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Dirichlet-ford domains and double Diriohlet domains.

1 Introduction

Fundamental domains in hyperbolic spaces, or spaces of constant curvature in general, have been studied for a long time. Together with Poincare's Polyhedron Theorem, they are often used to find presentations of discrete groups. Fundamental domains also are of use in the construction of discrete groups. Special attention goes to fundamental domains in hyperbolic 2- and 3-space, as they are strongly related to discrete subgroups of [PSL.sub.2](R) and [PSL.sub.2](C).

A major difficulty one encounters is the effective construction of such a domain. The two most known and used constructions in hyperbolic space are the Ford and Dirichlet fundamental domains. The Ford fundamental domain of a group [GAMMA] is defined in terms of the isometric spheres of the elements of r. The Dirichlet domain of [GAMMA] is based on the bisectors of some chosen center and its images by [GAMMA]. It is well-known that in the ball model of hyperbolic 2- or 3-space, the Dirichlet fundamental domain and the Ford fundamental domain of some discrete group [GAMMA] are the same (see for example [4, Theorem 9.5.2]). In the upper half-space model however, this is, in general, not the case. Hence an interesting topic is to study when the Dirichlet and the Ford fundamental domain coincide also in the upper half-space model. This is what we call a DF domain, i.e. a fundamental domain in [H.sup.2] or [H.sup.3] that is a Dirichlet and a Ford domain at the same time.

One major problem in the construction of a Dirichlet fundamental domain is the choice of an adequate center. In general changing the center, changes the shape of the fundamental domain completely, as is nicely shown by Martin Deraux's animation [2]. This is different for the hyperbolic reflection groups. Their fundamental domain is canonical and the choice of the center plays no role at all. The fundamental domain is the same for every chosen center, see [18, Exercise 7.1.1]. So one may wonder which other Dirichlet fundamental domains have multiple centers. We call them double Dirichlet domains. In some sense characterizing a group acting discontinuously on hyperbolic space by having a double Dirichlet domain comes down to study 'how close' the group is to a hyperbolic reflection group.

One of the most common examples of a discontinuous action on hyperbolic space is the action of [PSL.sub.2](Z) on hyperbolic 2-space [H.sup.2]. The probably best known fundamental domain for this action is the triangle in the upper half-plane model with vertices [infinity], [1/2] + [[square root of 3]/2]i and -[1/2] + [[square root of 3]/2] i. This is in fact both a Ford domain and a Dirichlet domain with center ti for every t > 0. So this is a first example of a DF-domain and a double Dirichlet domain.

In this paper we continue the investigations initiated by Lakeland in [14] on DF-domains and double Dirichlet domains. The main reason in [14] to study these domains is answering a question raised by Agol-Belolipetsky-Storm-Whyte in [3]: the existence of a maximal arithmetic hyperbolic reflection group which is not congruence. The author constructs a non-congruence arithmetic group [[GAMMA].sub.ref] and, by using the theory of DF-domains, he proves that [[GAMMA].sub.ref] is a maximal reflection group.

The main theorem of [14, Theorem 5.3] states that a finitely generated, finite coarea Fuchsian group [GAMMA] admits a DF-domain if and only if [GAMMA] is an index 2 subgroup of a reflection group. It also is proved that a Kleinian group [GAMMA] has a generating set consisting of elements whose traces are real ([14, Theorem 6.3].) We give a new and independent criterion for the result of [14] that also applies to Kleinian groups. Note that all the groups we are working with are non-cocompact. See also Remark 2.1. Our criterion is of algebraic nature and easily can be checked once a set of side-pairing transformations is given:

Theorem 1.1. Let [GAMMA] be a non-cocompact cofinite discrete subgroup of [PSL.sub.2](C), acting on [H.sup.2], respectively [H.sup.3], and [P.sub.0] = i, respectively [P.sub.0] = j. Suppose that the stabilizer of [P.sub.0] in [GAMMA] is trivial. Then, [GAMMA] admits a IDF domain F with center [P.sub.0] if and only if for every side-pairing transformation [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] of F we have that d = [bar.a].

Moreover, if [gamma] is a cofinite Fuchsian group, then [??] = <[sigma], [GAMMA]> is a reflection group and [??] = <[tau], [GAMMA]> is a Coxeter group, where [sigma] is the reflection in the imaginary axis and [tau] is the linear operator represented by the matrix [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], and both groups contain [GAMMA] as a subgroup of index two.

For such groups [GAMMA], as an immediate consequence, one obtains, in the Kleinian case, that the traces of the generating elements are real (see Corollary 4.4). Moreover, as an application, we get most of the results on DF and double Dirichlet domains obtained in [14]. Also, some of the proofs of [14] can be simplified using Theorem 1.1.

To prove the above theorem, we make use of concrete formulas given in [12]. In [12], the authors develop explicit formulas for the bisectors of i and [gamma](i) for some [gamma] [member of] [PSL.sub.2](R) not fixing i, or j and [gamma](j) for some [gamma] [member of] [PSL.sub.2](C) not fixing j. These bisectors are indeed necessary to determine the Dirichlet fundamental domain in hyperbolic 2-space [H.sup.2] and hyperbolic 3-space [H.sup.3] respectively. These fundamental domains are then used to tackle the non-trivial problem of describing units in an order of a non-commutative non-split division algebra or of a 2-by-2 matrix ring over a quadratic imaginary extension of the field Q. As the unit groups of some of these orders may be considered as discrete subgroups of [SL.sub.2](C), fundamental domains and Poincare's Polyhedron Theorem are of potential use to determine these unit groups. First attempts to this were done by Pita, del Rio and Ruiz in [16,17], where the authors use Ford domains to get presentations of some small subgroups of congruence subgroups of Bianchi groups and by Corrales, Jespers, Leal and del Rio in [9], where a presentation for the unit group of a "small" non-commutative division algebra (a quaternion algebra) is given using Dirichlet domains. As an application, one obtains a description of subgroups of finite index in the unit group of an integral group ring of some finite groups. Making use of concrete formulas, the authors obtain in [12] a more general approach than the isolated cases described above.

