The Existence of Quasimeromorphic Mappings in Dimension 3

We prove that a Kleinian group $G$ acting upon $\mathbb{H}^{3}$ admits a non-constant $G$-automorphic function, even if it has torsion elements, provided that the orders of the elliptic (i.e torsion) elements are uniformly bounded. This is accomplished by developing a technique for meshing distinct fat triangulations while preserving fatness. We further show how to adapt the proof to higher dimensions.


Introduction
The object of this article is the study of the existence of G-automorphic quasimeromorphic mappings (in the sense of Martio and Srebro -see [MS1]) f : H n → R n , R n = R n {∞}; i.e. such that (1.1) f (g(x)) = f (x) ; ∀x ∈ H n ; ∀g ∈ G ; were G is Kleinian group acting upon H n . Our principal goal is to prove the following: Theorem 1.1. Let G be a Kleinian group with torsion acting upon H n , n ≥ 3. If the elliptic elements (i.e. torsion elements) of G have uniformly bounded orders, then there exists a non constant G-automorphic quasimeromorphic mapping f : H n → R n .
In this paper we restrict ourselves to the proof of the theorem in the classical case (i.e. n = 3) only. This restriction is motivated by two reasons: (a) the proof in the 3-dimensional case employs mainly elementary tools and (b) it develops and uses a technique for for meshing distinct fat triangulations while preserving fatness, technique that is relevant in Computational Geometry and Mathematical Biology. The proof of the general case is presented in [S1] and it is based upon a more general result concerning the existence of fat triangulations for manifolds with boundarysee [S2].
The question whether quasimeromorphic mappings exist was originally posed by Martio and Srebro in [MS1] ; subsequently in [MS2] they proved the existence of fore-mentioned mappings in the case of co-finite groups i.e. groups such that V ol hyp (H n /G) < ∞ (the important case of geometrically finite groups being thus included). Also, it was later proved by Tukia ([Tu]) that the existence of nonconstant quasimeromorphic mappings (or qm-maps, in short) is assured in the case when G acts torsionless upon H n . Moreover, since for torsionless Kleinian groups Date: 20.9.2003. This paper represents part of the authors Ph.D. Thesis written under the supervision of Prof. Uri Srebro. G, H n /G is a (analytic) manifold, the next natural question to ask is whether there exist qm-maps f : M n → R n ; where M n is an orientable n−manifold. The affirmative answer to this question is due to K.Peltonen (see [Pe]); to be more precise she proved the existence of qm-maps in the case when M n is a connected, orientable C ∞ -Riemannian manifold. In contrast with the above results it was proved by Srebro ([Sr]) that, for any n ≥ 3, there exists a Kleinian group G ⊲< H n s.t. there exists no non-constant, G−automorphic function f : H n → R n . More precisely, if G (as above) contains elliptics of unbounded orders (with non-degenerate fixed set), then G admits no non-constant G−automorphic qm-mappings. Since all the existence results were obtained in constructive manner by using the classical "Alexander trick" (See [Al]), it is only natural that we try to attack the problem using the same method. For this reason we present here in succinct manner Alexander's method: One starts by constructing a suitable triangulation (Euclidian or hyperbolic) of H n or of M n = H n /G. Since H n and M n are orientable, an orientation consistent with the given triangulation (i.e. such that two given n-simplices having a (n − 1)-dimensional face in common will have opposite orientations) can be chosen. Then one quasiconformally maps the simplices of the triangulation into R n in a chess-table manner: the positively oriented ones onto the interior of the standard simplex in R n and the negatively oriented ones onto its exterior. If the dilatations of the qc-maps constructed above are uniformly bounded (as is the case of compact manifolds M n ), then the resulting map will be quasimeromorphic. The dilatations of each of the qc-maps above is dictated by the proportions of the respective simplex (see [Tu] , [MS2]), and since the dilatation is to be uniformly bounded, we are naturally directing our efforts in the construction of a fat triangulation, where: Definition 1.2. A k-simplex τ ⊂ R n (or H n ); 2 ≤ k ≤ (n−1) is f -fat if there exists f ≥ 0 s.t. the ratio r R ≥ f ; where r denotes the radius of the inscribed sphere of τ (inradius) and R denotes the radius of the circumscribed sphere of τ (circumradius). A triangulation (of a submanifold of R n or H n ) T = {σ i } i∈I is f -fat if all its simplices are f-fat. A triangulation T = {σ i } i∈I is fat if there exists f ≥ 0 s.t. all its simplices are f -fat; ∀i ∈ I.
