#### Abstract

Given a fundamental polyhedron for the action of , a classical
*kleinian* group, acting in -dimensional hyperbolic space, and , a finite index
subgroup of , one obtains a fundamental domain for pasting copies of
by a Schreier process. It also generalizes the side pairing
generating theorem for exact or inexact polyhedra. It is proved
as well that the general Mรถbius group acting in
is transitive on โ-spheresโ. Hence, describing the hyperbolic -planes in the upper half space
model intrinsically, and providing also an alternative proof of the transitive
action on them. Some examples are given in
detail, derived from the classical modular group and the Picard group.

#### 1. Introduction

The action of Fuchsian (and Kleinian) groups in the hyperbolic plane (and space) has shown to be a central theme in several areas in mathematics, and some in physics, in the last one hundred years. In particular, the knowledge of fundamental domains is a powerful tool to understand the geometry and topology of the corresponding Riemann surfaces and hyperbolic 3-manifolds (or orbifolds).

There are several papers describing ways of constructing fundamental domains. Some of the classical constructions are referred to as the Dirichlet and Ford domains, see for instance [1]. Poincarรฉ's theorem is perhaps the most important result of the subject, for a proof see [2]; also to see the usefulness of this result see for example [3]. In the particular case of modular subgroups, Kulkarni [4], using Farey sequences, showed a way to construct fundamental domains for some subgroups of the classical modular group. Also, one of us has constructed Ford domains for the Hecke congruence subgroups, see for instance [5]. There are many other papers on the subject, we just mention another recent work for cyclic subgroups, appearing in [6].

It is quite interesting to place this geometrical studies of fundamental domains in the hyperbolic plane, or space, in a more general setting, say in the -dimensional hyperbolic space. The book of Ratcliffe [7] makes a broad description of the action of discrete groups acting as isometries in . However, in several important results about fundamental polyhedra, he restricts to a particular case, namely, what he calls exact fundamental polyhedron, meaning that the * sides* of the polyhedron are * maximal* in the sense of being the biggest convex subsets in the boundary of the polyhedron, that are contained in hyperbolic -planes, see [7, Chapter 6]. However inexact polyhedra are important in the literature and occur naturally, see for instance [3], see also Examples 7.1 and 7.2.

In this paper we first generalize a fundamental result, we prove that for all convex fundamental polyhedron, exact or inexact, the side pairing generates the group, Theorem 4.1. Ratcliffe [7, Theorem ] only does it, in the exact case. This proof also provides a formal account of the brief analysis about this theorem appearing in Maskit's book [8]. We remark also that the existence of side pairings is part of the definition of fundamental polyhedra in [8]. In this paper, such hypothesis is a consequence of the fact that our domain is a convex fundamental polyhedron, that is, a locally finite convex fundamental domain (Lemma 2.3).

In the process we also generalize a theorem appearing in [1] that says that the action in of the group of Mรถbius transformations is transitive in โspheres,โ where โsphereโ means a codimension one sphere or plane. We actually proof that this group is transitive on โ-spheres,โ where a โ-sphereโ is an affine subspace, or a sphere intersected with an affine subspace of dimension (Theorem 3.3). In particular, we get an intrinsic description of the hyperbolic -planes in the upper half space model and provide an alternative proof (that does not use the hyperboloid model), of the transitive action on them, Corollaries 3.5, 3.6, and 3.7.

Given an index (finite) subgroup of a Fuchsian (or Kleinian) group , and a fundamental polygon for the action of in the hyperbolic plane (or space), it is algebraically natural to construct a fundamental set for by taking copies of . However, the geometry of how to get a connected open set is not so simple. Moreover, even in dimension 2, apparently there is not a comprehensive general formal theorem or algorithm to describe the new domain (see [9โ11]).

One of our main results states that given groups of isometries acting in hyperbolic -space , (finite index ) and a fundamental polyhedron for , one gets a fundamental domain for sewing copies of , according to Schreier rules, Theorem 5.2.