The outline of the paper is as follows. For the sake of completeness, we record in Section 2 some fundamentals on hyperbolic geometry, fundamental domains and on Fuchsian and Kleinian groups. In Section 3, we recall a result from [12] and develop a necessary proposition to prove our main result and its corollaries. In particular, we give conditions for an isometric sphere, in the upper half-space (-plane) model, to be a bisector. In the last section, we consider DF domains and double Dirichlet domains, prove Theorem 1.1 and show some corollaries.

2 Background

We begin by recalling basic facts on hyperbolic spaces and we fix notation. Standard references are [4, 8, 10, 11, 15, 18]. By [H.sup.2] and [H.sup.3] we denote the upper half-space model of hyperbolic 2- and 3-space. As is common, we identify [H.sup.2] with the subset {x + ri [member of] C | x [member of] R, r [member of] [R.sup.+]} of the complex numbers C and [H.sup.3] with the subset {z + rj [member of] H | z [member of] C, r [member of] [R.sup.+]} of the classical (real) quaternion algebra H = H(-1,-1/R) Denote by [Iso.sup.+] ([H.sup.i]) the group of orientation-preserving isometries of [H.sup.i] for i = 2, 3. It is well known that [Iso.sup.+]([H.sup.2]) is isomorphic with [PSL.sub.2](R) and that [Iso.sup.+]([H.sup.3]) is isomorphic with [PSL.sub.2](C).

Throughout, we will use the notation [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] both for an element of [SL.sub.2] (R) or [SL.sub.2](C) as well as for its natural image in [PSL.sub.2](R) or [PSL.sub.2](C). Moreover, abusing notation, we use the same letter for both the matrix in [SL.sub.2](R) or [SL.sub.2](C) and the Mobius transformation acting on [H.sup.2] or [H.sup.3] respectively. For [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] in [SL.sub.2](R) or [SL.sub.2](C), we write a = a([gamma]), b = b([gamma]), c = c([gamma]) and d = d([gamma]) when it is necessary to stress the dependence of the entries on the matrix [gamma].

We now describe the action of [PSL.sub.2](R) and [PSL.sub.2](C) on hyperbolic space. We do this in detail for the 3-dimensional case, the 2-dimensional case being done in a similar way. The action of [PSL.sub.2](C) on [H.sup.3] is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where (aP + b)[(cP + d).sup.-1] is calculated in H. Explicitly, if P = z + rj and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] then

[gamma](P) = [(az + b)([bar.cz] + [bar.d]) + a[bar.c][r.sup.2]/[[absolute value of cz + d].sup.2] + [[absolute value of c].sup.2][r.sup.2]] + (r/[[absolute value of cz +d].sup.2] + [[absolute value of c].sup.2][r.sup.2])j.

The hyperbolic distance [rho] in [H.sup.3] (or [H.sup.2] respectively) is determined by cosh [rho](P, P') = 1 + [d[(P, P').sup.2]/2rr'], where d is the Euclidean distance and P = z + rj and P' = z' + r'j are two elements of [H.sup.3] (respectively P = x + ri and P' = x' + r'i are two elements of [H.sup.2]).

Finally, recall that a group [GAMMA] is said to act discontinuously on a proper metric space X if for every compact subset K of X, K [intersection] [gamma](K) [not equal to] [empty set] for only finitely many [gamma] [member of] [GAMMA]. A well-known theorem states that if X is a proper metric space, then a group [GAMMA] acts discontinuously on X if and only if [GAMMA] is a discrete subgroup of Iso (X). For more details on this, see [18, Theorem 5.3.5]. A fundamental domain for a discontinuous group action of a group [GAMMA] on a metric space X, is a closed set F [subset or equal to] X such that the border of F has measure 0, the union of the images of F under [GAMMA] is the space X and the images of F[degrees], the interior of F, by different elements of [GAMMA] are pairwise disjoint. We call F a convex fundamental polyhedron, if F is a fundamental domain that is a convex polyhedron. The group [GAMMA] is said to be cofinite if F has finite volume. It is well-known that cofiniteness implies geometrical finiteness which, in dimension 2 and 3, implies that the F has finitely many sides. For more details, we refer to [7] and [13]. If F is a fundemantal polyhedron for a discontinuous group action [GAMMA], then, for every side S of F, there exists an element [[gamma].sub.S] [member of] [GAMMA] such that S = F [intersection] [[gamma].sub.S](F). If for every side S the element [[gamma].sub.S] is the reflection in the hyperplane <S>, then r is called a reflection group with respect to F. Reflection groups are particular cases of Coxeter groups. Formally a Coxeter group is a group [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] with [m.sub.ii] = 1 and [m.sub.ij] [greater than or equal to] 2 for i [not equal to] j. Note that [m.sub.ij] = [infinity] is possible and just means that there is no relation of the form [([r.sub.i][r.sub.j]).sup.m] between [r.sub.i] and [r.sub.j]. More details on this may be found in [18, Section 7.1].

In this paper we work with two different constructions of fundamental polyhedra, known as Dirichlet and Ford fundamental domain. We recall their construction and how they can be used to give a presentation for the considered groups, the so called Poincare method (for details see for example [4] or [18]). Let [GAMMA] be a discrete subgroup of [Iso.sup.+]([H.sup.3]). Let [[GAMMA].sub.j] be the stabilizer in [GAMMA] of j [member of] [H.sup.3] and let [F.sub.j] be a convex fundamental polyhedron of [[GAMMA].sub.j]. Put [D.sub.[gamma]](j) = {u [member of] [H.sup.3] | [rho](u, j) [less than or equal to] [rho](u, [gamma](j))} and set [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. The border [partial derivative][D.sub.[gamma]](j) = {u [member of] [H.sup.3] | [rho](u, j) = [rho](u, [gamma](j))} is the hyperbolic bisector of the geodesic linking j to [gamma](j). This is called a Poincare bisector. Note that [??] is stable under the action of [[GAMMA].sub.j]. Moreover [??] is a convex polyhedron such that [[union].sub.n] [[gamma].sub.n] ([??]) = [H.sup.3], where [[gamma].sub.n] are the coset representatives of [[GAMMA].sub.j] in [GAMMA]. By [4, Theorem 9.6.1], the set