The idea of the proof of Theorem 1.1. is first to build two fat triangulations: T e of a certain closed neighbourhood N e of the fixed set of G in H n ; and T c of H n \ N e ; and then to "mash" the two triangulations into a new triangulation, while retaining their fatness 1 . The first triangulation is constructive and is based upon the geometry of the elliptic transformations. The existence of the second triangulation is assured by Peltonen's result. Unfortunately, these two triangulations are not G-invariant, so they are unsuited for our purpose of building a G-automorphic function. However, they induce fat triangulations: T * e on (N e ∩ H n )/G , and T p on M c = (H n \ N e )/ G, where M c is a differential manifold with boundary ∂M c = (∂N e H n )/ G. Fortunately, we are provided with a ready made method of mashing triangulations, so we can direct our efforts towards the task of "fattening" the simplices of the "intermediate zone"; task which will be carried out in Section 4.
The "mashing" method mentioned above is based on a result and, even more, on the technique used in its proof, due to Munkres: Theorem 1.3. Let M n be a C r -manifold with boundary. Then any C r -triangulation of ∂M n can be extended to a C r -triangulation of M n , 1 ≤ r ≤ ∞.
Because of its importance to our own construction, we shall present the basic idea of the proof of Theorem 1.3. in the next Section. This paper is organized as follows: in Section 2 we present the necessarily background on elliptic transformations and present in a nutshell the main techniques we employ: the Alexander trick, Peltonen's method and the Proof of Munkres' Theorem. In Section 3 we show how to choose and triangulate the closed neighbourhood of the N of the fixed set of G, and how to select the "intermediate zone" where the two different triangulations overlap. Section 4 is dedicated to the main task of fattening the common triangulation. Finally, in Section 5 we indicate the way of adapting our construction to higher dimensions.

Preliminaries
2.1. Elliptic Transformations. We shall restrict ourselves mainly to the 3-dimensional case, for, as we have already stated, this will be the direction in which our main efforts will be directed. Let us first recall the basic definitions and notations: A transformation f ∈ Isom (H n is a hyperbolic line and will be denoted by A(f ) -the axis of f. If A is an axes of an elliptic of order m, then A is called an m − axes. If the discrete group G is acting upon H 3 , then by the discreteness of G, there exists no accumulation point of the elliptic axes in H 3 . Moreover, if G contains no elliptics with intersecting axes, then the distances between the axes are, in general, bounded from bellow. To be more precise, the following holds: GM1]). Let G be a discrete group G acting upon H 3 , and let f, g ∈ G be s.t. ord(f ) ≥ 3 or ord(g) ≥ 3; and s.t. A(f ) ∩ A(g) = ∅. Then ∃ δ > 0, that is independent of G, f, g s.t.
where dist hyp denotes the hyperbolic distance in H 3 .
It is extremely important to notice that the results above do not include the case when all the elliptic transformations of G are of order 2, since they depend intrinsically upon: The theorem above is of little avail in the case of two half-turns (that is elliptics of order 2); since any two half-turns (in H 3 ) generate a discrete group. (See [S3] for a proof of this "folkloric" result from an unpublished paper by Jørgensen.) Indeed, examples of discrete groups of isometries of hyperbolic 3-space can be constructed, such that the distances between the axes of the order 2 elliptics are not bound from bellow (see [S3]).
In the presence of node points (i.e. intersections of axes) the situation is more complicated. Fortunately, there are only a few types of such possible intersections -for the orders of the elliptic axes meeting at a node point must satisfy certain conditions (determined by the Euler number for the orbifold 2 ); namely, the possible local situations in orbifold are: either (a) Dihedral, i.e. of the type (2, 2, n) , n ≥ 2; or one of the following exceptional types: (b) Tetrahedral, i.e. of type (2, 2, 3) ; (c) Octahedral , i.e. of type (2, 3, 4) ; (d) Icosahedral, i.e. of type (2, 3, 5); (see Remark 2.3. The reduced number of possibilities is rather fortunate, for the computation of the distances between node points is more difficult than that of distances between disjoint axes. (For more specific information in this direction of study see [DM] and [Med] 3 ; and more recently [GM1] , [GM1] and [GMMR]).
Since our main interest lies in Kleinian groups acting upon H 3 whose elliptic elements have orders bounded from above, the following theorem is highly relevant: Theorem 2.4 ( [FM]). Let G be finitelly generated Kleinian group acting on H 3 . Then the number of conjugacy classes of elliptic elements is finite.
For a sketch of an alternative proof of this Theorem see Appendix.
