טכניון מכון טכנולוגי לישראל
הטכניון מכון טכנולוגי לישראל - בית הספר ללימודי מוסמכים  
Ph.D Thesis
Ph.D StudentEmil Saucan
SubjectThe Existence of Quasimeromorphic Mappings
DepartmentDepartment of Mathematics
Supervisor Professor Emeritus Srebro Uri (Deceased)


Abstract

      It is classical that every Riemann surface carries non-constant  meromorphic functions, implying that every Fuchsian group G has non-constant G-automorphic (meromorphic) functions.

      In higher dimensions n ≥ 3 the only locally conformal mappings are restrictions of Möbius transformations, and since they are injective, an extension of the classical existence theorem requires to look at quasimeromorphic mappings.

      Following partial results by Martio, Srebro and Tukia on the problem of existence or non-existence of non-constant

quasimeromorphic G-automorphic mappings (G being a discrete Möbius group acting on the hyperbolic space Hn we now give a complete characterization of all discrete Möbius groups G acting on hyperbolic space Hn,  that admit non-constant G-automorphic quasimeromorphic mappings, for any n ≥ 2.

      Following results by Tukia and Peltonen on the existence of non-constant quasimeromorphic mappings on complete CRiemannian manifolds, we now prove the existence of such mappings on manifolds with boundary, of lower differentiability class. Since the proofs are based on the existence of fat triangulations, we extend a classical result of Munkres by showing that every manifold C1 Mn with boundary consisting of finitely many compact components has a fat triangulation. We also prove that any fat triangulation of ∂Mn can be extended to Mn.