Hyperbolic topology and bounded locally homeomorphic quasiregular mappings in 3-space
We use our new type of bounded locally homeomorphic quasiregular mappings in the unit 3-ball to address long standing problems for such mappings, including the Vuorinen injectivity problem. The construction of such mappings comes from our construction of non-trivial compact 4-dimensional cobordisms...
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Apanasov, B.N. 2020-06-13T08:18:40Z 2020-06-13T08:18:40Z 2019 Hyperbolic topology and bounded locally homeomorphic quasiregular mappings in 3-space / B.N. Apanasov // Український математичний вісник. — 2019. — Т. 16, № 1. — С. 10-27. — Бібліогр.: 26 назв. — англ. 1810-3200 2000 MSC. 30C65, 57Q60, 20F55, 32T99, 30F40, 32H30, 57M30 https://nasplib.isofts.kiev.ua/handle/123456789/169429 We use our new type of bounded locally homeomorphic quasiregular mappings in the unit 3-ball to address long standing problems for such mappings, including the Vuorinen injectivity problem. The construction of such mappings comes from our construction of non-trivial compact 4-dimensional cobordisms M with symmetric boundary components and whose interiors have complete 4-dimensional real hyperbolic structures. Such bounded locally homeomorphic quasiregular mappings are defined in the unit 3-ball B³ ⊂ R³ as mappings equivariant with the standard conformal action of uniform hyperbolic lattices Г ⊂ IsomH³ in the unit 3-ball and with its discrete representation G = ρ(Г) ⊂ IsomH⁴. Here, G is the fundamental group of our non-trivial hyperbolic 4-cobordism M = (H⁴∪Ω (G))/G, and the kernel of the homomorphism ρ: Г → G is a free group F₃ on three generators. en Інститут прикладної математики і механіки НАН України Український математичний вісник Hyperbolic topology and bounded locally homeomorphic quasiregular mappings in 3-space Article published earlier |
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Hyperbolic topology and bounded locally homeomorphic quasiregular mappings in 3-space |
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Hyperbolic topology and bounded locally homeomorphic quasiregular mappings in 3-space Apanasov, B.N. |
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Hyperbolic topology and bounded locally homeomorphic quasiregular mappings in 3-space |
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Hyperbolic topology and bounded locally homeomorphic quasiregular mappings in 3-space |
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hyperbolic topology and bounded locally homeomorphic quasiregular mappings in 3-space |
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Apanasov, B.N. |
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Apanasov, B.N. |
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We use our new type of bounded locally homeomorphic quasiregular mappings in the unit 3-ball to address long standing problems for such mappings, including the Vuorinen injectivity problem. The construction of such mappings comes from our construction of non-trivial compact 4-dimensional cobordisms M with symmetric boundary components and whose interiors have complete 4-dimensional real hyperbolic structures. Such bounded locally homeomorphic quasiregular mappings are defined in the unit 3-ball B³ ⊂ R³ as mappings equivariant with the standard conformal action of uniform hyperbolic lattices Г ⊂ IsomH³ in the unit 3-ball and with its discrete representation G = ρ(Г) ⊂ IsomH⁴. Here, G is the fundamental group of our non-trivial hyperbolic 4-cobordism M = (H⁴∪Ω (G))/G, and the kernel of the homomorphism ρ: Г → G is a free group F₃ on three generators.
|
| issn |
1810-3200 |
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https://nasplib.isofts.kiev.ua/handle/123456789/169429 |
| citation_txt |
Hyperbolic topology and bounded locally homeomorphic quasiregular mappings in 3-space / B.N. Apanasov // Український математичний вісник. — 2019. — Т. 16, № 1. — С. 10-27. — Бібліогр.: 26 назв. — англ. |
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2025-11-26T15:34:35Z |
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1850626745132122112 |
| fulltext |
Український математичний вiсник
Том 16 (2019), № 1, 10 – 27
Hyperbolic topology and bounded
locally homeomorphic quasiregular
mappings in 3-space
Boris N. Apanasov
(Presented by V. Gutlyanskĭı)
Paper dedicated to the memory of my colleague and friend,
brilliant mathematician Bogdan Bojarski
Abstract. We use our new type of bounded locally homeomorphic
quasiregular mappings in the unit 3-ball to address long standing prob-
lems for such mappings, including Matti Vuorinen injectivity problem.
The construction of such mappings comes from our construction of non-
trivial compact 4-dimensional cobordisms M with symmetric boundary
components and whose interiors have complete 4-dimensional real hy-
perbolic structures. Such bounded locally homeomorphic quasiregular
mappings are defined in the unit 3-ball B3 ⊂ R
3 as mappings equivari-
ant with the standard conformal action of uniform hyperbolic lattices
Γ ⊂ IsomH3 in the unit 3-ball and with its discrete representation
G = ρ(Γ) ⊂ IsomH4. Here G is the fundamental group of our non-
trivial hyperbolic 4-cobordism M = (H4 ∪ Ω(G))/G and the kernel of
the homomorphism ρ :Γ → G is a free group F3 on three generators.
2010 MSC. 30C65, 57Q60, 20F55, 32T99, 30F40, 32H30, 57M30.
Key words and phrases. Bounded quasiregular mappings, Vuorinen
problem, local homeomorphisms, hyperbolic group action, hyperbolic
manifolds, cobordisms, group homomorphism, deformations of geometric
structures.
1. Introduction
The theory of quasiregular mappings, initiated by the works of
M. A. Lavrentiev and later by Reshetnyak and Martio, Rickman and
Väisälä, shows that they form (from the geometric function theoretic
point of view) the correct generalization of the class of analytic functions
Received 27.03.2019
ISSN 1810 – 3200. c© Iнститут прикладної математики i механiки НАН України
B. N. Apanasov 11
to higher dimensions. In particular, Reshetnyak proved that non-constant
quasiregular mappings are (generalized) branched covers, that is, con-
tinuous, discrete and open mappings and hence local homeomorphisms
modulo an exceptional set of (topological) codimension at least two, and
that they preserve sets of measure zero. For the theory of quasiregular
mappings, see the monographs [20, 21] and [26].
