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. 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 $B^3\subset {\mathbb{R}}^3$ as mappings equivariant with the standard conformal action of uniform hyperbolic lattices $\Gamma \subset \mathrm{Isom}{H}^3$ in the unit 3-ball and with its discrete representation $G=\rho (\Gamma )\subset \mathrm{Isom}{H}^4$. Here $G$ is the fundamental group of our non-trivial hyperbolic 4-cobordism $M=(H^4\cup \Omega (G))/G$ and the kernel of the homomorphism $\rho :\Gamma \to G$ is a free group $F_3$ on three generators.

Clarification of the previous submission. 13 pages, 2 figures. arXiv admin note: text overlap with arXiv:1510.08951, arXiv:1611.00432

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