Quasiregular Mappings and Discrete Group Actions

Abstract

We develop a new tool based on quasiconformal dynamics and conformal dynamics of discrete group actions in 3-geometries to construct new types of quasiregular and quasisymmetric mappings in space. This tool has close relations to new effects in Teichmüller spaces of conformally flat structures on closed hyperbolic 3-manifolds/orbifolds and non-trivial hyperbolic 4-cobordisms, to the hyperbolic and conformal interbreedings as well as to non-faithful discrete representations of uniform hyperbolic 3-lattices. We demonstrate several applications of this tool and new types of quasiregular mappings in space. Leaving such applications to geometry and topology of manifolds to another our papers [10, 11], here we continue a series of applications of our constructions to long standing problems for quasiregular mappings in space, including M.A. Lavrentiev surjectivity problem, Pierre Fatou problem on radial limits and Matti Vuorinen injectivity and asymptotics problems for bounded quasiregular mappings in the unit 3-ball (cf. [4, 7-9]).