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

is a fundamental domain of [GAMMA], which we call the Dirichlet fundamental domain with center j. Moreover, it may be shown that F is a convex polyhedron and if [GAMMA] is geometrically finite then a finite set of generators for [GAMMA] consists of the elements [gamma] [member of] [GAMMA] such that F [intersection] [gamma](F) is a side of the polyhedron together with [[GAMMA].sub.j], i.e. [GAMMA] = <[[GAMMA].sub.j], [gamma] | [gamma](F) [intersection] F is a side> (see [18, Theorem 6.8.3]). Let us denote the bisector [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] of the geodesic linking j to [[gamma].sup.-1] (j) by [[summation].sub.[gamma]]. It is easy to compute that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. From this it follows that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Note that the same construction can be done in [H.sup.2], replacing the point j [member of] [H.sup.3] by i [member of] [H.sup.2].

Let [GAMMA] be a discrete subgroup of [PSL.sub.2](C) and denote by [[GAMMA].sub.[infinity]] the stabilizer in [GAMMA] of tire point [infinity]. Denote a convex fundamental polyhedron of [[GAMMA].sub.[infinity]] by [F.sub.[infinity]]. For [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], denote the isometric sphere of [gamma] by [ISO.sub.[gamma]]. Note that these are the points P [member of] [H.sup.3] such that [[absolute value of cP + d].sup.2] = 1. Denote the set {P [member of] [H.sup.3] | [[absolute value of cP + d].sup.2] [greater than or equal to] 1} by [ISO.sup.[greater than or equal to].sub.[gamma]]. By the same reasoning as above, if [[GAMMA].sub.[infinity]] contains a parabolic element then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

is a fundamental domain of [GAMMA] called the Ford fundamental domain of [GAMMA]. Again, if F is a polyhedron and if [GAMMA] is geometrically finite then [GAMMA] = <[[GAMMA].sub.[infinity]], [gamma] | [gamma](F) [intersection] F is a side>. And also in this case one can easily show that, for every [gamma] [member of] [GAMMA]\[[GAMMA].sub.[infinity]], [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Again the same construction is possible in [H.sup.2].

Remark 2.1. If we talk about Ford domains, we implicitly assume the discrete subgroup r to have a parabolic element fixing the point to.

3 Poincare bisectors and isometric spheres

For completeness' sake, we first recall in this section the authors' result from [12] that is needed to prove our main result. Based on this result we prove a proposition that will be used later. As before, we develop the theory in dimension 3, but everything can be applied to dimension 2 as well. Let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Recall that [gamma] [member of] [PSU.sub.2](C) if and only if [gamma](j) = j (see [4,10]). As the Poincare bisector can only exist if y G ^, the case y G PSU2 (C) is excluded in the following. In the ball model of the hyperbolic space, it is well-known that the isometric sphere of y equals the bisector of the geodesic segment linking 0 and its image by y-1. We will not go into more details on this but refer the interested reader to [4, Section 9.5] for dimension 2. In [12, Theorem 3.1], the authors give an independent proof of this in dimension 3 (which is of course adaptable to dimension 2).

In the upper half-space [H.sup.3], an isometric sphere is not necessarily a Poincare bisector. As explained in Section 2, we denote the isometric sphere of [gamma] by [ISO.sub.[gamma]] and the Poincare bisector of the geodesic linking j to [[gamma].sup.-1](j) by [[summation].sub.[gamma]]. This bisector may be a Euclidean sphere or a plane perpendicular to [partial derivative][H.sup.3]. If it is a Euclidean sphere, we denote its center by [P.sub.[gamma]] and its radius by [R.sub.[gamma]].

The following result gives concrete formulas for the Poincare bisectors in the upper half-space model.

Proposition 3.1. [12, Proposition 3.2]

Let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], with [gamma] [not member of] [PSU.sub.2](C).

1. [[SUMMATION].sub.[gamma]] is a Euclidean sphere if and only if [[absolute value of a].sup.2] + [[absolute value of c].sup.2] [not equal to] 1. In this case, its center and its radius are respectively given by [P.sub.[gamma]] = [-([bar.a]b + [bar.c]d)/[[absolute value of a].sup.2] + [[absolute value of c].sup.2] - 1] and [R.sup.2.sub.[gamma]] = [1 + [[absolute value of [P.sub.[gamma]]].sup.2]/[[absolute value of a].sup.2] + [[absolute value of c].sup.2]].

2. [[summation].sub.[gamma]] is a plane if and only if [[absolute value of a].sup.2] + [[absolute value of c].sup.2] = 1. In this case Re([bar.v]z) + [[[absolute value of v].sup.2]/2] = 0, z [member of] C is a defining equation of [[summation].sub.[gamma]], where v = [bar.a]b + [bar.c]d.

The next proposition gives some information on the relation between [ISO.sub.[gamma]] and [[summation].sub.[gamma]], for some [gamma] [member of] [PSL.sub.2](C)\[PSU.sub.2](C) with c([gamma]) [not equal to] 0. Again the case [gamma] [member of] [PSU.sub.2](C) is excluded because otherwise [[summation].sub.[gamma]] does not exist. This proposition will be useful in the study of DF domains and double Dirichlet domains.

Proposition 3.2. Let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

1. If c [not equal to] 0 and [[absolute value of a].sup.2] + [[absolute value of c].sup.2] [not equal to] 1 then [ISO.sub.[gamma]] = [[summation].sub.[gamma]] if and only if d = [bar.a]. In this case, tr([gamma]) [member of] R and if b [not equal to] 0, c = [lambda][bar.b], with [lambda] [member of] R.

2. If c = 0 and [[absolute value of a].sup.2] = 1, roe also have that a = [bar.d], tr([gamma]) [member of] R and c = [lambda][bar.b], with [lambda] [member of] R.