Remark 2.5. The Theorem above is not true for groups of isometries of H n , n ≥ 4 ; indeed there exist counterexamples, one due to Mess and Feighn ( [FM]) and another due to Kapovitch and Potyagailo ([KP]). It should be remarked that both of the examples cited above produce (albeit different) conjugacy classes of elliptics of the same order. Considering this and the goal of our investigation, the following recent result is highly relevant: Theorem 2.6 ( [H]). There exists a discontinuous group Γ < Isom(H n ) , n ≥ 4 ; such that: (i) Γ contains elliptics of arbitrary large orders and (ii) V ol(N ε (M Γ )) < ∞ 4 , where M Γ = H Γ /Γ, and H Γ denotes the convex core (in H n ) of the limit set Λ(Γ) ⊂ H n , and N ε represents the ε-neigbourhood of M Γ .
Remark 2.8. The branching set of h is the 1-skeleton of the triangulation.
2.3. Peltonen's Technique. Peltonen's method is an extension of one due to Cairns, developed in order to triangulate C 2 -compact manifolds ( [Ca3]). It is based on the subdivision of the given manifold into a closed cell complex generated by a Dirichlet (Voronoy) type partition whose vertices are the points of a maximal set that satisfy a certain density condition. We give below a sketch of the Peltonen's method, refereing the interested reader to the authoritative [Pe] for the full details. 5 The construction devised by Peltonen consists of two parts: Part 1 This part of the proof proceeds in two steps: Step A We build an exhaustation {E i } of M n , generated by the pair (U i , η i ), where: (1) U i is th relatively compact set E i \ E i−1 and (2) η i is a number that controls the "fatness" of the simplices of the triangulation of E i , that will be constructed in Part 2, such that they don't differ to much on adjacent simplices, i.e.: (i) The sequence (η i ) i≥1 descends to 0 ; (ii) 2η i ≥ η i−1 .
(2) Prove that the Dirichlet complex {γ i } defined by the sets A i is a cell complex and every cell has a finite number of faces (so it can be triangulated in a standard manner).
Part 2 Consider first the dual complex Γ and prove that it is a Euclidian simplicial complex with a "good" density, then project Γ on M n (using the normal map). Finally, prove that the resulting complex can be triangulated by fat simplices.
4 That Γ is almost geometrically finite. 5 A rather detailed exposition of the main steps of the proof one can be find in [S3] 2.4. Munkres' Theorem. The basics steps in the proof 6 of Theorem 1.3. are as follows: a) Prove that you can triangulate a smooth manifold without boundary in the following way: approximate M n locally by a locally finite Euclidian triangulation, by means of the secant map (see [Mun], p. 90). Modify these local triangulations coordinate chart by chart, so they will be P L-compatible wherever they overlap.
To extend the triangulation globally, we work in R n , by using the coordinate charts and maps. Here again we have to approximate the given triangulation by a P Lmap, s.t. the given triangulation and the one we produce will be compatible. b) Triangulate a product neighbourhood P(∂M n ) of ∂M n , P(∂M n ) ⊂ M n , in a standard way, and mash it together with the triangulation of the non-bounded manifold int M , by using the same method as above.
It is important to emphasize that for most of the process the technique sketched above not only preserves the fatness of the simplices, but actually takes care that the said fatness will occur.

Constructing and Intersecting Triangulations
3.1. Geometric Neigbourhoods. If there are no elliptics with intersecting axes, a standard choice for a regular neighbourhood of an m-axes will be -for obvious geometric reasons -a doubly-infinite regular hyperbolic m-prism (henceforth called a geometric neighbourhood), and a fundamental domain will be a prismatic "slice", i.e. a fundamental region for the action of C m on {m} × A(f ), where {m} denotes the regular hyperbolic polygon with m sides and C m denotes the cyclic group of order m, i.e. the rotation group of {m}. In order to triangulate the geometric neighbourhood, we divide it in a finite number of radial strata of equal width ̺ = δ/κ 0 , and further partition it into "slabs" of equal hight h. Each prismatic fundamental region thus obtained naturally decomposes into three congruent tetrahedra, generating a C m -invariant triangulation of the geometric neigbourhood. (See Fig. 2 for a representation in the ball model of H 3 in the case m = 4.) The fatness of the triangulation of the geometric neighbourhood thus depends upon δ, κ 0 and h, enabling one to control the initial fatness of the geometric triangulation by means of the parameters ̺ and h.
