Ph.D Thesis | |

Ph.D Student | Saucan Emil |
---|---|

Subject | The Existence of Quasimeromorphic Mappings |

Department | Department of Mathematics |

Supervisor | PROFESSOR EMERITUS Uri Srebro (Deceased) |

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 *H*^{n} we now give a complete characterization of all
discrete Möbius groups *G* acting on hyperbolic space *H*^{n},
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 *C*^{∞ }Riemannian 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 *C*^{1 }*M*^{n} with boundary consisting of finitely
many compact components has a fat triangulation. We also prove that any fat
triangulation of ∂*M*^{n} can be extended to *M*^{n}.