Proof. First note that in 1, the case c = 0 is excluded so that [ISO.sub.[gamma]] exists and, by Proposition 3.1, the condition [[absolute value of a].sup.2] + [[absolute value of c].sup.2] [not equal to] 1 guarantees that [[summation].sub.[gamma]] is a sphere. If we denote the center of [ISO.sub.[gamma]] by [[??].sub.[gamma]], then [absolute value of [[??].sub.[gamma]] - [P.sub.[gamma]]] = [absolute value of -[d/c] + [bar.a]b + [bar.c]d/[[absolute value of a].sup.2] + [[absolute value of c].sup.2] -1] = [[absolute value of d - [bar.a]]/[absolute value of c]([[absolute value of a].sup.2] + [[absolute value of c].sup.2] - 1)]. Hence [ISO.sub.[gamma]] = [[summation].sub.[gamma]] implies that d = [bar.a] and therefore be = [[absolute value of a].sup.2] - 1 [member of] R. The latter implies that tr([gamma]) = a + [bar.a] [member of] R and that b = 0 or c = [lambda][bar.b] for some [lambda] [member of] R. To prove the converse, suppose that d = [bar.a]. Then by the above [P.sub.[gamma]] = [[??].sub.[gamma]]. Moreover if [P.sub.[gamma]] = -[d/c], then, by Proposition 3.1, [R.sub.[gamma]] = 1/[absolute value of c] and hence [ISO.sub.[gamma]] = [[summation].sub.[gamma]]. This proves the first item.

In the second item we have that the isometric sphere does not exist and [[summation].sub.[gamma]] is a plane (and not a sphere). The conditions c = 0 and [[absolute value of a].sup.2] = 1 imply that ad = 1 and a[bar.a] = 1. Hence d = [bar.a] and tr([gamma]) [member of] R. As c = 0, the equality c = [lambda][bar.b], with [lambda] [member of] R is trivially true.

Remark 3.3. Note that Proposition 3.2 does not treat the cases c [not equal to] 0 and [[absolute value of a].sup.2] + [[absolute value of c].sup.2] = 1 and c = 0 and [[absolute value of a].sup.2] [not equal to] 1. In the first case the isometric sphere exists but [[summation].sub.[gamma]] does not have the form of a sphere. In the second case [[summation].sub.[gamma]] exist in the form of a sphere, but the isometric sphere does not exist. So in both cases it does not make sense to compare the isometric sphere with [[summation].sub.[gamma]].

4 DF Domains and Double Dirichlet Domains

The goal of this section is to prove Theorem 1.1 and give some consequences that reprove and complement some results in [14]. The following definitions are taken from [14].

Definition 4.1. A Dirichlet fundamental domain which is also a Ford domain in [H.sup.n] is called a DF-domain. A Dirichlet fundamental domain which has multiple centers is called a double Dirichlet Domain.

Throughout this section, we work in [H.sup.2] and [H.sup.3] and we assume, without loss of generality, that the stabilizer of i, or j respectively, in [GAMMA], is trivial. The latter is possible by conjugating, if needed, the group [GAMMA] by an adequate affine subgroup of [PSL.sub.2](R), or [PSL.sub.2](C) respectively. Indeed, denote by A the subgroup of [PSL.sub.2](R), or [PSL.sub.2](C) consisting of upper triangular matrices. Consider the conjugated group [tau][GAMMA][[tau].sup.-1] of [GAMMA] for some [tau] [member of] A and let [P.sub.0] [member of] {i, j}, according to the space being [H.sup.2] or [H.sup.3] respectively. Then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and thus if the stabilizer of [[tau].sup.-1] ([P.sub.0]) is trivial in [GAMMA], the stabilizer of [P.sub.0] is trivial in [tau][GAMMA][[tau].sup.-1]. Let F be some fundamental domain for [GAMMA]. By definition every point in the interior of F has trivial stabilizer. As A acts transitively on the upper half-plane, there exists [tau] [member of] A such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is trivial. Moreover, if [GAMMA] contains a parabolic element of the form [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], then the conjugate [tau][GAMMA][[tau].sup.-1] also contains such a parabolic element. So, instead of proving the results for [GAMMA], we will prove them for a group conjugated to [GAMMA] with trivial stabilizer of [P.sub.0]. It is easy to see that if [tau][GAMMA][[tau].sup.-1] has a double Dirichlet domain, [GAMMA] has a double Dirichlet domain. Similarly if [tau][GAMMA][[tau].sup.-1] has a DF domain, [GAMMA] has a DF domain.

We first give two lemmas on Fuchsian groups.

Lemma 4.2. The following properties are equivalent for 1 [not equal to] [gamma] [member of] [PSL.sub.2](R).

1. a([gamma]) = d([gamma]).

2. [gamma] = [sigma] [omicron] [[sigma].sub.[gamma]], where [sigma] denotes the reflection in the imaginary axes, i.e., [sigma](z) = -[bar.z] and [[sigma].sub.[gamma]] is the reflection in [[summation].sub.[gamma]].

3. [[summation].sub.[gamma]] is the bisector of the geodesic linking ti and [[gamma].sup.-1](ti), for all t > 0.

4. There exists 0 < [t.sub.0] [not equal to] 1 such that [[summation].sub.[gamma]] is the bisector of the geodesic segment linking [t.sub.0]i and [[gamma].sup.-1]([t.sub.0]i).

Proof. We first prove that 1 implies 2. Suppose that a([gamma]) = d([gamma]) and suppose first that c([gamma]) = 0. Without loss of generality, we may take a([gamma]) = d([gamma]) = 1. By Proposition 3.1, [[summation].sub.[gamma]] is the line given by the equation x = -[b/2] and thus [[sigma].sub.[gamma]](z) = [sigma](z + b([gamma])) = [sigma]([gamma](z)). If c([gamma]) [not equal to] 0, by Proposition 3.2, we have that [[summation].sub.[gamma]] = [ISO.sub.[gamma]]. Hence the reflection [[sigma].sub.[gamma]] in [[summation].sub.[gamma]] is given by [[sigma].sub.[gamma]](z) = [P.sub.[gamma]] -[ ([[absolute value of c].sup.2][sigma] (z - [P.sub.[gamma]])).sup.-1] = [sigma]([gamma](z)). In either case, we have that [gamma] = [sigma] [omicron] [[sigma].sub.[gamma]].

