Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 31, 2006, 131–142

THE EXISTENCE OF QUASIMEROMORPHIC MAPPINGS

Emil Saucan

Technion, Department of Mathematics and Department of Electrical Engineering Haifa, Israel; semil@tx.technion.ac.il

Abstract. Let G be a Kleinian group G acting on B^{n}, n ≥ 2. We show that if the orders of the elliptic elements in G which have non-degenerate fixed set are bounded, then G carries non-constant G-automorphic quasimeromorphic mappings. This together with an earlier non-existence theorem by Srebro gives a complete characterization of Kleinian groups that admit non-constant quasimeromorphic automorphic mappings.

1. Introduction

Definition 1.1. Let D ⊆ R^{n} be a domain; n ≥ 2 and let f: D → R^{n} be a continuous mapping. f is called

(1) quasiregular if and only if (i) f belongs to W_{loc}^{1,n}(D) and (ii) there exists K ≥ 1 such that:

(1.1)  |f′(x)|^{n} ≤ KJ_{f}(x) a.e.

where f′(x) denotes the formal derivative of f at x, |f′(x)| = sup_{|h|=1} |f′(x)h|, and where J_{f}(x) = det f′(x).

(2) quasiconformal if and only if f: D → f(D) is a quasiregular homeomorphism.

(3) quasimeromorphic if and only if f: D → R̂^{n}, R̂^{n} = R^{n} ⋃{∞} is quasiregular, where the condition of quasiregularity at f^{−1}(∞) can be checked by conjugation with auxiliary Möbius transformations. The smallest number K that satisfies (1.1) is called the outer dilatation of f. One can extend the definitions above to oriented, connected 𝒞^{∞} Riemannian

manifolds as follows:

Definition 1.2. Let M^{n}, N^{n} be oriented, connected 𝒞^{∞} Riemannian nmanifolds, n ≥ 2, and let f: M^{n} → N^{n} be a continuous function. f is called locally quasiregular if and only if for every x ∈ M^{n}, there exist coordinate charts (U_{x}, φ_{x}) and (V_{f(x)}, ψ_{f(x)}), such that f(U_{x}) ⊆ V_{f(x)} and g = ψ_{f(x)} ∘ f ∘ φ_{x}^{−1} is quasiregular.

If f is locally quasiregular, then T_{x}f: T_{x}(M^{n}) → T_{f(x)}N^{n} exist for a.e. x ∈ M^{n}.

2000 Mathematics Subject Classification: Primary 30C65, 57R05, 57M60.

This paper represents part of the authors Ph.D. Thesis written under the supervision of Professor Uri Srebro.

Definition 1.3. Let M^{n}, N^{n} be oriented, connected 𝒞^{∞} Riemannian nmanifolds, n ≥ 2, and let f: M^{n} → N^{n} be a continuous function. f is called quasiregular if and only if

(i) f is locally quasiregular and

(ii) there exists K, 1 ≤ K < ∞, such that

(1.2)  |T_{x}f|^{n} ≤ KJ_{f}(x)

for a.e. x ∈ M^{n}.

Recall that a group G of homeomorphisms acts properly discontinuously on a locally compact topological space X if and only if the following conditions hold for any g ∈ G_{x}, x ∈ X: (a) the stabilizer G_{x} = {g ∈ G | g(x) = x} of x is finite; and (b) there exists a neighbourhood V_{x} of x, such that (b_{1}) g(V_{x}) ∩ V_{x} = ∅, for any g ∈ G \ G_{x}; and (b_{2}) g(V_{x}) ∩ V_{x} = V_{x}.

Definition 1.4. A discontinuous group of orientation-preserving isometries of B^{n} is called a Kleinian group.

It is well known that a discontinuous group is discrete (see [Ms]).

Definition 1.5. Let f: B^{n} → R̂^{n}, and let G be a Kleinian group acting upon B^{n}. The function f is called G-automorphic if and only if:

(1.3)  f(g(x)) = f(x); for any x ∈ B^{n} and for all g ∈ G;

Recall the definition of elliptic transformations:

Definition 1.6. A Möbius transformation f: B^{n} → B^{n}, f ≠ Id is called elliptic if and only if f has a fixed point in B^{n}.

The existence of non-constant automorphic meromorphic functions in dimension n = 2 represents a classical result which follows from the existence of meromorphic functions on Riemann surfaces (see [Fo], [K]).