Remark 3.1. One can easily modify the construction above and produce instead of a C m -invariant triangulation, a D m -invariant one, where D m denotes the dihedral group of order m, i.e. the full-symmetry group of {m}, thus allowing one to consider groups that contain orientation-reversing isometries of H 3 .
For the choice of geometric neigbourhoods for the node points, the natural choice is that of an Archimedean solid which is a natural carrier of the symmetry group of the desired type: D n , T, O or I (or rather for its spherical counterpart -see [Cox]).
3.2. Mashing Triangulations. We start by showing first how to construct the desired fat triangulation and how to produced the quasimeromorphic mapping ensuing from it in the basic case of groups who's elliptic elements axes do not intersect. Moreover, let us presume here that there exist a least an elliptic element of order ≥ 3. Since G is a discrete group, G is countable so we can write G = {g j } j≥1 and The steps in building the fat desired fat triangulation are as follows: (2) Consider the following quotients: The lifting of f * to a mapping f : H 3 → R 3 produces the desired G-invariant quasimeromorphic mapping.
Remark 3.2. The choice of "δ/4" instead of "δ" in the definition of the geometric neighbourhoods N i is dictated by the following Lemma: Rat]) Let X be a metric space, and let Γ < Isom(X) be a discontinuous group. Then, for any x ∈ X and any r ∈ (0, δ/4): where: Γ x is the stabilizer of x, π denotes the natural projection, δ := d(x, Γ(x)\{x}), and where the metric on X/Γ is given by Note Instead of the triangulation scheme presented above, scheme that follows closely the Proof of Theorem 1.3. , we could have used in this case the natural triangulation of geometric neigbourhoods in H 3 , to devise a simpler method for mashing triangulations, as follows: (1) Consider again the geometric neighbourhood with its natural fat triangulation.
(2) Replace the neighbourhoods N i = N i,1/4 by the neighbourhoods , and triangulate N ′ i in such a manner that the simplices of the triangulation of ∂N ′ i are also simplices of the triangulation of int N i . (3) Consider instead of N e the following manifold: The role of the triangulation T p is played by T ′ p , which consists of the simplices produced by Peltonen's method and those simplices resulting from those of the geometric triangulation of T i = N i,1/4 \N i,3/16 . (6) The desired triangulation T ′ * are composed of those of N i,1/4 , those of the original M c and those obtained by mashing the two triangulations of the tubes T i .
Remark 3.4. The second construction, besides being more simple and geometrically intuitive, reduces more rapidly the original problem to that of mashing and uniformly fattening two locally finite Euclidian triangulations.
In the case when there exist intersecting elliptic axes, the following modification of our construction is required: instead of δ one has to consider δ * = min (δ, δ 0 ), where δ 0 represents the minimal distance between node-points. 7 We still have to deal with the case when all the elliptic transformations are half-turns, since, as we have seen, no minimal distance between the axes can be computed in this case. However, by the discreetness of G it follows that there is no accumulation point of the axes in H 3 . Let D = {d ij | d ij = dist hyp (A i , A j )} denote the set of mutual distances between the axes of the elliptic elements of G. Then, since G is countable, so will be D, thus D = {d k } k≥1 . Then the set of The fatness of the simplices of the geometric triangulation of A G can be controlled, as before, by a proper choice of h and ̺.
Remark 3.5. The existence of N ♮ e is easy to justify geometrically if one uses the upper-half space model of H 3 : up to conjugation one can choose ∞ to be an accumulation point for A G , therefore the axes accumulated at this point are represented as parallel Euclidian half-lines, perpendicular to the plane R 2 . Consider a family of disjoint Euclidian cylinders i represent two parallel generators of C i , let h i be the hyperbola of vertex p i and asymptotes l ± i , and let H i the hyperboloid of rotation with axes A i and generatrix l + i . Then {int H i | i ≥ 1} represents a proper geometric neighbourhood for the set of axes accumulating at ∞.

Fattening Triangulations
4.1. Preliminaries. We have seen that we reduced the problem to that of "fattening" the intersection of two 3-dimensional finite, fat Euclidian triangulations. We do this piecemeal, first fattening the 2-simplices, then the 3-simplices. It is natural to do so, for the following holds: Pe]). If an n-dimensional simplex is fat, than all its k-dimensional faces, 2 ≤ k ≤ n − 1 are fat.