Here we address some properties of bounded quasiregular mappings
f : B3 → R3 in the unit ball B3 and well known problems on such
quasiregular mappings. These results will be heavily based on our recent
construction Apanasov [9] (Theorem 4.1) of surjective locally homemo-
rphic quasiregular mappings F : S3\S∗ → S3 with topological barriers
at points of a dense subset S∗ ⊂ S2 ⊂ S3. Due to its importance for
understanding of results of this paper, in the Appendix we will give some
details of this construction based on properties of non-trivial compact 4-
dimensional cobordisms M4 with symmetric boundary components – see
[7–9]. The interiors of these 4-cobordisms have complete 4-dimensional
real hyperbolic structures and universally covered by the real hyperbolic
space H4, while the boundary components of M4 have (symmetric) 3-
dimensional conformally flat structures obtained by deformations of the
same hyperbolic 3-manifold whose fundamental group Γ is a uniform lat-
tice in IsomH3. Such conformal deformations of hyperbolic manifolds
are well understood after their discovery in [3], see [6]. Nevertheless
till recently such “symmetric” hyperbolic 4-cobordisms were unknown de-
spite our well known constructions of non-trivial hyperbolic homology 4-
cobordisms with very assymmetric boundary components – see [10] and
[4–6].
The above subset S∗ of the boundary sphere S2 = ∂B3 is a count-
able Γ-orbit of a Cantor subset with Hausdorff dimension ln 5/ ln 6 ≈
0.89822444 (where a uniform hyperbolic lattice Γ conformally acts in the
unit ball B3). All its points are essential singularities of the bounded
locally homeomorphic quasiregular mapping f : B3 → R3 defined as the
restriction to the unit ball B3 of our quasiregular mapping F : S3\S∗ →
S3. This bounded quasiregular mapping f : B3 → R3 has no radial
limits at all points x ∈ S∗ ⊂ S2 = ∂B3 and gives an advance to the
well known Pierre Fatou’s problem on the correct analogue for higher-
dimensional quasiregular mappings of the Fatou’s theorem [13] on radial
limits of a bounded analytic function of the unit disc. There are several
results concerning radial limits of mappings of the unit ball. The most
recent progress is due to Kai Rajala who in particular proved that radial
limits exist for infinitely many points of the unit sphere, see [19] and
references there for some earlier results in this direction.
12 Bounded Locally Homeomorphic Quasiregular Mappings
Another application of our construction Apanasov [9] (Theorem 4.1)
of surjective locally homemorphic quasiregular mappings F : S3\S∗ →
S3 is to a well known open problem on injectivity of quasiregular map-
pings in space formulated by Matti Vourinen in 1970–1980s (see Vuorinen
[24–25], [26] (page 193, Problem 4) and Problem 7.66 in the Hayman’s
list of problems [11, 16]). The problem asks whether a proper quasiregu-
lar mapping f in the unit ball Bn, n ≥ 3, with a compact branching set
Bf ⊂ Bn is injective. It is false when n = 2. The conjecture is known to
be true in the special case f(Bn) = Bn, n ≥ 3 – see Vuorinen [24].
In Section 2 we show that the quasiregular mapping f : B3 → R3
defined as the restriction to the unit ball B3 of our quasiregular map-
ping F : S3\S∗ → S3 from Apanasov [9] (Theorem 4.1) is essentially a
counter-example to this conjecture. This mapping f is essentially proper
in the sense that any compact C ⊂ f(B3) has a compact subset C ′ ⊂ B3
such that f(C ′) = C. This mapping f is bounded locally homeomorphic
but not injective quasiregular mapping in the unit ball. Its restriction to
a round ball Br ⊂ B3 of radius r < 1 arbitrary close to one gives (after
re-scaling) a proper bounded quasiregular mapping of the round ball B3
serving as a counter-example to this Vuorinen conjecture (Theorem 2.2).
The last task of this paper is to investigate the asymptotic behavior
of bounded locally homeomorphic quasiregular mappings in the unit ball
in smaller balls Bn(r) ⊂ Bn of radius r close to one. There is an open
Matti Vuorinen conjecture that in dimension n ≥ 3 it is not possible that
for y ∈ f(Bn) and all r ∈ (1/2, 1), the cardinality
card(Bn(r) ∩ (f−1(y))) >
1
(1− r)n−1
(1.1)
In Section 3 we show that this question is closely related to the growth
function of the kernel (a free group of rank 3) F3 ⊂ Γ of the homomor-
phism ρ :Γ → G of our hyperbolic lattice Γ ⊂ IsomH3 to the constructed
discrete group G ⊂ IsomH4 – see [9]. We conclude that our bounded
locally homeomorphic quasiregular mappings in the unit ball B3 satisfies
this conjecture.
1.1. Acknowledgments
The author is grateful to Matti Vuorinen for attracting our attention
to these problems and for fruitful discussions.
2. Not injective bounded quasiregular mappings
Here we apply our construction [9] (see Appendix: Theorems 4.1 and
4.5) of bounded locally homeomorphic quasiregular mapping F :B3 → R3
B. N. Apanasov 13
to solve the Matti Vourinen open problem on injectivity of quasiregular
mappings in 3-dimensional space. This well known problem was for-
mulated by Matti Vourinen in 1970–1980s as result of investigations of
quasiregular mappings in space (see Vuorinen [24–25], [26] (page 193,
Problem 4) and Problem 7.66 in the Hayman’s list of problems [11, 16]).
The problem asks whether a proper quasiregular mapping f in the
unit ball Bn, n ≥ 3, with a compact branching set Bf ⊂ Bn is injective.
