Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 43, 2018, 579-596

TOPOLOGICAL BARRIERS FOR LOCALLY HOMEOMORPHIC QUASIREGULAR MAPPINGS IN 3-SPACE

Boris N. Apanasov

University of Oklahoma, Department of Mathematics
Norman, OK 73019, U.S.A.; apanasov 'at' ou.edu

Abstract. We construct a new type of locally homeomorphic quasiregular mappings in the 3-sphere and discuss their relation to the Lavrentiev problem, the Zorich map with an essential singularity at infinity, the Fatou's problem and a quasiregular analogue of domains of holomorphy in complex analysis. 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 locally homeomorphic quasiregular mappings are defined in the 3-sphere S³ as mappings equivariant with the standard conformal action of uniform hyperbolic lattices Γ ⊂ Isom H³ in the unit 3-ball and its complement in S³ and with its discrete representation G = ρ (Γ) ⊂ Isom H⁴. 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 Mathematics Subject Classification: Primary 30C65, 57Q60, 20F55, 32T99, 30F40, 32H30, 57M30.

Key words: Quasiregular mappings, local homeomorphisms, Fatou's problem, domains of holomorphy, hyperbolic group action, hyperbolic manifolds, cobordisms, group homomorphism, deformations of geometric structures.

Reference to this article: B. N. Apanasov: Topological barriers for locally homeomorphic quasiregular mappings in 3-space. Ann. Acad. Sci. Fenn. Math. 43 (2018), 579-596.

Full document as PDF file

https://doi.org/10.5186/aasfm.2018.4337