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

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.