Suppose now that [gamma] = [sigma] [omicron] [[sigma].sub.[gamma]] and let u [member of] [[summation].sub.[gamma]]. Then [rho](u, [[gamma].sup.-1](ti)) = [rho](u, [[sigma].sub.[gamma]] [omicron] [sigma](ti)) = [rho](u, [[sigma].sub.[gamma]](ti)) = [rho]([[sigma].sub.[gamma]](u), ti) = [rho](u, ti) and hence [[summation].sub.[gamma]] is the bisector of the geodesic linking ti and [[gamma].sup.-1](ti). This proves that 2 implies 3. Obviously 3 implies 4.

We now prove that 4 item implies 1. Let u [member of] [[summation].sub.[gamma]]. Then we have that [rho](u, [t.sub.0]i) = [rho](u, [[gamma].sup.-1] ([t.sub.0]i)) and hence [rho](u, [t.sub.0]i) = [rho]([gamma](u), [t.sub.0]i). Since [gamma] is a Mobius transformation we have that Im ([gamma](z)) = [absolute value of [gamma]'(z)]Im(z). Using this and the explicit formula of the hyperbolic distance in the upper half-plane model (see Section 2), we obtain that [absolute value of [gamma]'(u)][[absolute value of [t.sub.0]i - u].sup.2] = [[absolute value of [t.sub.0]i - [gamma](u)].sup.2]. It follows that Re[(u).sup.2][absolute value of [gamma]'(u)] - Re[([gamma](u)).sup.2] = (1 - [absolute value of [gamma]'(u)])[t.sup.2.sub.0] + ([absolute value of [gamma]'(u)] - 1)[absolute value of [gamma]'(u)]Im[(u).sup.2]. We may write this as an equation of the type [alpha][t.sup.2] = [beta] having t = [t.sub.0] as a solution. However as u [member of] [[summation].sub.[gamma]], by definition [rho](u, i) = [rho](u, [[gamma].sup.-1](i)) and hence also t = 1 is also solution of the given equation. Thus we have that [alpha] = [beta] and [alpha]([t.sup.2.sub.0] - 1) = 0. It follows that [alpha] = 0 and thus [absolute value of [gamma]' (u)] = 1, for all u [member of] [[summation].sub.[gamma]], i.e. [[summation].sub.[gamma]] = [ISO.sub.[gamma]]. Applying Proposition 3.2, we obtain that a([gamma]) = d([gamma]).

Recall that one says that an angle [alpha] is a submultiple of an angle [beta] if either there is a positive integer n such that n[alpha] = [beta] or [alpha] = 0.

Lemma 4.3. Let [GAMMA] be a cofinite discrete subgroup of [PSL.sub.2](R). Suppose that i [member of] [H.sup.2] has trivial stabilizer and let F be its Dirichlet fundamental polygon with center i. Let [[gamma].sub.k] be the side-pairing transformations of F for 1 [less than or equal to] k [less than or equal to] n. If, for every 1 [less than or equal to] k [less than or equal to] n, a([[gamma].sub.k]) = d([[gamma].sub.k]), then [GAMMA] is the subgroup of the orientation-preserving isometries of a discrete reflection group.

Proof. First note that, as a([[gamma].sub.k]) = d([[gamma].sub.k]), we have by Lemma 3.2 that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. This means that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] has the same radius as [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and their centers are the same in absolute value, but have opposite sign, and this for every 1 [less than or equal to] k [less than or equal to] n. Hence F is symmetric with respect to the imaginary axis [summation]. Consider the polygon P, whose sides are [summation] and the [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. We claim that all the dihedral angles of P are submultiples of [pi]. First we prove this statement for the dihedral angles between two sides of F. For this, consider a vertex [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. By the symmetry of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is a finite sequence of sides of F determined by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], according to the definition of [18, Chapter 6.8]. By [18, Theorem 6.8.7] the dihedral angle at the vertex [E.sub.k] is a submultiple of [pi] and this is true for every 1 [less than or equal to] k [less than or equal to] n. Let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] be the side of F with [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is a vertex of P and consider the angle [theta] between [summation] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. If [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], then the side [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] has the same radius but [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Hence, if [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] intersects the imaginary axis [summation], then so does [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [theta] is half the the angle between [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], which is a submultiple of [pi] by the previous. If [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], then [summation] is perpendicular to [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Thus in both cases the angle between [summation] and the adjacent side in P is a submultiple of [pi]. This proves the claim. Finally, by [18, Theorem 7.1.3], the group [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], where [[sigma].sub.k] denotes the reflection in [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], is a discrete reflection group with respect to P. The result then follows by Lemma 4.2.

We are now ready to prove Theorem 1.1.

Proof of Theorem 1.1. Let F be a DF domain, in [H.sup.2] or [H.sup.3] respectively, for [GAMMA] with center [P.sub.0] [member of] {i, j}. Let [[PHI].sub.0] be a set of side-pairing transformations. Consider [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] with c [not equal to] 0. As F is a Ford domain, F [intersection] [[gamma].sup.-1](F) [subset or equal to] [ISO.sub.[gamma]]. As F is a also a Dirichlet domain, F [intersection] [[gamma].sup.-1](F) [subset or equal to] [[summation].sub.[gamma]]. Thus [ISO.sub.[gamma]] = [[summation].sub.[gamma]] and, as [ISO.sub.[gamma]] is a sphere, [[summation].sub.[gamma]] is a sphere and hence [[absolute value of a].sup.2] + [[absolute value of c].sup.2] [not equal to] 1. Thus, by Proposition 3.2, d = [bar.a]. We now consider the case when c = 0. Then, as F is a Ford domain, F [intersection] y(F) is a plane coming from the convex fundamental polyhedron of [[GAMMA].sub.[infinity]]. As F is a also a Dirichlet domain, EY is a plane and hence [[absolute value of a].sup.2] = [[absolute value of a].sup.2] + [[absolute value of c].sup.2] = 1 and thus the second item of Proposition 3.2 allows to conclude.