We exhibit also some examples in dimension two and three derived from the two of the most important groups in mathematics, namely, the Picard group and its subgroup the classical modular group. Example 6.4 also appears in [9], though not explained in a complete formal way. The picture in Example 6.1 also appears in [10, page 258]. In general, the process shows that nonconvex domains may arise, see the details in Example 6.3. However, all our examples are not only fundamental domains but fundamental polygons or polyhedra. The last two examples (three dimensional) do not seem to appear elsewhere in the literature, it is quite interesting that these examples are inexact (in the sense of Ratcliffe), though they arise in a fairly natural way. Our motivation for the study of (3 dim.) fundamental domains is derived, in part, because, as Marden [12] writes โโฆprovide โconcreteโ models of 3-manifoldsโฆ.โ Also, in the presence of elliptic elements of the orbifolds defined by them.

#### 2. Preliminaries

We will be working in -dimensional hyperbolic space, specifically, we will use the upper half space model: provided with the riemannian metric defined by

It is well known that the group of isometries consists of the Poincarรฉ's extensions of the group of Mรถbius transformations acting in , that is, any finite composition of reflections in โspheres" (spheres or planes) of codimension one in , see for instance [1, Chapter ]. For the lower dimensional cases the conformal subgroups of isometries are in dimension two and the Poincarรฉ's extensions of in dimension three.

There are several equivalent ways to give a topology to the general Mรถbius group acting in , , so that it becomes a topological group, one of them using the hyperboloid model, see [1, Chapter , Section ]. Hence, it makes sense to call a subgroup of โPoincarรฉ's extensions of โdiscrete or not. Moreover, it can be proved that in all dimensions that discreteness is equivalent to a discontinuous action in , see for instance [7, Theorem ].

*Definition 2.1. * Let be a group of isometries acting in , a fundamental domain for the action of is an open subset of such that (1)any two points are not -equivalent,(2)given , there exist and , such that , where denotes the closure of in ,(3) has zero Lebesgue -dimensional measure.

The fundamental domain is called locally finite if any compact subset in meets only a finite number of images of . The importance of this property is derived from the fact that for this domains, the quotient of under the action of the group is homeomorphic to the corresponding quotient of the domain see [7, pages 237โ239].

It turns out that a locally finite convex fundamental domain for a discrete group of isometries acting in is of the form where are half spaces in , and such that only finitely many of them meet any compact set. For a proof of this fact we refer to Ratcliffe book pages 246-247. For this reason a locally finite convex fundamental domain is called a convex fundamental polyhedron. In contrast with [7, page 247], we will take to be an open set to match our discussion to domains or regions, which are open sets, as in the undergraduate courses.

After these results Ratcliffe's book restricts the discussion to โexactโ convex fundamental polyhedra, that is when any maximal convex subset of the boundary is of the form where means closure of in hyperbolic space. However, our results do not require the exactness condition, and a maximal convex subset in the boundary of , say might not be equal to a set of the form , but perhaps a finite (or even numerable) family of these sets (see, for instance, [1, page 253]), that is, where is finite or and has dimension for each , where means, as in [7], the smallest hyperbolic -plane where lies (see [7, pages 131โ137]).

Thereby, our definition of * side* of will be different to that one of Ratcliffe, as we will take our convex fundamental polyhedra not necessarily * exact*. The motivation of this more general point of view is that there are important examples in the literature of inexact polyhedra, see, for instance, [3].

*Definition 2.2. * A side of is a set of the form
where the hyperbolic plane has dimension .

We remark, in contrast with Maskit's book [8, page 69] that includes the side pairings as part of the definition of fundamental polyhedron, that this property is a consequence of being a convex fundamental polyhedron as proved in the next result.

Lemma 2.3. * Given a convex fundamental polyhedron for a discrete group of isometries acting in , there exists a side pairing, that is for every side there exists a side and an element such that . It is also satisfied the following condition: and . *

* Proof. *If is a side,
forms also part of the boundary of , and since Mรถbius transformations that are Poincarรฉ's extensions of elements in send any -hyperbolic plane into another -hyperbolic plane, see [1, pages 28โ30]. It follows that the dimension of is , hence is a side of . The other statements are straightforward.