The mapping f(z) = z2, where z ∈ B2, shows that the conjecture is
false when n = 2. The conjecture is known to be true in the special case
f(Bn) = Bn, n ≥ 3 – see Vuorinen [24]. Here we give a counter-example
to this conjecture for n = 3.
Proposition 2.1. Let the uniform hyperbolic lattice Γ ⊂ IsomH3 and
its discrete representation ρ :Γ → G ⊂ IsomH4 with the kernel as a free
subgroup F3 ⊂ Γ be as in Theorem 4.1. Then the bounded locally home-
omorphic quasiregular mapping F : B3 → R3 constructed in Theorem
4.5 as a Γ-equivariant mapping in (4.3) is an essentially proper bounded
quasiregular mapping in the unit 3-ball B3 which is locally homeomorphic
(BF = ∅) but not injective.
Proof. The discrete group G = ρ(Γ) ⊂ IsomH4 ∼= Möb(3) has its
invariant bounded connected component Ω1 ⊂ Ω(G) ⊂ S3 where its
fundamental polyhedron P1 is quasiconformally homeomorphic to the
convex hyperbolic polyhedron P0 fundamental for our hyperbolic lattice
Γ ⊂ IsomH3 conformally acting in the unit ball B3(0, 1)), φ−1
1 :P0 → P1.
This homeomorphism φ−1
1 from (4.2) maps polyhedral sides of P0 to
the corresponding sides of the polyhedron P1 and preserves all dihedral
angles.
Our bounded locally homeomorphic quasiregular mapping F :B3 →
Ω1 ⊂ R3 defined in (4.3) as the equivariant extention of this homeomor-
phism φ−1
1 maps the tessellation of B3 by compact Γ-images of P0 to the
tessellation of Ω1 by compact G-images of P1. This shows that for any
compact subset C ⊂ Ω1 = F (B3) (covered by finitely many polyhedra
g(P1), g ∈ G), there is a compact subset C ′ ⊂ B3 (covered by finitely
many corresponding polyhedra γ(P1), γ ∈ Γ) such that F (C ′) = C.
On the other hand, this locally homeomorphic quasiregular mapping
F is not injective in B3. In fact, for any element γ 6= 1 in the kernel
F3 ⊂ Γ of the homomorphism ρ : Γ → G the image F (P0) = P1 of the
fundamental polyhedron P0 of the lattice Γ is the same as the image
F (γ(P0)) of the translated polyhedron γ(P0), P0 ∩ γ(P0) = ∅.
One may restrict our not injective essentially proper bounded quasi-
regular mapping F in the unit 3-ball B3 from Proposition 2.1 to a round
14 Bounded Locally Homeomorphic Quasiregular Mappings
ball Br ⊂ B3 of radius r < 1 arbitrary close to one. The composition of
this restriction Fr :Br → R3 with the stretching of Br to the unit ball
B3, i.e. the mapping
f :B3 → R
3 , f(x) = Fr(rx) (2.1)
is a proper bounded locally homeomorphic quasiregular mapping of the
unit ball B3. For any point y ∈ f(B3) the number Ny of its pre-images
in B3, i.e. the cardinality of the set {x ∈ B3 :f(x) = y} is finite. In fact
due to Vuorinen Lemma 9.22 in [26], this number Ny is independent of
y ∈ f(B3) in the image set and is set as Nf . The number Nf of such
pre-images of y ∈ f(B3) is determined by the number of images γ(Pker),
γ ∈ F3 ⊂ Γ, of a fundamental polyhedron Pker ⊂ H3 in our round ball
Br ⊂ B3 of radius r < 1 defining the mapping f in (2.1). Here F3 is
the free subgroup F3 ⊂ Γ in the uniform hyperbolic lattice Γ ⊂ IsomH3
(the kernel of the discrete representation ρ : Γ → G ⊂ IsomH4 from
Theorems 2.1 and 4.1), and Pker ⊂ H3 is its fundamental polyhedron in
the hyperbolic space H3. Making the radius r < 1 sufficiently close to
1 (i.e. changing our quasiregular mapping f :B3 → R3), one can make
the number Nf arbitrary large. This proves the following (the Vuorinen
conjecture’ counter-example):
Theorem 2.2. There are proper bounded quasiregular mappings f :B3 →
R3 without branching sets (Bf = ∅) which are locally homeomorphic but
not injective. Their pre-images {x ∈ B3 :f(x) = y} are finite and can be
made arbitrary large.
3. Asymptotics of bounded quasiregular mappings in the
unit ball and growth in free groups
Here we investigate the asymptotic behavior of bounded locally home-
omorphic quasiregular mappings f in the unit ball. The question is
how many pre-images of a point y ∈ f(Bn) do we have in smaller balls
Bn(r) ⊂ Bn of radius r close to one. There is an open Matti Vuorinen
conjecture that in dimension n ≥ 3 it is not possible that for y ∈ f(Bn)
and all r ∈ (1/2, 1), the cardinality of such pre-image in Bn(r) is bigger
than (1− r)1−n – see (1.1).
As we show this question for our bounded quasiregular mappings in
the unit ball B3, F :B3 → R3, constructed in Theorem 4.5 is closely re-
lated to the growth function of the kernel F3 ⊂ Γ of the homomorphism
ρ : Γ → G of our uniform hyperbolic lattice Γ ⊂ IsomH3 to the con-
structed discrete group G ⊂ IsomH4 – see Proposition 4.3 and Lemma
4.4 in Appendix. Here F3 is a free group on 3 generators.
B. N. Apanasov 15
For free groups Fm on m generators one can use well known facts
about their growth functions, cf. [14]. The growth function γG,Σ of a
group (G,Σ) with a generating set Σ counts the number of elements in
G whose length (in the word metric) is at most a natural number k:
γG,Σ(k) = card{g ∈ G : |g|Σ ≤ k} . (3.1)
Lemma 3.1. A free group Fm on m generators (for any free system Σ
of generators) has 2m(2m − 1)k−1 elements of length k, and its growth
function:
γFm(k) = 1 +
m
m− 1
((2m− 1)k − 1). (3.2)
Proof. Clearly in a free group Fm on m generators we have the number
of elements with length i equals to ci = card{g ∈ Fm : |g| = i} =
2m(2m−1)i. Therefore the growth function γFm(k) = 1+2m+2m(2m−
1)+ . . .+2m(2m−1)k−1. This gives the value γFm(k) in the Lemma.