In particular, in order that an n-dimensional simplex be fat, its 2-simplices have to be fat. Note that for triangles, with angles α , β , γ , and r and R as above, the conditions: r/R ≥ f and min {α , β , γ} ≥ ϕ, where ϕ = ϕ(f ) is an angle depending on f , are equivalent. Thus we start "fattening" 2-simplices, by ensuring that where the minimum is taken over all the triangles of the resulting triangulation, and where ϕ 0 is the minimal angle of the original triangulations -ensured by their uniform fatness. From now on, let S = {s i } i∈I and Σ = {σ j } j∈J stand for the simplices of the triangulations of T i and M ′ c , respectively. 8 Since the intersections of tetrahedra can be rather unruly, we simplify the situation by requiring that the simplicies of one of the triangulations be much smaller than those of the other: (4.2) diam s i ≤ 1 10 k0 diam σ i ; ∀i ∈ I, ∀j ∈ J ; were k 0 is to be determined later. By using general position arguments 9 (see [Hu] , [Mun] ), the relative positions of the s i 's and the σ j 's are now reduced to the following relevant possibilities: (a) but we are not in one of the previous cases (see Fig. 3).

4.2.
Fattening 2-dimensional Triangulations. We start our triangulation "fattening" process by dealing with the 2-dimensional case first: Let σ j0 ∈ Σ be such that there exists a regular neighbourhood N j0 of σ j , triangulated by elements of S = {σ i }. Now S is partitioned by σ j into three disjoint families S 0,1 , S 0,2 , S 0,3 , where:  It is easy to assure -by eventual further subdivision and ε-moves 10 -that S 0,3 ∩ S 1,3 = ∅ ; where S 1,3 is the family corresponding to S 0,3 , induced by σ j1 , that is adjacent to σ j0 . The intersections belonging to the family S 0,2 are the principal generators of "unfatness", for ∠(e 0,l , e i,m ) may be arbitrarily small, where e i,m ; m = 1, 2, 3 are the edges of s i (see Fig. 3).  (we always choose the acute angle) (see Fig. 3). Now it is not possible that two consecutive of the angles φ 0,p , φ 0,m , φ 0,n are smaller than φ 0 : indeed, let us suppose that both φ 0,m < φ 0 and φ 0,n < φ 0 . Then φ 0,1 > π − 2φ 0,2 , so φ 0,2 + φ 0,3 < 2φ 0 , and thus either φ 0,2 < φ 0 or φ 0,3 < φ 0 , in contradiction to the fatness of s i . The fact above implies that we obtain two quadrilaterals which contain the "bad" points ν 0,p and ν 0,n in their (respective) interiors, let them be: Q 1 = ν 0,1 ν 0,4 ν 0,5 ν 0,2 and Q 1 = ν 0,2 ν 0,3 ν 0,6 ν 0,1 . We erase the segments ν 0,q ν 0,p , ν 0,2 ν 0,p , . . . , ν 0,n ν 0,r and we replace them with segments that will "fattily" triangulate the quadrilaterals in question. These triangles have "big" angles (since their angles are belonging to fat triangles or are the sum of two such angles). We distinguish between two cases: (a) Q i is convex, and (b) Q i is not convex. We first take care of the simpler case. (a) Let Q be a convex quadrangle such that all its angles are ≥ φ 0 and let − → l i,k be the ray interior to By its very definition this quadrangle has the property that, for any ν * 0,p ∈ int Q we have that: Also: Fig. 5) But one one of the angles ∠A i−1 A i A i+1 and ∠A i A i+2 A i−1 } belongs to one of the original fat triangles, so: So, from (4.1.) and (4.3.) it follows that the triangles △A i ν * 0 A i+1 ; i = 0, . . . , 3 (mod4) are fat. (b) In this case, rather than tracking back our steps through the same argument as in the previous case; we prefer to dissect Q into two triangles and one convex quadrilateral, in the following way: if A 0 A 0 A 0 A 3 is such that ∠A 3 A 0 A 1 > π and such that ν 0,p ∈ A 2 , then consider the bisectors A 0 B 12 of and ∠A 1 A 0 A 2 and A 0 B 23 of ∠A 2 A 0 A 3 . (See Fig. 6.) then the same argument we used for the original triangulation shows that ν 0,p−1 and ν 0,p+1 are ≥ φ 0 . If we can employ either one of the following two methods to remedy the situation: (a) use the general position technique again and bring the new triangulation to the required position; or (b) consider, instead of the bisector A 0 B 23 the meridian A 0 M 23 (M 23 ∈ A 2 A 3 ) -see [S3] for details.