#### 3. Hyperbolic -Planes

We will adopt Ratcliffe's definition of hyperbolic -plane, that is, in the hyperboloid model, the intersection of a dimensional time-like vector space of with the hyperboloid see [7, page 70], and in the upper half space the corresponding image under the * canonical* isometry (via the ball model) see [7, page 137].

We introduce some notation, similar to the one used in Beardon's book [1, Chapter ].

A sphere in is a set of the form a plane is defined by where , and . The vector is one of the normals to the plane.

We will call โsphereโ in , to either a sphere or a plane of codimension one in , as above. Corollary 3.7 will provide a more natural definition for a -sphere, though is very useful to proceed as follows.

*Definition 3.1. * A โ-sphereโ in , is the intersection of โspheresโ which are orthogonal two by two and intersect in more that one point.

*Definition 3.2. * A โ-sphereโ is orthogonal to a sphere provided that the โspheresโ defining are orthogonal to .

As in Beardon's book [1] orthogonality of โspheresโ is determined by the normals. Note that in general it is not always possible, in elementary terms to measure orthogonality with the tangent spaces, as there might not be enough space for the dimensions, so one has to use the normals in some cases.

Note that the general Mรถbius group acting in preserves the family of โ-spheres.โ This last follows because the transformations in this family preserve โspheresโ and using the elegant inversive product, it can be proved that orthogonality is also preserved, see [1, pages 20โ30].

Theorem 3.3. * acts transitively on โ-spheres.โ *

*Proof. * Let , be two โ-spheres.โ Using two Mรถbius transformations, one may suppose , . Take for instance reflections in spheres that pass through points in , . We may also apply translations, which are Mรถbius transformations, and suppose, . Then, it follows easily that and are vector spaces of dimension . Moreover, by elementary linear algebra, there is an orthogonal transformation sending the normal vectors (which might be taking orthonormal) defining , to those defining . Thereby, if is such orthogonal transformation, since and , the theorem follows.

We remark that this result generalizes [1, Theorem , page 28].

Lemma 3.4. *Given a โ-sphere,โ such that does not contain , might also be expressed as
** is a affine space of dimension and is a sphere (cod. 1), whose center is in . *

* Proof. * It is enough to prove that the intersection of two spheres, and , can be expressed as , where is a plane and is a sphere whose center is in . The proof of this fact follows from the Pythagorean theorem and the Euclidian cosine rule, by noting that the points in the intersection project always to the same point in the interval , see Figure 1.

It should be clear now, that a convenient way to describe hyperbolic -planes in , , is by the use of hemispheres and hemiplanes of dimension (codimension ), which are orthogonal to .

It follows easily from [7, Theorem , page 137], that if is a hyperbolic -plane in , such that (where denotes the euclidian closure in ), one has that is of the form where , , the collection of vectors is orthonormal, and the th coordinate of the vector is zero, that is lives in .

Observe that by completing to an orthonormal basis of , one gets an orthonormal basis of (with ). And, an easy calculation shows that where denotes the affine space in defined by .

Hence, is the intersection in of planes, all orthonormal among themselves and all orthogonal with . Orthogonality is measured, of course, with the normals. For example, a geodesic line in is the intersection of two orthogonal geodesic planes in . Hence, is a โ-sphereโ orthogonal to .

Note that is transitive in points in , since we may use translations and reflections on spheres in (now of dimension ) to send to any finite point. Thereby, if is now a hyperbolic -plane such that , by picking a suitable such that sends a point of to , one has that where are planes as described in (3.5) above, and is the Poincarรฉ extension of . Consequently, since Mรถbius transformations are bijective, and preserve โspheresโ and also the inversive product (hence orthogonality) (see [1, Chapter , pages 20โ35]): Therefore, is the intersection with of โspheresโ all orthogonal among themselves and all orthogonal with . And it is also a โ-sphereโ intersected with .