In the embedded Cayley graph ϕ(K(Γ,Σ)) ⊂ B3 (i.e. the graph
that is dual to the tessellation of B3 by convex hyperbolic polyhedra
γ(P0), γ ∈ Γ), we consider its subgraph (a tree) corresponding to our
free group F3 ⊂ Γ on 3 generators (the kernel of the homomorphism ρ).
The embedding ϕ of the Cayley graph K(Γ,Σ) is a Γ-equivariant proper
embedding. It is a pseudo-isometry, i.e. for the word metric (∗, ∗) on
K(Γ,Σ) and the hyperbolic metric d in the unit ball B3, there are positive
constants K and K ′ such that (a, b)/K ≤ d(ϕ(a), ϕ(b)) ≤ K · (a, b) for
all a, b ∈ K(Γ,Σ) satisfying one of the following two conditions: either
(a, b) ≥ K ′ or d(ϕ(a), ϕ(b)) ≥ K ′.
LetD be the maximum of hyperbolic length of generators of the kernel
F3 ⊂ Γ. All vertices of our tree subgraph corresponding to elements in
F3 of length at most k are in the hyperbolic ball centered at 0 ∈ B3 with
radius R = Dk. This hyperbolic ball corresponds to the Euclidean ball
B3(0, r) ⊂ B3(0, 1) of radius r = (eR − 1)/(eR + 1).
Multiplying (1.1) by (1− r)n−1, we see that we need to estimate the
asymptotics of
(1− r)n−1 card(Bn(r) ∩ (F−1(y))). (3.3)
for arbitrary small ǫ = (1−r), or for arbitrary large λ = ln((2/(1−r))−1).
In the case of our free group F3 on 3 generators (the kernel of the
homomorphism ρ), Lemma 3.1 shows that the growth function γF3(k) =
1+3(5k−1)/2. This reduces the asymptotics of (3.3) to the asymptotics of
3(5λ/D−1)/e2λ for arbitrary large λ. Since the last expression tends to 0
when λ tends to ∞, we conclude that our bounded locally homeomorphic
16 Bounded Locally Homeomorphic Quasiregular Mappings
quasiregular mappings F :B3 → Ω1 ⊂ R3 in the unit ball B3 satisfy the
Vuorinen conjecture (1.1).
Remark 3.2. There is an important observation. If in our analysis of the
asymptotics of (3.3) (and in our construction of groups Γ and G) the ker-
nel of the corresponding homomorphism ρ :Γ → G ⊂ IsomH4 were a free
subgroup Fm on a big number m of generators, then our last expression
would tend to ∞ when λ tends to ∞. This would provide a way to con-
structing a similar bounded locally homeomorphic quasiregular mapping
in the unit ball giving a possible counter example to (1.1).
4. Appendix: Hyperbolic 4-cobordisms and deformations
of hyperbolic structures
For the readers convenience, here we provide essential details of our
construction [9] of .locally homeomorphic quasiregular surjective map-
pings F :S3\S∗ → S3 based on the properties of non-trivial “symmmetric”
hyperbolic 4-cobordisms M4 = (H4 ∪Ω(G))/G constructed in Apanasov
[7]. Properties of the fundamental group π1(M
4) ∼= G ⊂ IsomH4 of such
“symmmetric” hyperbolic 4-cobordisms M4 = H4/G acting discretely in
the hyperbolic 4-space H4 and in the discontinuity set Ω(G) ⊂ ∂H4 = S3
are very essential for our construction of the quasiregular mapping F .
We start with our construction [7] of such discrete group G ⊂ IsomH4
and the corresponding discrete representation ρ :Γ → G of a uniform hy-
perbolic lattice Γ ⊂ IsomH3. These discrete groups G and Γ produce a
non-trivial (not a product) hyperbolic 4-cobordismsM4 = (H4∪Ω(G))/G
whose boundary components N1 and N2 are topologically and geomet-
rically symmetric to each other. These N1 and N2 are covered by two
G-invariant connected components Ω1 and Ω2 of the discontinuity set
Ω(G) ⊂ S3, Ω(G) = Ω1 ∪Ω2. The conformal action of G = ρ(Γ) in these
components Ω1 and Ω2 is symmetric and has contractible fundamental
polyhedra P1 ⊂ Ω1 and P2 ⊂ Ω2 of the same combinatorial type allowing
to realize them as a compact polyhedron P0 in the hyperbolic 3-space,
i.e. the dihedral angle data of these polyhedra satisfy the Andreev’s
conditions [1]. Nevertheless this geometric symmetry of boundary com-
ponents of our hyperbolic 4-cobordism M4(G)) does not make the group
G = π1(M
4) quasi-Fuchsian, and our 4-cobordism M4 is non-trivial.
Here a Fuchsian group Γ ⊂ IsomH3 ⊂ IsomH4 conformally acts
in the 3-sphere S3 = ∂H4 and preserves a round ball B3 ⊂ S3 where
it acts as a cocompact discrete group of isometries of H3. Due to the
Sullivan structural stability (see Sullivan [22] for n = 2 and Apanasov [4],
Theorem 7.2), the space of quasi-Fuchsian representations of a hyperbolic
B. N. Apanasov 17
lattice Γ ⊂ IsomH3 into IsomH4 is an open connected component of
the Teichmüller space of H3/Γ or the variety of conjugacy classes of
discrete representations ρ :Γ → IsomH4. Points in this (quasi-Fuchsian)
component of the variety correspond to trivial hyperbolic 4-cobordisms
M(G) where the discontinuity set Ω(G) = Ω1 ∪ Ω2 ⊂ S3 = ∂H4 is the
union of two topological 3-balls Ωi, i = 1, 2, and M(G) is homeomorphic
to the product of N1 and the closed interval [0, 1].
