On Value Distributions for Quasimeromorphic Mappings on H-type Carnot Groups

In the present paper we define quasimeromorphic mappings on homogeneous groups and study their properties. We prove an analogue of results of L. Ahlfors, R. Nevanlinna and S. Rickman, concerning the value distribution for quasimeromorphic mappings on H-type Carnot groups for parabolic and hyperbolic type domains. Introduction The classical value distribution theory for analytic functions w(z) studies the system of sets za of a domain Gz where the function w(z) takes the value w = a for an arbitrary a. A central result in the distribution theory is the Picard theorem, stating that a meromorphic function in the punctured plane assumes all except for at most two values a1, a2, a1 6= a2 infinitely often. In an equivalent way, we can say that an analytic function w(z) : R2 → R 2 \ {a1, a2} must be constant if a1 and a2 are distinct points in R 2. We mention results of J. Hadamard, E. Borel, G. Julia, A. Beurling, L. V. Ahlfors and others [1, 2, 7, 28] in general value distribution theory, going far beyond Picard-type theorems. Nevertheless, in those extensions and deep generalizations the nature of conformal mappings was not actively involved. New ideas of the function theory and potential theory point of view were incoming by R. Nevanlinna [44, 45]. The most important achievements of the Nevanlinna theory were not only analytic deep results, but its geometric aspects and relations with Riemannian surfaces of analytic functions. Such principal notions, as a characteristic function, a defect function, a branching index connect the asymptotic behavior of an analytic function w(z) with properties of the Riemannian surface which is the conformal image of the domain of w(z). A natural generalization of an analytic function of one complex variable to the Euclidean space of the dimension n > 2 was firstly introduced and studied by Yu. G. Reshetnyak in 1966—1968 [50, 51, 52]. In some sense this is a quasiconformal mapping admitting branch points. Such mappings were called in Russian school the mappings with bounded distortion. The main contribution of Yu. G. Reshetnyak to the foundation of this theory is a discovery of a connection between mappings with bounded distortion and non-linear partial differential equations. Yu. G. Reshetnyak has proved also that an analytic definition of mappings with bounded distortion implies the topological properties: the continuity, the openness, and the discreteness. Later these mappings, under the name quasiregular mappings, were investigated intensively by O. Martio, S. Rickman, J. Väisälä, F. W. Gehring, M. Vuorinen, B. Bojarski, T. Iwaniec and others [5, 6, 21, 41, 42, 58, 60, 77]. Briefly a quasiregular 2000 Mathematics Subject Classification. 32H30, 31B15, 43A80.