So we get a simple proof of the transitivity of the general group of isometries in hyperbolic -planes, different to the one in [7], that uses Lorentzian matrices.

Corollary 3.5. * The group (Poincarรฉ's extensions of ) acts transitively on the family of hyperbolic -planes. *

Corollary 3.6. * A hyperbolic -plane in is a โ-sphereโ orthogonal to intersected with . *

Corollary 3.7. * A hyperbolic -plane in is the intersection with , of either an affine dimensional space orthogonal to , or the intersection of an affine space of dimension orthogonal to , with a sphere whose center is in the intersection of this affine space with . *

#### 4. Side Pairing Generates

The following result extends Theorem , page 253 in Ratcliffe's book [7], since he only proves it for exact polyhedra. Maskit [8] actually gives an intuitive proof, also he assumes the existence of the side pairing as part of the definition of convex polyhedron.

Theorem 4.1. * Let be a group of isometries acting in and a convex fundamental polyhedron for the action of , then the side pairing transformations of generate . Including the case where is inexact. *

We need a connectedness lemma before proving our result.

Lemma 4.2. *Assuming the hypothesis of the Theorem 4.1, if denotes the subset of consisting of the union of the boundaries of , and all its images, that are contained in hyperbolic -planes, , one has that for any , and , there is a path in joining to . *

*Proof. * Take the geodesic joining and , and an open cover of this curve with balls of the form such that each ball intersect finitely many polyhedra, and whenever a polyhedron meets , it contains the point , see Figure 2.

By compactness we may cover by finitely open balls
and it is clear that if we may connect by a path any two points in any one of the sets ,, the lemma follows by transitivity, since the balls intersect in open sets, that is, if two balls intersect, they intersect in points that are not in . Also, it is enough to consider only balls that intersect more than one polyhedron.

Now for a fixed we take the complement of a smaller ball in it, for instance,
where denotes the hyperbolic distance.

We proceed by taking disjoint open tubular neighborhoods, of the same radius, say , of each of the components of . This can be achieved, since there are a finite number of components of in , see Figure 3. A tubular neighborhood of a set in is a set of the form .

Now, let be a component of , note that one may assume by maximality that . Using Corollary 3.5, we may send by an isometry , that is, a Poincarรฉ's extension of a transformation , to be the -hyperbolic plane in , defined by
see Figure 4.

The next step is to cover the set with a finite collection of Euclidian * cylinders* of the form , , where is an open two dimensional disc around the origin and , an open dimensional disc, for each , see Figure 4. Because of the compactness we may assume that these * cylinders* are contained in the image under of the tubular neighborhood of , and that
is an open cover of the set .

Although we are using now the Euclidian metric instead of the hyperbolic, this can be done by taking the image of the hyperbolic boundary of the tubular neighborhood, which is disjoint from . Note that the image under of the open tubular neighborhood is an open set in both metrics (since they define the same topology).

The sets are path connected in the euclidian metric, since we may connect all points, using paths of the form , for sufficiently small , or Euclidian convex combination segments. Thereby, they are also path connected in the hyperbolic metric (again, as both metrics define the same topology).

We can repeat this construction with all the components of . Finally, because of the compactness of the hyperbolic closure of , we may cover by a finite number of hyperbolic balls not meeting , and a finite number of open sets that are the preimages of the euclidian perforated * cylinders* defined above. Since any of these sets is path connected, by transitivity is path connected. Moreover, by hyperbolic convexity is also path connected.

* Proof of the Theorem 4.1. * Call the group generated by side pairing transformations. Given , we may select and , it follows by virtue of the lemma that there exists a path in joining and , such that does not intersect parts of the boundaries of , and its images, which are different to the relative interiors of the sides, in other words, does not meet dimensional * skeleton* in the boundaries of and its images.

Clearly by local finiteness, only intersects a finite number of polyhedra, say

Certainly the curve may pass from two fixed polyhedron say to more than once, however, since
is a compact set, and because and are convex, we may suppose by eliminating parts of , and renaming, that crosses at most once, from one fix polyhedron to another in the collection .