We may consider hyperbolic 4-cobordisms M(ρ(Γ)) corresponding to
uniform hyperbolic lattices Γ ⊂ IsomH3 generated by reflections. Natu-
ral inclusions of these lattices into IsomH4 act at infinity ∂H4 = S3 as
Fuchsian groups Γ ⊂ Möb(3) preserving a round ball B3 ⊂ S3. In this
case our construction of the mentioned discrete groups Γ and G = ρ(Γ)
results in the following (see Apanasov [7]):
Theorem 4.1. There exists a discrete Möbius group G ⊂ Möb(3) on the
3-sphere S3 generated by finitely many reflections such that:
1. Its discontinuity set Ω(G) is the union of two invariant components
Ω1, Ω2;
2. Its fundamental polyhedron P ⊂ S3 has two contractible compo-
nents Pi ⊂ Ωi, i = 1, 2, having the same combinatorial type (of a
compact hyperbolic polyhedron P0 ⊂ H3);
3. For the uniform hyperbolic lattice Γ ⊂ IsomH3 generated by re-
flections in sides of the hyperbolic polyhedron P0 ⊂ H3 and act-
ing on the sphere S3 = ∂H4 as a discrete Fuchsian group i(Γ) ⊂
IsomH4 = Möb(3) preserving a round ball B3 (where i : IsomH3 ⊂
IsomH4 is the natural inclusion), the group G is its image under
a homomorphism ρ :Γ → G but it is not quasiconformally (topolog-
ically) conjugate in S3 to i(Γ).
Construction: We define a finite collection Σ of reflecting 2-spheres
Si ⊂ S3, 1 ≤ i ≤ N . As the first three spheres S1, S2 and S3 we take
the coordinate planes {x ∈ R3 : xi = 0}, and S4 = S2(0, R) is the
round sphere of some radius R > 0 centered at the origin. The value
of the radius R will be determined later. Let B =
⋃
1≤i≤4Bi be the
union of the closed balls bounded by these four spheres, and let ∂B be
its boundary (a topological 2-sphere) having four vertices which are the
intersection points of four triples of our spheres. We consider a simple
closed loop α ⊂ ∂B which does not contain any of our vertices and which
symmetrically separates two pairs of these vertices from each other as
the white loop does on the tennis ball. This loop α can be considered
18 Bounded Locally Homeomorphic Quasiregular Mappings
as the boundary of a topological 2-disc σ embedded in the complement
D = S3\B of our four balls. Our geometric construction needs a detailed
description of such a 2-disc σ and its boundary loop α = ∂σ obtained as
it is shown in Figure 1.
The desired disc σ ⊂ D = S3 \B can be described as the boundary in
the domain D of the union of a finite chain of adjacent blocks Qi (regular
cubes) with disjoint interiors whose centers lie on the coordinate planes S1
and S2 and whose sides are parallel to the coordinate planes. This chain
starts from the unit cube whose center lies in the second coordinate axis,
in e2 ·R+ ⊂ S1∩S3. Then our chain goes up through small adjacent cubes
centered in the coordinate plane S1, at some point changes its direction
to the horizontal one toward the third coordinate axis, where it turns its
horizontal direction by a right angle again (along the coordinate plane
S2), goes toward the vertical line passing through the second unit cube
centered in e1 ·R+ ⊂ S2∩S3, then goes down along that vertical line and
finally ends at that second unit cube, see Figure 1. We will define the
size of small cubes Qi in our block chain and the distance of the centers
of two unit cubes to the origin in the next step of our construction.
Figure 1: Configuration of blocks and the white loop α ⊂ ∂B.
B. N. Apanasov 19
Figure 2: Big and small cube sizes and ball covering.
Let us consider one of our cubes Qi, i.e. a block of our chain, and
let f be its square side having a nontrivial intersection with our 2-disc
σ ⊂ D. For that side f we consider spheres Sj centered at its vertices
and having a radius such that each two spheres centered at the ends of
an edge of f intersect each other with angle π/3. In particular, for the
unit cubes such spheres have radius
√
3/3. From such defined spheres we
select those spheres that have centers in our domain D and then include
them in the collection Σ of reflecting spheres. Now we define the distance
of the centers of our big (unit) cubes to the origin. It is determined by
the condition that the sphere S4 = S2(0, R) is orthogonal to the sphere
Sj ∈ Σ centered at the vertex of such a cube closest to the origin. As in
Figure 2, let f be a square side of one of our cubic blocks Qi having a
nontrivial intersection fσ = f ∩ σ with our 2-disc σ ⊂ D. We consider
a ring of four spheres Si whose centers are interior points of f which
lie outside of the four previously defined spheres Sj centered at vertices
of f and such that each sphere Si intersects two adjacent spheres Si−1
and Si+1 (we numerate spheres Si mod 4) with angle π/3. In addition
these spheres Si are orthogonal to the previously defined ring of bigger
spheres Sj, see Figure 2. From such defined spheres Si we select those
spheres that have nontrivial intersections with our domain D outside the
previously defined spheres Sj, and then include them in the collection Σ
of reflecting spheres. If our side f is not the top side of one of the two
unit cubes we add another sphere Sk ∈ Σ. It is centered at the center
20 Bounded Locally Homeomorphic Quasiregular Mappings
of this side f and is orthogonal to the four previously defined spheres Si
with centers in f , see Figure 2.
Now let f be the top side of one of the two unit cubes of our chain.