The question whether quasimeromorphic mappings (or qm-mappings, in short) exist in any dimension n ≥ 3 was originally posed by Martio and Srebro in [MS1]; subsequently in [MS2] they proved the existence of the fore-mentioned mappings in the case of co-finite groups, i.e., groups such that Vol_{hyp}(B^{n}/G) < ∞ (the important case of geometrically finite groups being thus included). Also, it was later proved by Tukia ([Tu]) that the existence of non-constant qm-mappings is assured in the case when G acts torsionless upon B^{n}. Moreover, since for torsionless Kleinian groups G, B^{n}/G is an (analytic) manifold, the next natural question to ask is whether there exist non-constant qm-mappings f: M^{n} → R̂^{n}; where M^{n} is an orientable n-manifold. A partial affirmative answer to this question is due to Peltonen (see [Pe]); to be more precise she proved the existence of qm-mappings in the case when M^{n} is a complete, connected, orientable 𝒞^{∞} -Riemannian manifold.

Our main result is the following theorem:

Theorem 1.7. Let G be a Kleinian group G acting on B^{n}, n ≥ 2. If the orders of the elliptic elements of G which have non-degenerate fixed set are bounded, then G admits non-constant G-automorphic quasimeromorphic mappings.

In contrast with the above results it was proved by Srebro ([Sr]) that, if G is a Kleinian group acting on B^{n}, n ≥ 3, containing elliptic elements with non-degenerate fixed set, of arbitrarily large orders, then G does not admit nonconstant G-automorphic qm-mappings; and showed that such groups exist in all dimensions n ≥ 3.

This non existence result, together with Theorem 1.7 gives a complete characterization of those Kleinian groups which admit G-automorphic quasimeromorphic mappings. Namely:

Theorem 1.8. Let G be a Kleinian group acting on B^{n}. Then G admits non-constant automorphic qm-mappings if and only if :

(1) n = 2; or

(2) n ≥ 3, and the orders of the elliptic elements of G having non-degenerate fixed sets are uniformly bounded.

Remark 1.9. Given any finitely generated Kleinian group acting on B^{3} the number of conjugacy classes of elliptic elements is finite (see [FM]). However, this is not true for Kleinian groups acting upon B^{n}, n ≥ 4; (for counterexamples, see [FM], [Po] and [H]).

Remark 1.10. Hamilton ([H, Theorem 4.1]) constructed examples of Kleinian groups G acting on B^{4} such that there exists an infinite sequence {f_{n}}_{n∈N} ⊂ G of elliptic transformations, with ord (f_{n}) → ∞ and such that the fixed set of each f_{n} is degenerate. (For the relevant definitions, see Section 2 below.) (Here ord (f_{n}) denotes the order of f_{n}.)

Note that by Remark 1.9 we have the following corollary:

Corollary 1.11. Let G be a finitely generated Kleinian group acting upon B^{3} . Then there exists a non constant G-automorphic qm-mapping f: B^{3} → R̂^{3} .

The classical methods employed in proving the existence in the case n = 2 do not apply in higher dimensions—indeed, for n ≥ 4, B^{n}/G is not even a manifold, but an orbifold. Therefore, different methods are needed. Following other researchers, we shall employ the classical “Alexander trick” (see [Al]).

A uniform bound for the dilatations can be attained (see [MS2], [Tu]) if the considered triangulation is fat, i.e. such that each of its individual simplices may be mapped onto a standard n-simplex, by a L-bilipschitz map, followed by a homothety, for a fixed L. (For a precise definition of fatness see Section 3 below.)

The idea of the proof of Theorem 1.7 is, in a nutshell, as follows: Based upon the geometry of the elliptic transformations construct a fat triangulation 𝒯_{1} of N_{e}^{∗}, where N_{e}^{∗} is a certain closed neighbourhood of the singular set of B^{n}/G. Since M_{p} = (B^{n} \ Fix(G))/G, Fix(G) = {x ∈ B^{n} | there exists g ∈ G \ {Id}, g(x) = x} is an orientable analytic manifold, we can apply Peltonen’s result to gain a triangulation 𝒯_{2} of M_{p}. Therefore, if the triangulations 𝒯_{1} and 𝒯_{2} are chosen properly, each of them will induce a triangulation of N_{e}^{∗} \ N_{e}^{∗}′, for a certain N_{e}^{∗}′ ⊊ N_{e}^{∗} (see Section 2).

