Topological barriers for locally homeomorphic quasiregular mappings in 3-space

We construct a new type of locally homeomorphic quasiregular mappings in the 3-sphere and discuss their relation to the M.A.Lavrentiev problem, the Zorich map with an essential singularity at infinity, the Fatou’s problem and a quasiregular analogue of domains of holomorphy in complex analysis. 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 locally homeomorphic quasiregular mappings are defined in the 3-sphere $S^3$ as mappings equivariant with the standard conformal action of uniform hyperbolic 3-lattices $\mathrm{\Gamma }$ in the unit 3-ball and its complement in $S^3$ and with its discrete representation $G=\rho (\mathrm{\Gamma })$ in the group of isometries of $H^4$. 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.