Then, as before, we consider another ring of four spheres Sk. Their centers
are interior points of f , lie outside of the four previously defined spheres
Si closer to the center of f and such that each sphere Sk intersects two
adjacent spheres Sk−1 and Sk+1 (we numerate spheres Sk mod 4) with
angle π/3. In addition these new four spheres Sk are orthogonal to the
previously defined ring of bigger spheres Si, see Figure 2. We note that
the centers of these four new spheres Sk are vertices of a small square
fs ⊂ f whose edges are parallel to the edges of f , see Figure 2. We set
this square fs as the bottom side of the small cubic box adjacent to the
unit one. This finishes our definition of the family of twelve round spheres
whose interiors cover the square ring f\fs on the top side of one of the
two unit cubes in our cube chain and tells us which two spheres among
the four new defined spheres Sk were already included in the collection
Σ of reflecting spheres (as the spheres Sj ∈ Σ associated to small cubes
in the first step).
This also defines the size of small cubes in our block chain. Now
we can vary the remaining free parameter R (which is the radius of the
sphere S4 ∈ Σ) in order to make two horizontal rows of small blocks with
centers in S1 and S2, correspondingly, to share a common cubic block
centered at a point in e3 · R+ ⊂ S1 ∩ S2, see Figure 1.
We can use the collection Σ of reflecting spheres Si to define a discrete
reflection group G = GΣ ⊂ Möb(3). Important properties of Σ are: (1)
the closure of the disc σ ⊂ D is covered by balls Bj ; (2) any two spheres
Sj , Sj′ ∈ Σ either are disjoint or intersect with angle π/2 or π/3; (3)
the complement of all balls Bj, 1 ≤ j ≤ N , is the union of two disjoint
contractible polyhedra P1 and P2 of the same combinatorial type and
equal corresponding dihedral angles. So the discontinuity set Ω(G) ⊂ S3
of G consists of two invariant connected components Ω1 and Ω2 which
are the unions of the G-orbits of P̄1 and P̄2, and this defines a Heegaard
splitting of the 3-sphere S3 (see [9]):
Lemma 4.2. The splitting of the discontinuity set Ω ⊂ S3 of our discrete
reflection group G = GΣ ⊂ Möb(3) into G-invariant components Ω1 and
Ω2 defines a Heegaard splitting of the 3-sphere S3 of infinite genus with
ergodic word hyperbolic group G action on the separating boundary Λ(G).
To finish our construction in Theorem 4.1 we notice that the com-
binatorial type (with magnitudes of dihedral angles) of the bounded
component P1 of the fundamental polyhedron P ⊂ S3 coincides with
B. N. Apanasov 21
the combinatorial type of its unbounded component P2. Applying An-
dreev’s theorem on 3-dimensional hyperbolic polyhedra [1], one can see
that there exists a compact hyperbolic polyhedron P0 ⊂ H3 of the same
combinatorial type with the same dihedral angles (π/2 or π/3). So one
can consider a uniform hyperbolic lattice Γ ⊂ IsomH3 generated by
reflections in sides of the hyperbolic polyhedron P0. This hyperbolic lat-
tice Γ acts in the sphere S3 as a discrete co-compact Fuchsian group
i(Γ) ⊂ IsomH4 = Möb(3) (i.e. as the group i(Γ) ⊂ IsomH4 where
i : IsomH3 ⊂ IsomH4 is the natural inclusion) preserving a round ball
B3 and having its boundary sphere S2 = ∂B3 as the limit set. Obviously
there is no self-homeomorphism of the sphere S3 conjugating the action
of the groups G and i(Γ) because the limit set Λ(G) is not a topological
2-sphere. So the constructed group G is not a quasi-Fuchsian group. �
One can construct a natural homomorphism ρ : Γ → G, ρ ∈ R3(Γ),
between these two Gromov hyperbolic groups Γ ⊂ IsomH3 and G ⊂
IsomH4 defined by the correspondence between sides of the hyperbolic
polyhedron P0 ⊂ H3 and reflecting spheres Si in the collection Σ bound-
ing the fundamental polyhedra P1 and P2. Then we have:
Proposition 4.3. The homomorphism ρ ∈ R3(Γ), ρ : Γ → G, in The-
orem 4.1 is not an isomorphism. Its kernel ker(ρ) = ρ−1(eG) is a free
rank 3 subgroup F3 � Γ.
Its proof (see [9], Prop.2.4) is based on the following statement (see [9],
Lemma 2.5) in combinatorial group theory:
Lemma 4.4. Let A = 〈a1, a2 | a21, a22, (a1a2)2〉 ∼= B = 〈b1, b2 | b21, b22,
(b1b2)
2〉 ∼= C = 〈c1, c2 | c21, c22, (c1c2)2〉 ∼= Z2 × Z2, and let ϕ :A ∗B → C
be a homomorphism of the free product A ∗B into C such that ϕ(a1) =
ϕ(b1) = c1 and ϕ(a2) = ϕ(b2) = c2. Then the kernel ker(ϕ) = ϕ−1(eC)
of ϕ is a free rank 3 subgroup F3�A∗B generated by elements x = a1b1,
y = a2b2 and z = a1a2b2a1 = a1ya1.
4.1. Bending homeomorphisms between polyhedra
Here we sketch our construction of a quasiconformal homeomorphism
φ1 :P1 → P0 between the fundamental polyhedron P1 ⊂ Ω1 ⊂ Ω(G) ⊂ S3
for the group G action in Ω1 and the fundamental polyhedron P0 ⊂ B3 for
conformal action of our hyperbolic lattice Γ ⊂ IsomH3 from Theorem 4.1.
This mapping φ1 is a composition of finitely many elementary "bending
homeomorphisms". It maps faces to faces, and preserve the combinatorial
structure of the polyhedra and their corresponding dihedral angles.