‘Mash’ 𝒯_{1} and 𝒯_{2} (in N_{e}^{∗} \ N_{e}^{∗}′) i.e. ensure that the given triangulations intersect into a new triangulation 𝒯_{0} (see [Mun, Theorem 10.4]). Modify 𝒯_{0} to receive a new fat triangulation 𝒯 of B^{n}/G.

In the presence of degenerate components A_{k} = A(f_{k}) of the fixed set of G, where the transformations f_{k} may have arbitrarily large orders, a modification of this construction is needed; see Section 4.

Apply Alexander’s trick to receive a quasimeromorphic mapping f: B^{n}/G → R̂^{n}. The lift f̃ of f to B^{n} represents the required G-automorphic quasimeromorphic mapping.

In [S3] we showed how to build 𝒯_{1} using a generalization of a theorem of Munkres ([Mun, 10.6]) on extending the triangulation of the boundary of a manifold (with boundary) to the whole manifold. Munkres’ technique also provided us with the basic method of mashing the triangulations 𝒯_{1} and 𝒯_{2} . In this paper we present a more direct, geometric method of triangulating N_{e}^{∗} and mashing the two triangulations. We already employed this simpler method in [S1], where we proved Theorem 1.7 in the case n = 3. The original technique used in [S1] for fattening the intersection of 𝒯_{1} and 𝒯_{2} is, however, restricted to dimension 3. Therefore here we make appeal to the method employed in [S3], which is essentially the one developed in [CMS].

This paper is organized as follows: in Section 2 we show how to triangulate the closed neighbourhood N_{e}^{∗} of the singular set of B^{n}/G. Section 3 is dedicated to the main task of mashing the triangulations and fattening the resulting common triangulation. In Section 4 we show how to apply the main result in the construction of a G-automorphic quasimeromorphic mapping from B^{n} to R̂^{n}.

2. Geometric neigbourhoods

If G is a discrete Möbius group and if f ∈ G, f ≠ Id is an elliptic transformation, then there exists m ≥ 2 such that f^{m} = Id. The smallest m satisfying this condition is called the order of f, and it is denoted by ord(f). In the 3dimensional case the fixed point set of f, i.e. Fix(f) = {x ∈ B^{3} | f(x) = x}, is a hyperbolic line and will be denoted by A(f)—the axis of f. In dimension n ≥ 4 the fixed set (or axis of f) of an elliptic transformation is a k -dimensional hyperbolic plane, 0 ≤ k ≤ n − 2. An axis A is called degenerate if and only if dim A = 0. In dimensions higher than n = 3, different elliptics may have fixed sets of different dimensions.

If G is a discrete group, G is countable and so is the set of elliptics and the set of connected components of Fix(G), which we denote by {f_{i}}_{i≥1} and {C_{j}}, respectively.

Moreover, by the discreteness of G, the sets 𝒜 = {𝒜_{i≥0}}—and hence 𝒮 = {C_{j}}—have no accumulation points in B^{n}.

Figure 1. Geometric neighbourhood for n = 3 and m = 4. Here {4} denotes the regular (hyperbolic) polygon with 4 sides.

Hence we can choose disjoint, G-invariant neighbourhoods N_{j} and N′_{j} of C_{j}, N′_{j} ⊊ N_{j}. Indeed, first choose a neighbourhood N_{1} of C_{1} , such that \overline{N}_{1} ∩ ⋃_{j≥2} C_{j} = ∅; then recursively build a neighbourhood N_{k} of C_{k}, such that N_{k} ⊂ B^{n}\(N_{1}∪⋯∪N_{k−1}) and \overline{N}_{k}∩⋃_{j>k} C_{j} = ∅, for all k ≥ 2. Denote N_{e} = ⋃_{j∈N} N_{j},

N′_{e} = ⋃_{j∈N} N′_{j}. Define N_{e}^{∗} = (\overline{N}_{e} ∩ B^{n})/G, N_{e}^{∗}′ = (\overline{N}′_{e} ∩ B^{n})/G.

