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

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Abstract

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 Γ ⊂ Isom H3 in the unit 3-ball 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.

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Authors and Affiliations

  1. Department of Mathematics, University of Oklahoma, Norman, USA

    Boris N. Apanasov

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  1. Boris N. Apanasov
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Correspondence to Boris N. Apanasov.

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The paper dedicated to the memory of my colleague and friend, brilliant mathematician Bogdan Bojarski

Translated from Ukrains’kiĭ Matematychnyĭ Visnyk, Vol. 16, No. 1, pp. 10–27 January–March, 2019.

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Apanasov, B.N. Hyperbolic topology and bounded locally homeomorphic quasiregular mappings in 3-space. J Math Sci 242, 760–771 (2019). https://doi.org/10.1007/s10958-019-04514-4

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