Since this collection is finite, it is clear now that there is finite chain
so that each is in the family (4.5), and shares a side with for all . Finally, if is a side
is a side of , so is a side pairing generator of for all . In particular is a side pairing generator, and , since is a side pairing, inductively .

#### 5. Fundamental Polyhedra for Subgroups

In the following we consider an abstract group with generators , and a subgroup of finite index of . Let be a complete system of representatives of right cosets of in , where each word in the generators is written in a reduced form. A complete set of representatives is called a * Schreier system*, if the initial segment of any word , is also a representative. If a subgroup has a given system of representatives and is a word in , the assignment , where denotes the right coset representative of , is called * a representative function of right cosets*. It is clear that if and only if , also , if and are in the same right coset.

*Assertion 1. *.

* Proof. *It is enough to observe that the right coset and are the same. To prove this, note first that
Also, since , one has that for some , thereby

The next result appears in [13, page 93], however for the sake of completeness we include a proof.

Theorem 5.1. * Let be a group and a proper finite index subgroup of , then there exists a Schreier system of representatives of . *

* Proof. *We construct a Schreier system of representatives of in the following way. We may choose a representative for the coset to be the empty word .

As the subgroup is proper, there exists a generator of such that . We denote by the right coset of and choose this element as the representative. If the index of in is , the proof is over.

Suppose that . We assert that there exists a generator , such that . If for all generators , then one has that all the words of length are in . This last arises, because by virtue of Assertion 1, one has that
where , o . It follows then by induction that all the words of length bigger than are in , thereby has index , contradicting the hypothesis that . If is the right coset , we may choose this last word as a representative of . If the proof is finished, otherwise we proceed inductively obtaining a Schreier system of representatives for .

We apply now these general results to our specific context, that is, a group โand a finite index subgroupโof isometries acting in hyperbolic space, so that the side pairing transformations of a fundamental polyhedron generate .

Theorem 5.2. *Let be a group of isometries acting in , and a convex fundamental polyhedron for the action of . Suppose also, that is a proper finite index subgroup of . Then, there exists a Schreier system of right cosets representatives in for , determined by the side pairing generators, say
**
so that, if
**
one has that is a fundamental domain for . *

*Proof. * The existence of a such a Schreier system determined by the side pairing generators follows from Theorem 4.1. Now, given , there exists and such that . Also, is of the form for an , and . Hence, , and in the subgroup . Thereby, all orbits are represented in . Note that some points that are not interior points of , become interior points of , for instance points in the interior of the side shared by and .

We show now that any two points are not -equivalent. Suppose there exists such that , since is locally finite, given a compact neighborhood of , for instance , intersects a finite number of images of , say by . We may also assume that their closures contain , and that each such transformation is a member in the collection . This might be achieved, because is in the interior of
so a sufficiently small neighborhood intersects only images of of the form .

The same configuration arises for a small neighborhood of , which may be taken as . Call the images of that intersect the neighborhood . As above, we may assume that for all .

Now, for some and , , one has that . This follows because by pulling back the image by , of any one of the polyhedra around , that intersect , say , one gets a polyhedron around intersecting , say , So that , . Since, is a right coset system, and , it follows that , and .

#### 6. Two-Dimensional Examples

To illustrate our results in the Fuchsian case, it is convenient to recall the famous fundamental (Dirichlet, and Ford) domain for the classical modular group , that is, see Figure 5. A proof of this fact appears, for instance, in [1, pages 229-230].

Of course, the sides are the geodesic arcs and the vertices the points , , and . The transformations pairings clearly are As a consequence of Theorem 4.1, the transformations and generate .

Our 2-dimensional examples will be taken to be subgroups of the classical modular group which we will denote by (either or ). We will denote by the same symbol a modular transformation, or the corresponding matrix.