To produce the desired closed neighbourhood N_{e}^{∗} of the singular set of B^{n}/G and its triangulation 𝒯_{1} , we first consider the case where C_{i} = A(f), for some f ∈ G, and then construct a standard neighbourhood N_{f} = N(A(f)) of the axis of each elliptic element of G such that N_{f} ≃ A(f) × I^{n−k}, where A(f) = S^{k} and where I^{n−k} denotes the unit (n−k)-dimensional interval. The construction of N_{f} proceeds as follows: By [Cox, Theorem 11.23] the fundamental region for the local action of the stabilizer group of the axis of f, G_{f} = G_{A(f)} = {g ∈ G | g(x) = x} is a simplex or a product a simplices. Let 𝒮_{f} be the fundamental region (see Figure 2).

Then we can define the generalized prism (or simplotope—see [Som, VII.25]) 𝒮_{f}^{⊥}, defined by translating 𝒮_{f} in a direction perpendicular to 𝒮_{f}, where the translation length is dist_{hyp}(𝒮_{f}, A(f)). It naturally decomposes into simplices (see [Som, VII.25], [Mun, Lemma 9.4]). We have thus constructed an f -invariant triangulation of a prismatic neigbourhood N_{f} of A(f). We can reduce the mesh of this triangulation as much as required, while controlling its fatness by dividing 𝒮_{f} into similar simplices and partitioning N_{f} into a finite number of radial strata of equal width ϱ. In the special case when the minimal distance between axes δ = min{dist_{hyp}(A(f), A(g)) | g elliptic, g ≠ f} is attained we can chose ϱ = δ/κ_{0} , for some integer κ_{0} , and further partition it into ‘slabs’ of equal hight h. (In particular one can use this approach in the case when G acts on B^{3} and it contains no order two elliptics, since in this particular case, according to a result of Gehring and Martin [GM1], the minimum exists and is strictly positive.) Henceforth we shall call the neighbourhood thus produced, together with its fat triangulation, a geometric neighbourhood.

Figure 2. Canonical decomposition into simplices of 𝒮_{f}^{⊥}, for n = 3.

Since the stabilizer Stab(A_{1,...,k}) of the intersection of axes A_{1,...,k} = A_{i_{1}} ∩ ⋯ ∩ A_{i_{k}} is a finite subgroup of O^{+}(n), and since in any dimension there exist only a finite number of such groups of orders ≤ M_{0} , for any M_{0} ∈ N (see [Cox, Chapter 11]), the angles between the axes of transformations of orders ≤ m_{0} admit a bound α = α(m_{0}, n). Therefore, the intersection N(A_{1,...,k}) = N_{f_{1}} ∩ ⋯ ∩ N_{f_{k}} of the geometric neighbourhoods of several axes is also endowed with a natural fat triangulation, invariant under the group G = ⟨G_{f_{1}}, …, G_{f_{k}}⟩. (In the particular case n = 3 one can choose as a geometric neighbourhood of A a regular or a semi-regular polyhedron together with its interior (see Figure 3 below).

If q ∈ B^{n}, dim q = 0, is a degenerate element of the singular locus, we replace the tubular neighbourhood considered above by 𝒫_{q}∪int𝒫_{q}, where 𝒫_{q} is a regular polytope invariant under the stabilizer G_{q} of q in G, together with its canonical simplicial subdivision (see [Cox, 7.6]). Indeed, every finite group generated by reflections is the symmetry group of a regular polyhedron 𝒫 (see [Cox, p. 209]). Moreover, the rotation group of 𝒫 has order nl/2, where l is the number of faces of 𝒫 (see [Cox, pp. 227–231]).

Remark 2.1. As noted above, if G is a Kleinian group acting with torsion on B^{n}, then M_{p} = (B^{n} \Fix(G))/G is a complete orientable manifold. Moreover, since the isotropy groups of any point in Q_{G} = B^{n}/G are subgroups of O^{+}(n), it

Figure 3. A Euclidean semi-regular polyhedron and two of its fundamental tetrahedra (n = 3).

follows that B^{n}/G is complete orientable orbifold (see [Dr, p. 46]). The singular locus Σ_{Q_{G}} = Fix(G)/G of Q_{G} contains all the non-manifold points of Q_{G}, yet the two sets are not equal. Indeed, in dimension n = 2 (n = 3) any orbifold (orientable orbifold) is homeomorphic to a manifold. The local structure of Σ_{Q_{G}} at a point x_{Q} ∈ Q_{G} is determined by the stabilizer in G of its preimage in B^{n}, i.e. by the finite subgroups of O^{+}(n). (For instance, in dimension n = 3 only two infinite families and three more special cases of branching points (of Fix(G) and thus of Σ_{Q_{G}}) can occur—see [Th1, 5.6]). However, the global structure of Σ_{Q_{G}} can be very complicated (see [Th1, 5.6]).