22 Bounded Locally Homeomorphic Quasiregular Mappings
First we observe that to each cube Qj , 1 ≤ j ≤ m, used in the above
construction of the group G (see Figure 1), we may associate a round
ball Bj centered at the center of the cube Qj and such that its bound-
ary sphere is orthogonal to the reflection spheres Si from our generating
family Σ whose centers are at vertices of the cube Qj. In particular for
the unit cubes Q1 and Qm, the reflection spheres Si centered at their ver-
tices have radius
√
3/3, so the balls B1 and Bm (whose boundary spheres
are orthogonal to those corresponding reflection spheres Si) should have
radius
√
5/12. Also we add another extra ball B3(0, R) (which we con-
sider as two balls B0 and Bm+1) whose boundary is the reflection sphere
S2(0, R) = S4 ∈ Σ centered at the origin and orthogonal to the closest
reflection spheres Si centered at vertices of two unit cubes Q1 and Qm.
Our different enumeration of this ball will be used when we consider dif-
ferent faces of our fundamental polyhedron P1 lying on that reflection
sphere S4.
Now for each cube Qj , 1 ≤ j ≤ m, we may associate a discrete sub-
group Gj ⊂ G ⊂ Möb(3) ∼= IsomH4 generated by reflections in the
spheres Si ∈ Σ associated to that cube Qj - see our construction in Theo-
rem 4.1. One may think about such a group Gj as a result of quasiconfor-
mal bending deformations (see [4], Chapter 5) of a discrete Möbius group
preserving the round ball Bj associated to the cube Qj (whose center
coincides with the center of the cube Qj). As the first step in such defor-
mations, we define two quasiconformal “bending” self-homeomorphisms
of S3, f1 and fm+1, preserving the balls B1, . . . , Bm and the set of their
reflection spheres Si, i 6= 4, and transferring ∂B0 and ∂Bm+ 1 into 2-
spheres orthogonally intersecting ∂B1 and ∂Bm along round circles b1
and bm+1, respectively - see (3.1) and Figure 6 in [9].
In the next steps in our bending deformations, for two adjacent cubes
Qj−1 and Qj, let us denote Gj−1,j ⊂ G the subgroup generated by re-
flections with respect to the spheres Si ⊂ Σ centered at common vertices
of these cubes. This subgroup preserves the round circle bj = bj−1,j =
∂Bj−1 ∩ ∂Bj . This shows that our group G is a result of the so called
"block-building construction" (see [4], Section 5.4) from the block groups
Gj by sequential amalgamated products:
G = G1 ∗
G1,2
G2 ∗
G2,3
· · · ∗
Gj−2,j−1
Gj−1 ∗
Gj−1,j
Gj ∗
Gj,j+1
· · · ∗
Gm−1,m
Gm (4.1)
The chain of these building balls {Bj}, 1 ≤ j ≤ m, contains the
bounded polyhedron P1 ⊂ Ω1. For each pair Bi−1 and Bi with the
common boundary circle bi = ∂Bi−1 ∩ ∂Bi, 1 ≤ i ≤ m, we construct
a quasi-conformal bending homeomorphism fi that transfers Bi ∪ Bi−1
onto the ball Bi and which is conformal in dihedral ζi-neighborhoods
B. N. Apanasov 23
of the spherical disks ∂Bi\Bi−1 and ∂Bi−1\Bi - see (3.3) and Figure 7
in [9]. In each i-th step, 2 ≤ i ≤ m, we reduce the number of balls Bj
in our chain by one. The composition fm+1fifi−1 · · · f2f1 transfers all
spheres from Σ to spheres orthogonal to the boundary sphere of our last
ball Bm which we renormalize as the unit ball B(0, 1). We note that all
intersection angles between these spheres do not change. We define our
quasiconformal homeomorphism
φ1 :P1 → P0 (4.2)
as the restriction of the composition fm+1fmfm−1 · · · f2f1 of our bending
homeomorphisms fj on the fundamental polyhedron P1 ⊂ Ω1.
4.2. Bounded locally homeomorphic quasiregular mappings
Now we define bounded quasiregular mappings F : B3 → R3 as in
Theorem 4.1 in [9]:
Theorem 4.5. Let the uniform hyperbolic lattice Γ ⊂ IsomH3 and its
discrete representation ρ : Γ → G ⊂ IsomH4 with the kernel as a free
subgroup F3 ⊂ Γ be as in Theorem 4.1. Then there is a bounded locally
homeomorphic quasiregular mapping F : B3 → R3 whose all singular-
ities lie in an exceptional subset S∗ of the unit sphere S2 ⊂ R3 and
form a dense in S2 Γ-orbit of a Cantor subset with Hausdorff dimension
ln 5/ ln 6 ≈ 0.89822444. These (essential) singularities create a barrier
for F in the sense that at points x ∈ S∗ the map F does not have radial
limits.
Construction: We construct our quasiregular mapping F :B3 → Ω1 =
F (B3) in the unit ball B3 by equivariant extention of the above quasicon-
formal homeomorphism φ−1
1 :P0 → P1 which maps polyhedral sides of P0
to the corresponding sides of the polyhedr P1 and preserves combinatorial
structures of polyhedra as well as their dihedral angles:
F (x) = ρ(γ) ◦ φ−1
1 ◦ γ−1(x) if |x| < 1, x ∈ γ(P0), γ ∈ Γ (4.3)
The tesselations of B3 and Ω1 by corresponding Γ- and G-images of their
fundamental polyhedra Po and P1 are perfectly similar. This implies
that our quasiregular mapping F defined by (4.3) is bounded and locally
homeomorphic. It follows from Lemma 4.2 that the limit set Λ(G) ⊂ S3
of the group G ⊂ Möb(3) defines a Heegard splitting of infinite genus
of the 3-sphere S3 into two connected components Ω1 and Ω2 of the
discontinuity set Ω(G). The action of G on the limit set Λ(G) is an
ergodic word hyperbolic action. For this ergodic action the set of fixed
24 Bounded Locally Homeomorphic Quasiregular Mappings
points of loxodromic elements g ∈ G (conjugate to similarities in R3) is
dense in Λ(G). Preimages γ ∈ Γ of such loxodromic elements g ∈ G
for our homomorphism ρ : Γ → G are loxodromic elements in Γ with
two fixed points p, q ∈ Λ(Γ) = S2, p 6= q. This and Tukia’s arguments
of the group completion (see [23] and [4], Section 4.6) show that our
mapping F can be continuously extended to the set of fixed points of such
elements γ ∈ Γ, F (Fix(γ)) = Fix(ρ(γ)). The sense of this continuous
extension is that if γ ∈ Γ is a loxodromic preimage of a loxodromic
element g ∈ G, ρ(γ) = g, and if x ∈ S3\S2 tends to its fixed points p or
q along the hyperbolic axis of γ (in B(0, 1) or in its complement B̂(0, 1))
(i.e. radially) then lim|x|→1 F (x) exists and equals to the corresponding
fixed point of the loxodromic element g = ρ(γ) ∈ G. In that sense one
can say that the limit set Λ(G) (the common boundary of the connected
components Ω1,Ω2 ⊂ Ω(G)) is the F -image of points in the unit sphere
S2 ⊂ S3. So the mapping F is extended to a map onto the closure
Ω1 = Ω1 ∪ Λ(G) ⊂ R3.
