interest, see for instance [8,14,21].
In Section 5 we study the problem of discreteness of subgroups. This problem had not been considered in our previous papers. For instance we prove that the group A √ p , A i √ q , B with p, q ∈ N is discrete. Using ideas of T. Jörgensen [9] and of A.F.Beardon [1], R.Riley [21,Th.1] proved the following beautiful theorem: Theorem 1.1. (Riley) Let Γ(ζ) (|ζ| < 1) be a holomorphic family of subgroups of SL(2,C) which is non-elementary except possibly for countably many ζ. Then T := {ζ ∈ D : Γ(ζ) is discrete} is closed and the critical points are dense in the complement of T .
The boundary of T lies in the closure of the singular set S defined in Section 4. We will not use these results but they serve as a guide line for our Theorem 4.2 and Proposition 5.1. In Section 6 some subgroups are discussed.
The motivation to introduce the parametrized modular group [13] was the study of representations of the group of 2-bridge links. This problems was solved by Riley in a beautiful collection of paper [18,19,20]. The problem of the representations of 3-bridge links is not solved, but Riley gave some examples in his seminal paper [19]. This paper give us the motivation to study the generalized parametrized group [16] and now to specialized this group to the particular case considered in this paper. In [17] we apply our ideas in order to develop an algorithm to compute representations of 3-bridge knot groups and continue the work in [16]. Now we introduce some notation and review some of our previous results that are the motivation of this paper.
In [13] we studied the subgroup of SL(2, is the ring of polynomials in the variable ξ. For ζ ∈ C, let Π(ζ) be the subgroup of SL(2, C) obtained by substituting the indeterminate ξ by the number ζ and W (ζ) the matrix obtained by substituting ξ by ζ in an element W in Π. The modular group Γ, which have been amply studied, is Π(1) in our notation. The groups Π(2 cos(π/q)) (q ≥ 3) become the classical Hecke groups after projecting to PSL(2, C), and Π(ζ) for ζ ∈ R become the generalized Hecke groups. The group Π 1 generated by 1 ξ 0 1 , 1 0 ξ 1 is a free subgroup of Π [ξ] with index 4. The group Π 1 is conjugate to the much investigated two-parabolic group generated by ( 1 2λ 0 1 ) , ( 1 0 1 1 ). We studied algebraic and analytic properties of this group. First we provide the algebraic descriptions of the group Π = Π [ξ] and its subgroup Π 1 . We use combinatorial techniques to describe precisely the elements of both groups. To each element W in Π we associate a sequence of non zero integers and an inductive way to compute the polynomial that conforms the entries of the matrix W . We also describe some technical aspects of a word W with some particular type of associated sequences. We complete the algebraic aspects of Π by studying the set of words W such that for some ζ, ±W (ζ) becomes a relator in the group Π (ζ), i.e, W (ζ) = ±I. Then we consider analytic aspects of sets related to the groups Π and Π 1 . We define the singular set of Π, denoted S (Π), as the set of elements ζ ∈ C such that W (ζ) is not loxodromic, for some W ∈ Π. Notice that any relator provides elements in S (Π), so it is a "natural" transition to pass from studying the relators to study S (Π) and S (Π 1 ). For the proofs in this part we rely heavily on the description of the elements in Π and Π 1 . We give examples of singular sets for particular words, showing some pictures to illustrate the behavior of S (Π). We exhibit symmetry properties of S(Π) and estimate the logarithmic capacity. In a much more general context, the closure of the singular set has been studied in [21] and [12]. Very little is known about the set-theoretic properties of ∂S(Π). Is it connected?, does it have infinite linear measure or even positive Haussdorff dimension? A computer generated picture by Wright in [8, p.11] suggests that it is well behaved, another in [12] suggests that it is chaotic. The singular set S(Π) is the union of countably many analytic arcs. The closure of the singular set of analytic families of subgroups of PSL(2, C) has been much studied, see e.g. [9,21].