3. Mashing and fattening triangulations

We present the main steps of the Munkres ([Mun, Chapter 10]) and Cheeger ([CMS, pp. 432–440]) techniques, and we indicate how to adapt them to our particular setting. First let us establish some definitions and notation:

Definition 3.1. Let M^{n} be a PL-manifold. Two triangulations 𝒯_{1}, 𝒯_{2} of M^{n} intersect transversally if and only if for any p ∈ M^{n}, there exist neighbourhoods U_{1} , U_{2} , U_{3} of p in |𝒯_{1}|, |𝒯_{2}| and M^{n}, respectively, such that the triple (U_{1}, U_{2}, U_{3}) is PL-homeomorphic to a neighbourhood of 0 in (R^{n}×0, 0×R^{n}, R^{n}× R^{n}).

To ensure the fatness of the common triangulation we need to make appeal to a stronger notion of transversality, namely:

Definition 3.2. Let σ_{i} ∈ K, dim σ_{i} = k_{i}, i = 1, 2; such that diam σ_{1} ≤ diam σ_{2} . Denote by [σ_{i}] the affine subspace of R^{N} generated by σ_{i}, and let ⟨σ_{i}⟩ denote the subspace parallel to [σ_{i}], such that 0 ∈ ⟨σ_{i}⟩ ⊂ R^{N}; i = 1, 2. We say that σ_{1} , σ_{2} are δ -transverse if and only if

(i) dim([σ_{1}] ∩ [σ_{2}]) = max(0, k_{1} + k_{2} − n);

(ii) 0 < δ < ∡([σ_{1}], [σ_{2}]), where ∡([σ_{1}], [σ_{2}]) = ∡(⟨σ_{1}⟩, ⟨σ_{2}⟩), and where ∡(⟨σ_{1}⟩, ⟨σ_{2}⟩) = min_{(e_{1},e_{2})} arccos (e_{1}, e_{2}), e_{i} ∈ (⟨σ_{1}⟩ ∩ ⟨σ_{1}⟩)^{⊥} ∩ ⟨σ_{i}⟩, ∥e_{i}∥ = 1, i = 1, 2; where (e_{1}, e_{2}) denotes the standard inner product on R^{n}; and if σ_{3} ⊊ σ_{1} , σ_{4} ⊊ σ_{2} , such that dim σ_{3} + dim σ_{4} < n = dim K, then

(iii) dist(σ_{3}, σ_{4}) > δ · d_{1} , where d_{1} = diam σ_{1} . In this case we write: σ_{1} ⋔_{δ} σ_{2} .

Definition 3.3. Let τ ⊂ R^{n}; 0 ≤ k ≤ n be a k -dimensional simplex. The fatness φ of τ is defined as being:

(3.1)  φ = φ(τ) = inf_{{σ<τ \atop dim σ=l}} \frac{Vol(σ)}{diam^{l}σ} .

The infimum is taken over all the faces of τ, σ < τ, and Vol_{eucl}(σ) and diam σ stand for the Euclidean l-volume and the diameter of σ, respectively. (If dim σ = 0, then Vol_{eucl}(σ) = 1, by convention.)

A simplex τ is φ_{0} -fat, for some φ_{0} > 0, if φ(τ) ≥ φ_{0} . A triangulation (of a submanifold of R^{n}) 𝒯 = {σ_{i}}_{i∈I} is φ_{0} -fat if all its simplices are φ_{0} -fat. A triangulation 𝒯 = {σ_{i}}_{i∈I} is fat if there exists φ_{0} > 0 such that all its simplices are φ_{0} -fat.

Remark 3.4. There exists a constant c(k) that depends solely upon the dimension k of τ such that

(3.2)  \frac{1}{c(k)} · φ(τ) ≤ min_{{σ<τ \atop dim σ=l}} ∡(τ, σ) ≤ c(k) · φ(τ),

and

(3.3)  φ(τ) ≤ \frac{Vol(σ)}{diam^{l}σ} ≤ c(k) · φ(τ);

where ∡(τ, σ) denotes the (internal) dihedral angle of σ < τ. (For a formal definition, see [CMS, pp. 411–412], [Som].)

Remark 3.5. The definition above is the one introduced in [CMS]. For equivalent definitions of fatness, see [Ca1], [Ca2], [Mun], [Pe], [Tu].