We now prove the converse. Let F be a Dirichlet or Ford fundamental domain and let [[PHI].sub.0] be a set of side-pairing transformations, such that for every [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Suppose first that c [not equal to] 0, i.e. the isometric sphere associated to y exist. We claim that [[absolute value of a].sup.2] + [[absolute value of c].sup.2] = 1. By contradiction, suppose the contrary. As d = [bar.a] and det([gamma]) = 1, we have that b = -[bar.c]. Hence [[parallel][gamma][parallel].sup.2] = 2[[absolute value of a].sup.2] + 2[[absolute value of c].sup.2] = 2. By [4, Theorem 2.5.1], [gamma] [member of] [PSU.sub.2](C) which is in contradiction with the fact that the stabilizer of [P.sub.0] is trivial. Hence by Proposition 3.2, [ISO.sub.[gamma]] = [[sumamtion].sub.[gamma]]. Suppose now that c = 0. Then F [intersection] [gamma](F) is a plane coming from the convex fundamental polyhedron of [[GAMMA].sup.[infinity]]. The facts that d = [bar.a] and det([gamma]) = 1 imply that [[absolute value of a].sup.2] = 1 and thus [[summation].sub.[gamma]] is a plane given by the equation Re(a[bar.b]z) = -[[absolute value of b].sup.2]/2. By choosing the fundamental polyhedron of [[GAMMA].sub.[infinity]] well, one of its sides coincides with [[summation].sub.[gamma]].

To prove the last part of Theorem 1.1, suppose that [GAMMA] is Fuchsian. By Lemma 4.3 we have that [??] = <[sigma], [GAMMA]> is a reflection group containing [GAMMA] as a subgroup of index 2. Consider finally the group [??] := <[tau], [GAMMA]>. It is clear that [[tau].sup.2] = 1 and by computation [([tau][gamma]).sup.2] = 1 for all [gamma] [member of] [[PHI].sub.0]. Moreover, it is easy to compute that ([tau][gamma])([tau][gamma]') has order at least 2. It thus follows that [??] is a Coxeter group with [[??]: [GAMMA]] = 2.

Note that a presentation of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] can be obtained using [10, Theorem II.7.5]. Also this result simplifies a lot the proof of [14, Theorem 3.1], i.e. it easily follows that the orbifold of [GAMMA] is a punctured sphere in the Fuchsian case. Moreover, as is shown by the next corollary, [14, Theorem 7.3] follows easily from Theorem 1.1.

Corollary 4.4. Let [GAMMA] < [PSL.sub.2] (C) be a cofinite discrete group and suppose [GAMMA] admits a DF domain F. Then, for every side-pairing transformation [gamma], tr([gamma]) [member of] R and the vertical planes bisecting [[summation].sub.[gamma]] and [[summation].sub.[gamma]-1] (for [gamma] [not member of] [[GAMMA].sub.[infinity]]) all intersect in a vertical axis.

Proof. Without loss of generality, we may assume that [GAMMA] admits a DF domain with center j (see the beginning of the section). That tr([gamma]) [member of] R, for [gamma] a side-pairing transformation, is a direct consequence of Theorem 1.1 or of Proposition 3.2.

If [gamma] [not member of] [[GAMMA].sub.[infinity]], [[summation].sub.[gamma]] and [[summation].sub.[gamma]-1] are Euclidean spheres with center - [bar.a([gamma])]c([gamma]) and a([gamma])/c([gamma]) respectively. A simple computation then shows that the Euclidean bisector of these two points contains the point 0 and hence the vertical plane bisecting [[summation].sub.[gamma]] and [[summation].sub.[gamma]-1] contains the point j. Hence all these vertical planes intersect in a vertical line through j.

We now consider when a fundamental domain is a double Dirichlet domain. The next two corollaries of our main Theorem give an alternative way to [14, Section 4] to treat such domains.

Corollary 4.5. Let [GAMMA] bea cofinite Fuchsian group with trivial stabilizer of i [member of] [H.sup.2]. Then the following properties are equivalent.

1. [GAMMA] is the subgroup of orientation-preserving isometries of a Fuchsian reflection group containing the reflection in the imaginary axis.

2. [GAMMA] has a DF domain with center i.

3. [GAMMA] has a Dirichlet fundamental domain F with center i such that, for every side-pairing transformation [gamma], a(Y) = d([gamma]).

4. [GAMMA] has a Dirichlet fundamental domain F with i and [t.sub.0]i as centers, for some 1 [not equal to] [t.sub.0] > 0.

5. [GAMMA] has a Dirichlet fundamental domain F such that all the points of the geodesic through i and ti, for t > 0, are centers of F.

Proof. Theorem 1.1 shows that 2 and 3 are equivalent. The equivalence of 3,4 and 5 is given by Lemma 4.2. Moreover, by Lemma 4.3, 3 implies 1. We show that 1 implies 4. Fix a polygon P for the reflection group such that one of the sides is the imaginary axis [summation] and denote the reflection in [summation] by [sigma]. Let [[summation].sub.k] be a side of P. Denote by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] the reflection in [[summation].sub.k] and let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Then the result follows from Lemma 4.2.

The previous result can be generalized, by conjugating the group [GAMMA]. We then get the following result, where the Dirichlet fundamental domain has arbitrary center P [member of] [H.sup.2]. However, in that case, the third item has to be dropped. We also regrouped parts 4 and 5.

Corollary 4.6. Let [GAMMA] bea cofinite Fuchsian group with trivial stabilizer of P [member of] [H.sup.2]. The following properties are equivalent.