In [15] we studied free subgroups of index four of the parametrized modular group Π. We show that there are eight free subgroups, four of which are normal and four are non-normal. Then we studied the intersections of the normal subgroups. We give canonical presentations of these subgroups in terms of generators and relations. The derivation of our presentations relies on the results about the enumeration of the word in Π. We proved that the commutator subgroup Π has infinite index in Π, which is quite different in other contexts, for instance, the first three commutator subgroups of the Picard group have finite index. At the end of [15] we find connections between Π and the Picard group and other Bianchi groups and to a group from relativity theory. In order to establish the connections we needed to enlarge the groups Π(ζ) and consider groups generated by two or more parabolics. This was a motivation to study the generalized parametrized modular group in [16]. Given a set of polynomials p 1 , . . . , p m with complex coefficients and a indeterminate ξ which is the same for all µ, p µ = 0 (µ = 1, . . . , m), we define in [16] the generalized parametrized modular group Π = Π[p 1 , , . . . , p m ] = A(p 1 ), . . . , A(p m ), B . For ζ ∈ C, the notation Π(ζ) := Π[p 1 , . . . , p m ](ζ) ∈SL(2, C) (ζ ∈ C), means that the polynomials p µ are evaluated at ζ. If W = a b c d then, for instance, a = a(ξ) is a polynomial whereas a(ζ) is a complex number. We did not impose any restrictions on the polynomials p 1 , . . . , p m but we were able to show the existence of a simple algorithm to obtain the elements of Π. We show a way to describe the element in the group by a set of polynomial, but in general these polynomials are not uniquely determined. However, we have uniqueness under some special conditions on the p µ , (µ = 1, . . . , m). By imposing the restrictions on the polynomial we were able to proved similar results to the ones in [13]. These conditions are the motivation for the conditions we are imposing on the polynomial in the present paper. We discuss several concrete examples and its applications to knot theory. In many of our examples the p µ are complex numbers, and therefore, we obtain subgroups of SL(2, C); and PSL(2, C) is isomorphic to the group of orientation preserving isometries of the hyperbolic space H 3 . Our applications to knot theory use the fact that many knots K have groups with representations in PSL(2, C) and therefore S 3 − K admits the structure of a hyperbolic 3-manifold, [19]. The use of the indeterminate ξ, however, allows us to arrange matrix elements according to the degree of polynomials. We introduce the subgroup Π 1 of index 4 which is generated by the parabolic matrices A(z) and C(z) = BA(z)B −1 . For m = 1 and p 1 = ξ this generalizes the group studied in [14]. As an example, we consider two-bridge and three-bridge knots.
Using an idea of Riley [18] we show that at least some of these knots lead to subgroups of Π 1 generated by four or less parabolic matrices. An example is the "figure-eight knot" [11, p.60], the matrix group that represent its fundamental group is generated by A(1), C(ω), ω = e 2πi/3 which is a subgroup of Π 1 [1, ω].
2. The group Π 2.1. Let m ∈ N and t 1 , . . . , t m ∈ C \ {0} be given and let The set M does not depend on the order or the signs of the t µ . Let Note that A(z) n = A(nz). We shall study the group defined by the presentation where ξ will always be the indeterminate of polynomials in C[ξ]. The group Π does not depend on the order and signs of the parameters t 1 , .., t m .
We will often write W = a b c d . Multiplying by B it is easy to check that

Critical points
A point ζ 0 ∈ C is called a critical point if there exists a non-constant word V ∈ Π such that V (ζ 0 ) = I and we then say that the critical point ζ 0 is associated to V . Then we obtain the presentation Since Π has only a countable number of words it follows that B 4 = I is the only relation of Π(ζ) except for a countable number of ζ ∈ C. In this section we always write We shall present two methods to obtain critical points.
(i) The first method use the Riley operator W → W ∼ . By definition the Riley operator replaces all A(rξ) by A(−rξ) and all B ± by B ∓ without changing the order of these matrices in W . We have Proof. It follows from (8) that Since a − d is not constant by assumption, there exist one or more ζ 0 with a(ζ 0 ) − d(ζ 0 ) = 0 and therefore with V (ζ 0 ) = I.

For instance we have
The monic Chebyshev polynomials appear under various names.
The first possible case is n = 3. By (11) and (12) we have We want that W 3 (τ ) = I. This holds if and only if τ = −1.
As an example we consider the Picard group Π[1, i] with r 1 = 1 + i, r 2 = r ∈ Z, If a ± d is constant then a ± d ∈ {0, 2, −2}.  The singular set of Π is defined as and it is the union of countably many analytic arcs.
We obtain from (12) that W n (ζ nνj ) = I so that ζ nνj is a critical point. By (11) their union is dense in S 0 (W ). Now it follows from (16) that the critical points are dense in S and therefore dense in the closureS.
By induction it follows that The same is true with c(ζ) replaced by a(ζ), b(ζ) or d(ζ).
To conclude the proof we apply the above result to the matrices BW, BW B and BW using (7) and we obtain (21) (ii) If S 0 (W ) ⊂ {z ∈ C : |z| ≤ q} then the length satisfies Proof. (i) Using (6) the first assertion follows from a theorem of Fekete [5] and the fact that cap [−2, +2] = 1, see [13, p.122].
(ii) The second assertion is a consequence of an important result of Eremenko and Hayman [4, Th.2]: Let f be a polynomial of degree n and let We apply this result to the polynomial f (z) := tr W (z) and F 2 = S 0 (W ). By (25) we obtain len S 0 (w) = then the group Π(ζ) is not discrete in a neighbourhood of ζ 0 except that Π(ζ 0 ) may be discrete if c(ζ 0 ) = 0. The same holds for a, b, d instead of c.
are discrete.
is not discrete.
Proof. We need a classical result about diophantine approximation [10, Th.5]: If α ∈ R is irrational then there exist sequences p n , q n ∈ Z such that |q n α − p n | < 1/q n , q n → ∞ (n → ∞).
Hence the W n Π form an infinite system of disjoint cosets in Π 1 .