Nevertheless not all loxodromic elements γ ∈ Γ in the hyperbolic lat-
tice Γ ⊂ IsomH3 have their images ρ(γ) ∈ G as loxodromic elements.
Proposition 4.3 shows that ker ρ ∼= F3 is a free subgroup on three gener-
ators in the lattice Γ, and all elements γ ∈ F3 are loxodromic. Now we
look at radial limits limx→p F (x) when x radially tends to a fixed point
p ∈ S2 of this loxodromic element γ ∈ F3 ⊂ Γ.
LetK(Γ,Σ) be the Cayley graph for a group Γ with a finite generating
set Σ. Our lattice Γ ⊂ IsomH3 has an embedding ϕ of its Cayley graph
K(Γ,Σ) in the hyperbolic space H3 ∼= B3. For a point 0 ∈ H3 not fixed
by any γ ∈ Γ\{1}, vertices γ ∈ K(Γ,Σ) are mapped to γ(0), and edges
joining vertices a, b ∈ K(Γ,Σ) are mapped to the hyperbolic geodesic seg-
ments [a(0), b(0)]. In other words, ϕ(K(Γ,Σ)) is dual to the tessellation
of H3 by polyhedra γ(P0), γ ∈ Γ. The map ϕ is a Γ-equivariant proper
embedding: for any compact C ⊂ H3, its pre-image ϕ−1(ϕ(K(Γ,Σ))∩C)
is compact. Moreover for any convex cocompact group Γ ⊂ IsomHn this
embedding ϕ is a pseudo-isometry (see [12] and [4], Theorem 4.35), i.e.
for the word metric on K(Γ,Σ) and the hyperbolic metric d, there are
K > 0 and K ′ > 0 such that |a, b|/K ≤ d(ϕ(a), ϕ(b)) ≤ K · |a, b| for all
a, b ∈ K(Γ,Σ) such that either |a, b| ≥ K ′ or d(ϕ(a), ϕ(b)) ≥ K ′.
This implies (see [4], Theorem 4.38) that the limit set of any convex-
cocompact group Γ ⊂ Möb(n) can be identified with its group completion
Γ, Γ = K(Γ,Σ) \ K(Γ,Σ). Namely there exists a continuous and Γ-
equivariant bijection ϕΓ :Γ → Λ(Γ).
For the kernel subgroup F3 = ker ρ ⊂ Γ ⊂ IsomH3 and for the
above pseudo-isometric embedding ϕ, we consider its Cayley subgraph in
B. N. Apanasov 25
ϕ(K(Γ,Σ)) ⊂ H3 which is a tree - see Figure 5 in [9]. Since the limit
set of ker ρ = F3 ⊂ Γ corresponds to the “bondary at infinity” ∂∞F3 of
F3 ⊂ Γ (the group completion F3), it is a closed Cantor subset of the
unit sphere S2 with Hausdorff dimension ln 5/ ln 6 ∼ 0.89822444.
The Γ-orbit Γ(Λ(F3)) of our Cantor set is a dense subset S∗ of S2 =
Λ(Γ) because of density in the limit set Λ(Γ) of the Γ-orbit of any limit
point. In particular we have such dense Γ-orbit Γ({p, q}) of fixed points
p and q of a loxodromic element γ ∈ F3 ⊂ Γ (the images of p and q are
fixed points of Γ-conjugates of such loxodromic elements γ ∈ F3 ⊂ Γ).
On the other hand let x ∈ lγ where lγ is the hyperbolic axis in B(0, 1)
of an element γ ∈ F3 ⊂ Γ. Denoting dγ the translation distance of γ,
we have that any segment [x, γ(x)] ⊂ lγ is mapped by our quasiregular
mapping F to a non-trivial closed loop F ([x, γ(x)]) ⊂ Ω1, inside of a
handle of the handlebody Ω1 (mutually linked with Ω2 - similar to the
loops β1 ⊂ Ω1 and β2 ⊂ Ω2 constructed in the proof of Lemma 4.2).
Therefore when x ∈ lγ radially tends to a fixed point p (in fix(γ) ∈ S2) of
such element γ, its image F (x) goes along that closed loop F ([x, γ(x)]) ⊂
Ω1 because F (γ(x)) = ρ(γ)(F (x)) = F (x). Immediately it implies that
the radial limit limx→p F (x) does not exist. This shows that fixed points
of any element γ ∈ F3 ⊂ Γ (or its conjugates) are essential (topological)
singularities of our quasiregular mapping F . So our quasiregular mapping
F has no continuous extension to the subset S∗ ⊂ S2 which is a dense
subset of the unit sphere S2 = ∂B3 ⊂ S3. �
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Contact information
Boris Nikolaevich
Apanasov
Department of Mathematics,
University of Oklahoma, Norman, USA
E-Mail: apanasov@ou.edu
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