We still have to deal with Case (a) We repeat once more the procedure used for the subdivision of ∂S and F ront S that we employed in Case (b). We divide the edge e 2 into ρ 2 − 1 equal segments and consider the joins (cones) J(v 2 j , ε j+1 ) and J(w ′ k , v 2 k v 2 k+1 ) (see Fig.  9 (a)). Using the same arguments as before one easily checks that the resulting triangulation of s \ S will be fat. Moreover, although this procedure dramatically reduces the "fatness" of the next stratum of simplices of the family Σ, it leaves the other strata unchanged. This procedure takes care of the intersections of Σ with "the front wave" one proceeds along the same lines and then fits the new triangulation S ′ to S in a properly chosen tubular region J 1 , using, instead of σ, triangles s 0 ∈ S ∩ T 1 . We do have yet to contend with the problem posed by "corners" i.e. by triangles s ∈ S such that (i) s ∈ F ront S and (ii) ∂s ∩ ∂S = v (were v is a vertex). The two cases that ensue may be treated with the methods developed before. (See [S3] .) Remark 4.2. We can dispense with theses last considerations altogether by considering the following: we are concerned -in fact -only with patching together triangulations already contained in a δ 0 -neighbourhood of an axis. But the geometric triangulations employed partitioned these neighbourhoods into "levels" or "heights", so one can fit the triangulation of two consecutive levels -say "m" and "m + 1" -by considering the intermediary adjusting patch delimited by the levels "m − 1 2 " and "m + 1 2 ". (A finite number of further barycentric subdivisions may still be required.) This concludes the "fattening" process for the two dimensional triangulations.

4.3.
Fattening 3-dimensional Triangulations. We shall divide the "fattening" process of two intersecting 3-dimensional tetrahedra into two parts: A) The "fattening" of the 2-dimensional intersection between the triangles belonging to ∂S and a face f 123 = △v 1 v 2 v 3 of a tetrahedron σ ∈ Σ , and B) The extension of the new triangulation to a fat 3-dimensional triangulation. 4.3.1. Fattening 2-dimensional intersections. Let us start by noticing that, since the fatness of the tetrahedra s i ∈ S is bounded from below, so will be their dihedral angles 13 . It follows that even after the partition of f 123 into triangles {τ k } and quadrilaterals {η j }, to a triangulation {τ k }, the number of triangles around each vertex will be bounded by a natural number m 1 . We shall exploit this fact to our advantage, so that we will be able to replace {τ k } by a fat triangulation {τ * k } that is "close" to {τ k }. Indeed, if we denote byα 0 j , j = 1, . . . ,m 0 ,m 0 ≤ m ; the angles of the triangles {τ 0 k } that are respectively adjacent to the vertexν 0 = jτ ′ j (see Fig. 10), then there are two sources of "thiness": eitherα 0 j = α 0 j is smaller than ϕ 0 , for some j 0 ∈ {1, . . . ,m} or one of the anglesα 0 j1α 0 j1+1 produced by the division of the quadrilateral η 1 into two triangles:τ 0 j1 andτ 0 j1+1 . If we bisect the anglesα 0 j , j = 1, . . . ,m 0 ; we will receive anglesα 0 jk , j = 1, . . . ,m 0 ; k = 1, 2. Let us consider the anglesβ 0 jk , j = 1, . . . ,m 0 ; k = 1, 2 ; . 14 If each end every of the anglesβ 0 jk defined above is greater than ϕ 1 = ϕ 0 /10, then we desist and proceed towards part. But it may be that, for instance, bothα 0 j−1 andα 0 j are smaller than ϕ 0 /5. If this happens we continue the process of "mixing the angles". To be more precise: let us delate -for commodity reasons -the upper index "0" in the enumeration of the angles "β", and denote them byβ j , j = 1, . . . ,m 0 ; and let us form the sequence of anglesβ ′ j ,β ′′ j ,β ′′′ j , etc. , whereβ ′ j = 1 2 (β j−1 +β j+1 ),β ′′ j = 1 2 (β ′ j−1 +β ′ j ), and so on. But this process will halt -inasmuch as we are concerned -in a finite number of steps, for the following inequality holds: (4.13)βm 0 j > α 1 + · · · + αm 0 2m 0+1 ; j = 1, . . . ,m 0 ; that is: (4.14) βm 0+1 j > 2π 2m 0+1 > ϕ 0 10 k ; j = 1, . . . ,m 0 ; and so: wherek is the least natural power that satisfies the right-handed inequality in (4.12). We shall use the bound above in order to produce a fat triangulation. However, some care is needed in doing this, for in general, both the number of iterations used for each vertex and the number of bisectors b k j , j = 1, . . . ,m 0 ; k = 1, 2, 3 ; that intersect the triangleτ will be different, thus affecting the sizes of the anglesα k j . Let b denote the shortest of the segments b k j ∩τ , k = 1, 2, 3. (By elementary geometry, b should be the segment b k0 i0 that is the nearest to the shortest side of τ , whereα k0 is the smallest angle ofτ .) 