*Example 6.1. *Let be the subgroup generated by the squares of . There is another description of . We recall that has a presentation given by
Given a word in under this presentation, say , , one defines the exponent sum in the obvious way, for instance has exponent sum . Hence, one may define the parity of a word (reduced or not) to be even or odd, if the exponent sum is even or odd. On the other hand, an element in may be described by two different words, however one word is obtained from the other, by inserting or deleting a relator, or by inserting (or deleting) a symbol of the form or (see [13, pages 12โ15]). It follows then, since has even parity, that the parity of an element in is well defined. Under this definition, it is evident that all the elements in have even parity. To prove that both descriptions are the same, we use the following fact.

*Assertion 2. *.

* Proof. * Certainly, the union is disjoint, since the words in have odd parity. To prove the equality we use induction on the length of any element
where or , and the word is reduced.

If , one has that
this follows because implies . Also

Assuming the assertion is true for the words of length . Let , be a reduced word of length . If , the assertion follows easily from the case . Otherwise, , and by the induction hypothesis, it is of the form , where is an element in . Finally, we just have to check for the case , this follows because .

Consequently, consists exactly in all the words in the modular group which have even parity. In this example is clear from the assertion that the pair is a Schreier system. Hence, it follows from Theorem 5.2 that a fundamental domain for is given by see Figure 6. It also follows from Theorem 4.1 that is generated by and .

It is an interesting fact that is the only subgroup, of index two, of the classical modular group cf. [10, page 366]. We use the fact that for any convex fundamental polygon for a Fuchsian group, the angle sum of a cycle is where ord is the order of the cycle (cf. [1, pages 221-222]). It then follows that has two elliptic classes of order three represented in by and .

It is well known that every Fuchsian group acting in defines a Riemann surface where denotes the orbit of , cf. [1, Chapter 6 and pages 206โ210]. Moreover, one may include the cusps, and get a compact Riemann surface, see [14, pages 17โ20]. We recall that the Euler characteristic of denoted by is given by where denotes the genus. Also, there is a result in basic algebraic topology which says that given an spherical complex obtained from points by adjoining cells of dimension , , one has that cf. [15, page 101].

In the case of , there is one cell, two sides, and three vertices (including the cusp). It follows then from (6.11) that the genus of the corresponding Riemann surface is zero, thereby the surface is a sphere with one cusp and two marked points of order three.

*Example 6.2. * We recall that the Hecke congruence subgroups are defined by
It is well known that these subgroups (which are not normal) have index
see for instance [16, page 79].

We construct a fundamental domain for . It follows from (6.14) that the index is equal to three. It is no difficult to check that a Schreier system of representatives for is given by

Thereby, it follows from Theorem 5.2 that is a fundamental domain for . One easily obtains that the region appears as in Figure 7. To see this, it is enough to locate . This can be done by noting that the image under of the boundary of is contained in the geodesics , and , see Figure 8.

We refer to Figure 7 to obtain the pairing transformations. Clearly is a generator. Note that , and as above . Therefore, the parabolic transformation pairs the sides and . An easy calculation shows that this transformation belongs to the subgroup. Observe also that Thereby, the elliptic transformation of order two pairs and . A calculation shows that this transformation belongs to . It follows then from Theorem 4.1 that is a set of generators for . The corresponding Riemann surface has genus zero, since the number of cells is one, there are three sides and four vertices. Also, there are two cusps and a marked point of order two. Note that the cycle of vertices associated to is accidental, since the angle sum is , cf. [1, page 221].

*Example 6.3. *In a similar way to the previous example, we construct a fundamental domain for . Using (6.14), one has that , hence we have to find a Schreier system with 4 elements, we may take the identity and as the first two. Since does not belong to the subgroup, one checks that it may be taken as another representative of the system.

The proof of the Schreier theorem suggests that we can take either one of the two representatives , or , as the last representative, indeed a calculation show that both elections yield a Schreier system. However, in order to get a convex region we choose and choose our Schreier system to be

Theorem 5.2 asserts that the fundamental domain for is We claim that this region is the one described in Figure 9.

