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

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 $\Gamma\subset \operatorname{Isom} H^3$ in the unit 3-ball and with its discrete representation $G=\rho (\mathrm{\Gamma })\subset \mathrm{Isom}{H}^4$. Here $G$ is the fundamental group of our non-trivial hyperbolic 4-cobordism $M=(H^4\cup \mathrm{\Omega }(G))\mathrm{/}G$ and the kernel of the homomorphism $\rho!:! \Gamma\rightarrow G$isafreegroup$F_3$ on three generators.