1. [GAMMA] is the subgroup of orientation-preserving isometries of a Fuchsian reflection group containing the reflection in the vertical line through P.

2. [GAMMA] has a DF domain with center P.

3. [GAMMA] has a Dirichlet fundamental domain F such that all the points of the geodesic through P and P + i are centers of F.

From Corollary 4.4, it follows that that all examples given in Section VII.3 in the book of Elstrodt, Grunewald and Mennicke [10] are groups whose Ford domain is also a Dirichlet domain. Note that this does not follow immediately from the results of [14].

Hence an interesting question is to analyse when the Bianchi groups have a DF domain. This question can be linked to the following result by Belolipetsky and Mcleod [5, Theorem 2.1]: for the ring of integers O in Q([square root of - d]) (with d a positive square free integer), the Bianchi group [PSL.sub.2](O) extended by two reflections is a reflection group if and only if d [less than or equal to] 19 and d [not equal to] 14,17. To make a link to Belilopetsky's and Mcleod's result, we next show the following lemma.

Lemma 4.7. Let d be a positive square free integer and let O be the ring of integers of Q([square root of -d]). Assume [not equal to] 1,3 and let [GAMMA] denote the Bianchi Group [PSL.sub.2]((9). Denote by [[sigma].sub.x] and [[sigma].sub.y] the reflections in the hyperplanes x = 0 and y = 0 in [H.sup.3], i.e. for P = z + rj [member of] [H.sup.3], [[sigma].sub.x](P) = -[bar.P] and [[sigma].sub.y](P) = [bar.z] + rj. Suppose that [GAMMA] has a DF domain and, moreover, that here is a set of side-pairing transformations [gamma] for [GAMMA] that have lower left matrix entry real or purely imaginary. Then <[GAMMA], [[sigma].sub.x], [[sigma].sub.y]> is a reflection group.

Proof. First note that the side-pairing transformations of a Dirichlet domain of center j in the case of a Bianchi group are not uniquely determined, as the group has a non-trivial stabilizer of j, namely [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Nevertheless, if the Bianchi group [GAMMA] has a DF domain, it is possible to choose the side-pairing transformations in such a way that the isometric sphere equals the bisector, i.e. d = [bar.a] by Proposition 3.2. Moreover, in this lemma we suppose the matrix entry c of each side-pairing transformation to be real or purely imaginary.

Let F be the Dirichlet domain of [GAMMA] as described in [12]. Suppose moreover F is a DF domain and let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] be a side-pairing transformation of F that does not fix to with c [member of] R or c [member of] iR. We first note that F is symmetric with respect to the hyperplane x = 0. Indeed if [gamma] [member of] [GAMMA]\[[GAMMA].sub.[infinity]], then [ISO.sub.[gamma]] and [ISO.sub.[gamma]-1] have the same radius and [[sigma].sub.x]([P.sub.[gamma]]) = [P.sub.[gamma]-1]. Moreover, if [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] Too then also [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Thus we also have symmetry with respect to the hyperplane y = 0. Let [[sigma].sub.[gamma]] be the reflection in the isometric sphere [ISO.sub.[gamma]] of [gamma]. Note that [ISO.sub.[gamma]] has center -[bar.a]/c and radius 1/[absolute value of c]. We compute [gamma][[sigma].sub.[gamma]](P) for P = z + rj [member of] [H.sup.3].

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Suppose now that c [member of] IR or c G [member of] iR. Then [gamma][[sigma].sub.[gamma]]{P) = -[bar.P] or [gamma][[sigma].sub.[gamma]]{P) = [bar.z] + rj and hence [gamma] = [[sigma].sub.x][[sigma].sub.[gamma]] or [gamma] = [[sigma].sub.y][[sigma].sub.[gamma]]. We now have to distinguish two cases. Suppose first that d = 1, 2 mod 4. Then, by [12, Lemma 4.9], the vertical hyperplanes of F are given by x = [+ or -] 1/2 and y = [+ or -] [square root of d]/2. Define

[??] = F [intersection] {z + rj | Re(z) [greater than or equal to] 0, Im(z) [greater than or equal to] 0}.

Then the hyperplanes and spheres defining the border of [??] are the four hyperplanes given by x = 0, x = 1/2, y = 0 and y = [square root of 2]/2, the spheres [ISO.sub.[gamma]] having center in {z + rj | Re(z) [greater than or equal to] 0, Im(z) [greater than or equal to] 0} and the unit sphere having center in 0. This sphere comes from the unique stabilizer of j, given by the matrix [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. By the above, [[sigma].sub.x][gamma] or [[sigma].sub.y][gamma] is the reflection in [ISO.sub.[gamma]] if [gamma] does not fix to. If [gamma] fixes infinity then it is easy to see that [[sigma].sub.x][gamma] or [[sigma].sub.y][gamma] gives reflection in the vertcial hyperplanes. If [gamma](j) = j, then [[sigma].sub.x][gamma] = [[sigma].sub.y][gamma] = [gamma]. As F is symmetric with respect to x = 0 and y = 0, [??] has finite volume. We now show that all dihedral angles are submutliples of [pi]. Similar as in the proof of Lemma 4.3, by the symmetry of F and [18, Theorem 6.8.7], the dihedral angles of [??] are all submultiples of [pi]. Hence, by [18, Theorem 7.1.3], <[GAMMA], [[sigma].sub.x], [[sigma].sub.y]> is a reflection group with respect to [??].

Suppose now that d [equivalent to] 3 mod 4. Then, again by Lemma 4.3, the vertical hyperplanes of F form a hexagon. This is not suited for a reflection polyhedron. Therefore we define [??] in the following way.

[??] = (f [union] [theta](F)) [intersection] {z + rj | 0 [less than or equal to] Re(z) [less than or equal to] 1/2, 0 [less than or equal to] Im(z) [less than or equal to] (d + 1)[square root of d]/4d},

where [theta] is the translation given by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. In this way, the vertical planes forming the border of [??] form a rectangle. Again, similar as above, the dihedral angles of [??] are all submultiples of n and hence <[GAMMA], [[sigma].sub.x], [[sigma].sub.y]> is a reflection group with respect to [??].