To show this we just have to find the image of under . That is, one has to locate the image under of the hyperbolic triangle define by the geodesics joining and with , and the one joining and . One gets, that such triangle is determined by the geodesics joining and with zero, and the geodesic joining and , see Figure 10.

One of the obvious generators for is which pairs and . To find the parabolic pairing of and , using , and , one gets Moreover, a calculation shows that , thereby this map pairs and (see Figures 9 and 10).

Finally, the pairing of the sides and is a hyperbolic rotation of order three conjugate to . Indeed, note that where as in Figure 9. Hence, since the rotation around sends to , one gets A calculation also shows that One concludes that a set of generators for is given by

It also follows that has two nonequivalent cusps, one elliptic class of order three and none of order two. Our domain has also an accidental cycle, that is, of vertices which are not fixed points, of course, the angle sum is . Note also that the domain is not exact. Using again Euler's formula, the corresponding Riemann surface has genus zero, because there is one cell, the number of sides is three, and there are four vertices. See Figure 11.

We remark that in the last two examples a simpler exact polygon construction with no accidental vertices may be derived using the Ford domains cf. [5]. We conclude the 2-dimensional examples with a domain for the principal congruence subgroup .

*Example 6.4. *We obtain a fundamental domain for . The index of with respect to the modular group is six, see for instance [14]. We may choose and as the first two representatives of our system. Since our next representative has to be . We choose as the fourth representative in our Schreier system. A calculation show that this last element does not belong to the subgroup and does not belong to the same right coset of the previous ones. We take as the next element . Clearly, this element does not belong to the subgroup. It is also easily checked that this transformation is not in the same right cosets of the previous ones. Our last choice is , other easy calculations allows to conclude that a Schreier system is given by

It follows then from Theorem 5.2 that a fundamental domain for is the region as it is shown in Figure 12.

To check this we just have to find the polygons and . The first one can be derived from Figure 8. Since , we just have to find the image of under this last transformation. For this, note that is the triangle defined by the geodesics joining and with , and the geodesic joining and zero. Thereby, its image under is the triangle determined by the geodesics joining and to zero, and the geodesic joining and , as described in Figure 12.

To conclude, we find the pairings and describe the corresponding Riemann surface. Clearly pairs and , as in Figure 12. The parabolic transformation that pairs and is given by this follows because , and . Note also that belongs to , since is normal in .

Finally, to find the parabolic transformations that pairs to , one notes that Hence, This transformation belongs to because of the normality.

One concludes from Theorem 4.1 that a set of generators for is given by Note that has an accidental cycle.

The corresponding Riemann surface is sphere with three cusps (Figure 13). This follows because using (6.11) the genus of the surface is zero, since there is one cell, three sides, and four vertices. In these examples, we are considering, of course, the compactified surfaces. We finally remark that that is a free group of rank two, this follows easily from Theorem 4.1, and Poincarรฉ's theorem.

#### 7. Three-Dimensional Examples

For our three dimensional examples, we will use a fundamental domain for the Picard group derived from Poincarรฉ's theorem, that appears in [17, pages 58-59], see Figure 14, we will denote it by .

The Picard group is the set of unimodular matrices in whose entries are gaussian integers, that is, elements of . Alternatively, it is the group of isometries acting in determined by these matrices. We denote the Picard group by . We denote the side pairing generators of by , and , where the action of these transformations, in the extended complex plane, is given by
Note that the transformations ,โโ, and act in hyperbolic space as half rotations around the axes determined by , , and , respectively. We will use the same symbol, for either, the transformation acting in the extended complex plane, or the matrix in that determines it, or the corresponding transformation acting in (Poincarรฉ extension). By looking for cycles of geodesic segments in the boundary of , that is, compositions of side pairings of (strictly speaking, in a three-dimensional context, * face* pairing), that fix pointwise geodesic segments in the boundary of , one gets by virtue of Poincarรฉ's theorem (cf. [8, page 75]) a set of relations of the Picard group, namely,
cf. [17, pages 58-59]. We remark that there is a misprint in the book of Maclachlan and Reid [17], the upper right entry in the matrix should be and not , since the polyhedron stands in the half space .