15 Moreover, let us consider segments ν k ν k i ⊂ b k i , such that length(ν k ν k i ) = b/10 k2 =b where k 2 is chosen in such a manner that the quadrilateral ν k 1 ν l 1 y k 1 y l 1 is simple; where ν k 1 , ν l 1 denote the vertices closest to the edge ν k ν l and y k 1 , y l 1 play the same role in the adjacent triangleτ (see Fig. 11 (a)). It may be that we will have to consider ν k 1 ν l 1ỹ k 1ỹ l 1 (or any permutation of indices) -see Fig. 11 (b). (Herew k 1 plays in the triangleτ the rolẽ ν k 1 plays in the triangleτ .) In this case consider a pointŷ k ∈ intτ , such that: (i) length(ŷ k ) =b ; (ii 1 ) ∠w k 1νkŷk ≥ ϕ1 2 = ϕ 2 ; (ii 2 ) ∠ŷ kνkν k 1 ≥ ϕ1 2 = ϕ 2 . Remark 4.3. The existence of the positive integer k 2 with the desired proprieties is guaranteed by the fact that the anglesα k ,α l ,α k i ,α l i ,α m andα m are ≥ ϕ 1 , thus In consequence we are facing the situation depicted in Fig. 12 , where we also illustrated the conning of the segmentsν k iν k i+1 andν k 1ν l 1 from the barycenterb of τ ; k = 1, 2, 3; l = 1, 2, 3. Now we have to show the fatness of three types of polygons: 1b , k, l = 1, 2, 3 . We start by noticing that "fatness" of a quadrilateral means that: a) its angles are bounded from below, and: b) the ratios l λ l ι , λ, ι = 1, . . . , 4m between the lengths of its sides are also bounded from below. 15 The best possibility occurs, of course, when we have to use the bisection only once, so we will have only (ordinary) bisectors b 1 , b 2 , b 3 that will meet, of course, at the barycenterb ofτ . An easy computation (see [S3] ) shows that ; whereb = length(ν kỹk 1 ) . 16 Therefore, the arguments employed before show that Q kl is decomposable into fat triangles, so case 2) is dealt with. In a manner similar to that used in case 2) show that the triangles of types T ij and T kl are also fat; indeed: (4. 21) ∠ν l 1bν k 1 > ∠ν lbνk > ∠ν lνmνk ≥ ϕ 1 ; and so we dispose with cases 1) and 3) too, thus concluding the proof of part A). 16 and similar formulas hold for the other pairs of vertices.

4.3.2.
The extension to a fat 3-dimensional triangulation. We start by observing that, if u is a vertex of the simplex s i s.t. f 123 ∩ s i =τ -where f 123 is a face of the tetrahedron σ -then the triangles T kl , T ij and those produced by the subdivision of Q kl are also may also be conned from u. We want to show that the simplices thus generated -denoted by V δ ij = J(u δ , T ij ) , δ = 1, 2, 3, 4; etc. -have big angles. We shall justify this affirmation for tetrahedra of type V δ ij , the other cases being completely analogous. Indeed: a) The anglesν k i ,ν k j and ∠b are "big" by the very construction of △ν k iν k jb ; b) The plane angles around the vertex u δ are "big", since length(ν k iν k j ) ≥ c 1 b , and becauseν k iν k j is included in the plane ofτ ; c) The dihedral angles around u δ are also "big" (by the argument above and by the "tetrahedral sinus formula"). To sum up, the angles denoted by "L" in Fig. 13 are larger than some constant ϕ 2 . However, the basic type of small angles may still occur (see Fig. 14 , where they are denoted by "s"). In fact, the case of Fig. 3 (a) is associated with a low ratio "height/base side"; whereas, in this instance: (by the fatness of the tetrahedron u 1 u 2 u 3 u 4 ), so this case is excluded. Part of the angles covered by case (b) of Fig 14 are "big", for the following inequality (and its analogues hold)  To ensure the proper size of the last two angles one has to notice that the size ofe.g. ∠ν k u δν k i -inversely proportional to: (a) the distanceδ from u δ toν k lν k , and (b) the distance δ * fromū δ 18 toν k lν k . But, since the lengthsν k lν k are bounded from below, there exists a minimal uni- Figure 15.