The first step is that of mashing the triangulations 𝒯_{1} , 𝒯_{2} :

We approximate the triangulation 𝒯_{2} of M_{p} by a locally finite Euclidean triangulation, by means of the secant map (see [Mun, p. 90]). Also, the hyperbolic simplices of 𝒯_{1} can be approximated arbitrarily well by Euclidean simplices, by considering diam σ, σ ∈ 𝒯_{1} small enough (see [Tu]). Therefore the mashing and fattening of triangulations reduces to that of Euclidean ones.

Next we ensure that the given triangulations intersect into a new triangulation 𝒯_{0} . This is first done locally by modifying these local triangulations coordinate chart by chart, so they will be PL-compatible wherever they overlap. More precisely, we first apply infinitesimal moves of the vertices so that the two triangulations will intersect transversally. Next we perform suitable barycentric subdivisions of the closed, convex polyhedral cells γ̄ = σ̄_{1} ∩ σ̄_{2} , σ_{i} ∈ 𝒯_{i}, i = 1, 2; in the following manner: suppose each cell β ⊂ ∂γ already has a subdivision into simplices β_{i}, i = 1, …, p; choose an interior point p_{γ} ∈ intγ, construct the joins J(p_{γ}, β_{i}), i = 1, …, p; and consider all their simplices (see [Mun, 10.2–10.3]).

To extend the local triangulations to a global triangulation 𝒯_{0} , we work in R^{n}, by using the coordinate charts and maps. Here again we have to approximate the given triangulation by a PL-map, such that the given triangulation and the one we produce will be PL-compatible (see [Mun, Theorem 10.4]). The existence of the common triangulation 𝒯_{0} follows immediately (see [Mun, Theorem 10.5]).

We next present the main steps of the fattening process (for details see [CMS]): One begins by triangulating and fattening the intersection of two individual

simplices belonging to the two given triangulations, respectively. First one shows that if two individual simplices are fat and if they intersect δ -transversally, then one can choose the points p_{γ} such that the barycentric subdivision γ̄^{∗} will be composed of fat simplices. (See [CMS, Lemma 7.1].)

Next one shows that given two fat Euclidean triangulations that intersect δ - transversally, it is possible to infinitesimally move any given point of one of the triangulations such that the resulting intersection will be δ^{∗} -transversal, where δ^{∗} depends only on δ, the common fatness of the given triangulations, and on the displacement length (see [CMS, Lemma 7.3]).

By repeatedly applying this results to the simplices of dimensions 0, …, n, of the intersection of two fat triangulations, one can now prove the main fattening result:

Proposition 3.6 ([CMS, Lemma 6.3]). Let 𝒯_{1} , 𝒯_{2} be two fat triangulations of open sets U_{1}, U_{2} ⊂ R^{n}, B_{r}(0) ⊆ U_{1} ∩ U_{2} , having common fatness ≥ φ_{0} and such that d_{1} = inf_{σ_{1}∈𝒯_{1}} diam σ_{1} ≤ d_{2} = inf_{σ_{2}∈𝒯_{2}} diam σ_{2} . Then there exist φ_{0}^{∗} -fat triangulations 𝒯′_{1}, 𝒯′_{2} , φ_{0}^{∗} = φ_{0}^{∗}(φ_{0}), of open sets V_{1}, V_{2} ⊆ B_{r}(0), such that

(1) 𝒯′_{i}|_{B_{r−8d_{2}} (0)} = 𝒯_{i}|_{B_{r−8d_{2}} (0)} , i = 1, 2;

(2) 𝒯′_{1} and 𝒯′_{2} agree near their common boundary. Moreover:

(3) inf_{σ′_{1}∈𝒯′_{1}} diam σ′_{1} ≤ 3d_{1}/2, inf_{σ′_{2}∈𝒯′_{2}} diam σ′_{2} ≤ d_{2} .

We apply Proposition 3.6 above to our particular context in the following manner: Let 𝒯_{1} , 𝒯_{2} be the triangulations of N_{e}^{∗} \ N_{e}^{∗}′ constructed above. To gain a globally fat triangulation from the mashing of 𝒯_{1} and 𝒯_{2} , we start by partitioning N_{e}^{∗} \ N_{e}^{∗}′ into (almost) cubes Q. If the diameters of the sets Q are small enough we can apply Proposition 3.6, for Q instead B_{r}(0). Extend 𝒯_{0} by 𝒯_{2} on the face included in ∂N_{e} and by 𝒯_{1} on the other faces, to receive the desired triangulation 𝒯 . (Further fattening of the triangulations induced on the lower dimensional faces may be necessary. However, by the locally finiteness of the triangulation, the number of steps required for fattening the lower dimensional intersections is finite and depends solely upon the dimension n.) This gives the required globally fat triangulation of B^{n}/G.