*Example 7.1. * We construct and index two subgroup by considering the subgroup of squares in . Namely, let
By analyzing the intersections of with , an induction argument of on the reduced words in the generators provides a formal proof that this group is indeed proper.

It turns out that is an index two subgroup of . This follows since
where the union is disjoint. This follows by induction on reduced words on and . For words of length one, it is evident. Assuming the induction hypothesis, given
a reduced word on the generators, one has that either , and the conclusion follows, or . In the latter case one gets subcases, depending if , , , . In the first three subcases . If , since one gets that , similarly if . Since is a proper subgroup, , and the union in (7.4) is disjoint.

Certainly, a Schreier system is given by . Thereby, a fundamental domain for is obtain as in Figure 15. Note that the domain is a convex fundamental polyhedron which is * inexact*. The side (*face*) pairing generators are given by , , , , , .

*Example 7.2. * For our last example, we consider the ideal in determined by the gaussian multiples of , that is, the gaussian numbers of the form:
we denote by , this ideal. Note that , are not elements of , however , since .

Call the subgroup of defined by the matrices
such that . A trivial calculation shows that is indeed a subgroup. We prove now that has index three in . We claim that
is disjoint. These right cosets, indeed, do not intersect, since , , and are determined by matrices of the form
We use again an induction argument on reduced words to prove (7.8).

Since , , and are elements of , any word of length one is, either in , or in . Assuming that any reduced word is an element of , or , or , we prove for reduced words of length , say

There are three cases according as belongs to , or to , or to . In the first one, it is easily seen that , or .

In the second case, , we get subcases for , , , . If , if . For the case , a calculation shows, that at the matrix level,
hence . If , since , . Finally, if , a calculation shows that
thereby .

It remains only to prove the case that is . We take subcases, if . For , a calculation yields
so . For the next subcase, , one gets that
and . If , one calculates
getting . Finally, if
and .

Consequently, it follows from Theorem 4.1 that we may construct a fundamental domain, actually a polyhedron, for the subgroup of the Picard group by gluing properly three copies of the polyhedron , using the Schreier system , as it appears in the Figure 16.

To see this, observe first that the action of in the upper half space is the inversion on the unitary sphere, followed by the inversion on the plane. Note also that the action of the transformations and is the same in the upper half plane , and in the half plane . This follows because both actions are Poincarรฉ's extensions of the same transformations in .

Now, it is clear that the image of under is obtained by a half rotation around the geodesic in , so the new polyhedron shares a side (*face*) with , but now is attached to the origin, see Figure 16. To visualize is less evident. Since, are isometries and preserve geodesic planes, it will be sufficient to find the images of the vertices of the polyhedron under . Call these vertices , , , , and , as in Figure 16. It is clear now that and are the images of and , respectively, under the inversion in the plane. Also, using the fact of the action of in the half plane is the same as in , one gets that is located as appears in Figure 16 (see also Figure 9). Finally, the point described in Figure 16 has coordinates , and a calculation yields . Note, that he images of , , , and are in the plane, determined by the line .

Our fundamental domain obtained for is actually a polyhedron, it has seven sides * faces* and is certainly inexact. The reader is invited to describe the side pairing, this can be done as in our other examples, by conjugations. Note also the amazing coincidence of our Schreier system for , and our two-dimensional example for .

It also turns out, that an index six subgroup of is given by see [18]. Hence, this new group, thought as a transformation group, has index two in .

On the other hand, writing (in the plane, the translation ), by , one may think, comparing with our examples, that the subgroup of the Picard group, obtained by extending the modular group with the translation , has finite index. However, this group has infinite index in the Picard group, since one may construct a fundamental domain for this subgroup of infinite volume, see [3]. It will be interesting to find out, what is the index of the extension of the modular group by the translation , in the Picard group. Recent information of the Picard group, in relation with Jรธrgensen inequality, appears in [19].