Using Aurel Page's package KleinianGroups [1] and the algorithm described in [12], we determine the side-pairing transformations for the Dirichlet fundamental domain with center j for the Bianchi groups with d [less than or equal to] 19. This yields the following result.

Lemma 4.8. Let d and O be as in Lemma 4.7. The Bianchi group [PSL.sup.2](O) has a DF domain if and only if d [member of] {1, 2, 3, 5, 6, 7, 11, 15, 19}.

Combining Lemma 4.8 and Lemma 4.7, we thus get a part of [5, Theorem 2.1]. Indeed for all the values of d stated in Lemma 4.8, the condition that the matrix entry c is real or purely imaginary is fulfilled and hence for these d the Bianchi group is a subgroup of a reflection group. Note that we excluded d = 1 and d = 3 from Lemma 4.7 because in that case the stabilizer of j is more complicated. Nevertheless it can be easily verified, that also in these two cases, the Bianchi group [PSL.sup.2](O) is a subgroup of index 4 of a reflection group. This coincides in fact with a much earlier result of Bianchi. In [6], Bianchi showed already that the Bianchi groups extended by two reflections are reflection groups for d [less than or equal to] 19 and d [not equal to] 14,17. For d = 10 and d = 13, Bianchi states that the group is generated by a so-called improper reflection, see [6, Paragraph 17]. This is reflected in Lemma 4.8, which shows that the Bianchi group does not have a DF domain for these values of d.

Acknowledgment. The authors would like to thank the referee for pointing out the link between their results and [5]. Moreover they thank Aurel Page for its help with the KleinianGroups package.

References

[1] Aurei Page's Homepage, KleinianGroups Magma Package. http://www. normalesup.org/~page/Recherche/Logiciels/logiciels.html.

[2] Martin Deraux's Homepage, Dirichlet domains for the Modular Group. https ://www-fourier.ujf-grenoble.fr/~deraux/java/dirctr/.

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[6] L. Bianchi. Sui gruppi di sostituzioni lineari con coefficienti appartenenti a corpi quadratici immaginari. Math. Ann., 40(3):332-412,1892.

[7] B. H. Bowditch. Geometrical finiteness for hyperbolic groups. J. Funct. Anal., 113(2) :245-317,1993.

[8] M. R. Bridson and A. Haefliger. Metric spaces of non-positive curvature, volume 319 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles ofMathematical Sciences]. Springer-Verlag, Berlin, 1999.

[9] C. Corrales, E. Jespers, G. Leal, and A. del Rio. Presentations of the unit group of an order in a non-split quaternion algebra. Adv. Math., 186(2):498-524, 2004.

[10] J. Elstrodt, F. Grunewald, and J. Mennicke. Groups acting on hyperbolic space. Springer Monographs in Mathematics. Springer-Verlag, Berlin, 1998. Harmonic analysis and number theory.

[11] M. Gromov. Hyperbolic groups. In Essays in group theory, volume 8 of Math. Sci. Res. Inst. Publ., pages 75-263. Springer, New York, 1987.

[12] E. Jespers, S. O. Juriaans, A. Kiefer, A. de A. e Silva, and A. C. Souza Filho. From the Poincare theorem to generators of the unit group of integral group rings of finite groups. Math. Comp., 84(293):1489-1520,2015.

[13] M. Kapovich. Kleinian groups in higher dimensions. In Geometry and dynamics of groups and spaces, volume 265 of Progr. Math., pages 487-564. Birkhauser, Basel, 2008.

[14] G. S. Lakeland. Dirichlet-Ford domains and arithmetic reflection groups. Pacific J. Math., 255(2):417-437,2012.

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[16] A. Pita and A. del Rio. Presentation of the group of units of ZD- In Groups, rings and group rings, volume 248 of Lect. Notes Pure Appl. Math., pages 305-314. Chapman & Hall/CRC, Boca Raton, FL, 2006.

[17] A. Pita, A. del Rio, and M. Ruiz. Groups of units of integral group rings of Kleinian type. Trans. Amer. Math. Soc., 357(8):3215-3237,2005.

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E. Jespers S. O. Juriaans A. Kiefer A. de A. e Silva A. C. Souza Filho

Department of Mathematics, Vrije Universiteit Brussel, Pleinlaan 2,1050 Brussel, Belgium

emails: efjesper@vub.ac.be and akiefer@vub.ac.be

Instituto de Matematica e Estatistica, Universidade de Sao Paulo (IME-USP), Caixa Postal 66281, Sao Paulo, CEP 05315-970--Brasil

email: ostanley@usp.br

Departamento de Matematica Universidade Federal da Paraiba

e-mail: andrade@mat.ufpb.br

Escola de Artes, Ciencias e Humanidades, Universidade de Sao Paulo (EACH-USP), Rua Arlindo Bettio, 1000, Ermelindo Matarazzo, Sao Paulo, CEP 03828-000--Brasil

email: acsouzafilho@usp.br

* The first author is supported by Onderzoeksraad of Vrije Universiteit Brussel and Fonds voor Wetenschappelijk Onderzoek (Flanders), the second by FAPESP-Brazil (while visiting the Vrije Universiteit Brussel), the third by Fonds voor Wetenschappelijk Onderzoek (Flanders)-Belgium (while visiting Universitat Bielefeld) and the fourth by FAPESP and CNPq-Brazil, and the fifth by FAPESP (Fundacao de Amparo a Pesquisa do Estado de Sao Paulo), Proc. 2014/06325-1.

Received by the editors in January 2016--In revised form in April 2016.

Communicated by E. Caprace.

2010 Mathematics Subject Classification: Primary 20H10,30F40; Secondary 51M10,57M60.
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Author:Jespers, E.; Juriaans, S.O.; Kiefer, A.; de A. e Silva, A.; Filho, A.C. Souza
Publication:Bulletin of the Belgian Mathematical Society - Simon Stevin
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Date:Jul 1, 2016
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