But, since the lengthsν k lν k are bounded from below, there exists a minimal universal distance δ * 0 s.t. if δ * < δ * 0 then ∠ν k u δν k j > ϕ * 2 , for a suitable ϕ * 2 . By eventually decreasing δ * 0 to a new δ * 1 , we can ensure that the angles of type ∠ν k u δν k i are strictly grater than some ϕ * * 2 , ϕ * * 2 ≤ ϕ * * 2 , for eachν k ∈ St(u δ ) ∩ f 123 and for eachν k i , which concludes the proof of case(b). Indeed, we can vary the position ofū δ in the region Λū δ , such that all the required angles will be largesee  After disposing with this case, we can turn our attention to and to case (a) . Now, since | tan ∠ν k iν k j | = u δūδ u δūδ , where u δūδ ⊥ (ν k i ν kν k j ),ū δūδ ⊥ν k iν k j , and sincē u δū is known, we only have to ensure that u δūδ is bounded from below. For this, one essentially makes use of the very specific form of the triangulation of the geometric neighbourhood of an elliptic axes. Namely, instead of considering tetrahedra u δν k iν k jν k , we shall consider tetrahedra u + δν k iν k jν k where u + δ is the vertex corresponding to u δ in the next stratum of "s"type tetrahedra, of the family S. 19 Now the desired bound -for u + δū + δ u + δū + δ instead of u δūδ u δūδ -is achieved, whilst the very existence of the constants obtained before isn't changed, only their magnitude; we shall denote them by upper-right-superscript: e.g. ϕ ++ 2 . Remark 4.4. The method we have just used is adaptable to the general context: consider -instead of u δ -vertices u ++ δ , such that dist eucl (u δ , u ++ δ ) = h 0 , where h 0 = 1 3 h min , and h min = the minimal height of the simplices s + ∈ St(u δ ) . The points u ++ δ are to be chosen on the normal through u δ to the plane f 123 , so that the combinatorics of the triangulation will suffer no alteration. Clearly, uniform bounds will again be attained.
In order to conclude the proof of Theorem 1.1 we still have to provide for a fat triangulation around the node points. However, this missing case is easily dealt with by considering the intersections of the geometric neighbourhoods of the elliptic axes. Such an intersection will automatically inherit from the tubular neigbourhoods that generate it a natural, stratified, fat triangulation. (See Section 3.) So, the arguments involved in the proof of the restricted case of un-intersecting axes do apply here, too. Thus we conclude the Proof of Theorem 1.1 19 Of course -as before -we may have to subdivide the tetrahedra once or twice.

Higher Dimensions
Our approach to the extension of our results to dimension n ≥ 4 -which we will expose only briefly -is based upon reduction to smaller dimensions. To do this, let us observe that if S(n + 1) is a simplex in R n , then any sectioning hyperplane separates the vertices of S(n + 1) into two groups of p and n + 1 − p vertices, respectively. The notation will be as follows: (p, n + 1 − p) ≡ (n + 1 − p, p) for the polytopes of the section; and (p | n + 1 − p) and (n + 1 − p | p) for the resulting frusta. 20 The reduction to lower dimensions is permitted by: Lemma 5.1 ( [Som]). The frustum of type (p | 2) of S(p + q) is isomorphic to the section of type (p, q + 1) of S(p + q + 1).
The fattening algorithm is, in a nutshell, as follows: (i) Divide the polytopes obtained by sectioning into simplices: first the section polytope, then the faces that are part of the original S(n + 1), while ensuring fatness by the methods exposed in Section 3. If needed apply an inductive process on the dimension of the simplices. (ii) "Fatten" the frusta by proving the existence of a locus of points where from all the (n − k)-faces are seen at big (n − k)-dimensional angles. 21 Theorem 1.1 now follows. However, we have to understand much better the geometry of the elliptic locus of a Kleinian group with torsion, acting upon H n , n ≥ 4 . We are, however, fortunate, for the fixed set of an elliptic transformation is a kdimensional hyperbolic plane, 0 ≤ k ≤ n − 2 ; thus providing the fixed point set with a geometric neighbourhood -together with its natural fat triangulation. Some complications may arise because different elliptics may well have fixed loci of different dimensions 22 , so the respective simplices will have different dimensions and fatnesses. However this is easy to remedy by completing them to n-dimensional simplices, by "expanding" the low dimensional neighbouhoods to maximal dimension in a product manner. and since the double M is compact it follows, exactly as above, that π( M ) has only a finite number of conjugacy classes of elliptics, hence so has π 1 (M ).