4. The existence of quasimeromorphic mappings

We first prove the following lemma:

Lemma 4.1 ([MS1], [Pe]). Let M^{n} ⊂ R^{N} be an orientable n-manifold, let 𝒯 be a chessboard fat triangulation of M^{n}, let σ ∈ 𝒯 , σ = (p_{0}, …, p_{n}) and let τ_{0} = (p_{0,1}, …, p_{0,n}) denote the equilateral n-simplex inscribed in the unit sphere S^{n−1} . Then there exists a orientation-preserving homeomorphism h = h_{σ}: |σ| → R̂^{n} such that

(1) h(|σ|) = |τ_{0}|, if σ is positively oriented and h(|σ|) = R̂^{n} \ |τ_{0}|, otherwise.

(2) h(p_{i}) = p_{0,i}, i = 0, …, n.

(3) h|_{∂|σ|} is a PL-homeomorphism.

(4) h|_{int|σ|} is quasiconformal.

Proof. If det (p_{0}, …, p_{n}) > 0, then the PL-mapping h defined by condition (2) above also satisfies conditions (1), (3) and (4). If det(p_{0}, …, p_{n}) < 0, we define h as follows: h = φ^{−1} ∘ J ∘ φ ∘ h_{0} , where φ is the radial linear stretching φ: τ_{0} → R^{n}, J denotes the reflection in the unit sphere S^{n−1} and h_{0}: |σ| → |τ_{0}| is the orientation-reversing PL-mapping defined by condition (2). Recall that φ is onto and bilipschitz (see [MS2]). Moreover, by a result of Gehring and Väisalä, φ is also quasiconformal (see [V]). We can extend φ to R̂^{n} by defining φ(∞) = ∞. It follows that h indeed represents the required PL-homeomorphism.

The existence theorem of quasimeromorphic mappings now follows immediately:

Proof of Theorem 1.7. Let 𝒯 be the φ_{0}^{∗} -fat chessboard triangulation of B^{n}/G constructed in Section 3 above. Let f: B^{n}/G → R̂^{n} be defined by: f|_{|σ|} = h_{σ}, where h is the homeomorphism constructed in the lemma above. Then f is a local homeomorphism on the (n − 1)-skeleton of 𝒯̃ too, while its branching set B_{f} is the (n − 2)-skeleton of 𝒯̃ . By its construction f is quasiregular and its (outer) dilatation depends only φ_{0}^{∗} and on the dimension n (see [Tu, Lemma E]). The lift f̃ of f to B^{n} represents the required G-automorphic quasimeromorphic mapping.

In the case of degenerate components A_{k} of the fixed set 𝒮 , the proof is essentially the same as in the classical case of Riemann surfaces (see, e.g. [Fo, pp. 233–238]). More precisely, we proceed as follows: We excise from B^{n} disjoint ball neighbourhoods B_{k} of A_{k}. Let S_{k} = ∂B_{k}. Then each of the quotients S_{k}/G admits a fat triangulation 𝒯_{k}. The manifold (B^{n} \ ⋃_{k≥1} B_{k})/G admits a fat triangulation that extends the fat triangulation of S_{k} (see [CMS, p. 444] and [S3, Theorem 2.9]). We build the simplices P_{k} with vertex A_{k}/G and base

The existence of quasimeromorphic mappings 141

T_{kl}, where T_{kl} are simplices belonging to 𝒯_{k}. Then each of the simplices P_{k} can be quasiconformally mapped onto a half-space, with bounded dilatation which depends only on n and not on the angles at the vertices A_{i}, even if the orders of the transformations f_{k} are not bounded from above (see [Car, Theorem 3.6.10 and Theorem 3.6.13]).

Acknowledgment. The author would like to thank Professor Uri Srebro for posing this problem and for his enlightening remarks and suggestions. Many thanks are also due to Professor Pekka Tukia for his kind support and to Professor Bronislaw Wajnryb for his help and advice.

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Received 7 December 2004