Hyperbolic topology and bounded locally homeomorphic quasiregular mappings in 3-space

• Veröffentlicht in: Український математичний вісник
• Datum: 2019
• 1. Verfasser: Apanasov, B.N.
• Format: Artikel
• Sprache: English
• Veröffentlicht: Інститут прикладної математики і механіки НАН України 2019
• Online Zugang: https://nasplib.isofts.kiev.ua/handle/123456789/169429
• Назва журналу: Digital Library of Periodicals of National Academy of Sciences of Ukraine
• Zitieren: 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.

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.