We develop a new tool based on quasiconformal dynamics and conformal of discrete group actions in 3-geometries to construct types quasiregular quasisymmetric mappings space. This has close relations effects Teichmüller spaces conformally flat structures closed hyperbolic 3-manifolds/orbifolds non-trivial 4-cobordisms, the interbreedings as well non-faithful representations uniform 3-lattices. d...