Factorizations of Finite Mappings on Riemann Surfaces

This dissertation, "Factorizations of Finite Mappings on Riemann Surfaces" by Mingxi, Wang, 汪明晰, was obtained from The University of Hong Kong (Pokfulam, Hong Kong) and is being sold pursuant to Creative Commons: Attribution 3.0 Hong Kong License. The content of this dissertation has not been altered in any way. We have altered the formatting in order to facilitate the ease of printing and reading of the dissertation. All rights not granted by the above license are retained by the author. Abstract: Abstract of thesis entitled FACTORIZATIONS OF FINITE MAPPINGS ON RIEMANN SURFACES submitted by Mingxi Wang for the Degree of Master of Philosophy at The University of Hong Kong in November 2007 Theaimofthisthesisstudywastoestablishsomefundamentalresultsonthe factorizations(inthesameofcompositions)ofnitemappingsonRiemannsur- faces, especially the nite Blaschke products. One main result is that, although a nite Blaschke product may have many di(R)erent factorizations into prime nite Blaschke products, the number of the prime nite Blaschke products in each factorization is always the same. Another result is an explicit characteri- zation on the relationship between two di(R)erent prime factorizations of a nite Blaschke product, which tells us how one can pass from one factorization to the other one. This thesis is mainly motivated by Ritt's original results on factorizations of polynomials and F. Dorey and G. Whaples's di(R)erent proof of Ritt's second theorem (Theorem R2). One of Ritt's results is that the length of a prime factorization of a polynomial is an invariant. F. Dorey and G. Whaples's proof of this result is based on some local properties of the function elds at innity. Their proof suggests that the reason for having such an invariant is that the localpropertyofapolynomialaroundinnityissogoodthatitcangiveaglobal invariant. Therefore, it is of interest to investigate what good local properties can induce a global invariant in general. In this thesis, this problem for nite mappings between Riemann surfaces was studied. Ritt's original results on factorizations of polynomials and many other rel-evant standard materials were introduced briey in section 1.1 and 1.2. In section 1.3, a variation of Riemann-Hurwitz Formula was stated and proved. It was also shown that the Schreier Index Formula and the Riemann-Hurwitz Formula are indeed equivalent to each other. Section 1.4 includes all the results on lattices and group theory that will be used in chapter 2. Ritt's rst two theorems (Theorem R1 and R2) were generalized to nite mappingsbetweenRiemannsurfacesinsection2.1. Thestatingpointhereisthe belief that a good local property can induce a global invariant. To describe the desiredgoodlocalpropertyofanitemapping, loopgroupsratherthanfunction 0 0 elds were considered and Theorem R1 and Theorem R2 were established. As a corollary of Theorem R2, the length of any prime factorization of a nite Blaschke product is an invariant. In section 2.2, some geometric properties of a Blaschke product were recov- eredfromaspecialmonodromy, whichisessentialforthestudyoffactorizations of nite Blaschke products. The main tool for recovering this nite mapping is a fundamental technique in Riemannian geometry which says that the set of xed points of a family of isometries on a Riemannian manifold is totally geodesic. Finally, in section 2.3, by making use of the established results, the main theorem (Theorem 2.3.5) on the factorizations of nite Blaschke products was stated and proved. DOI: 10.5353/th_b3955778 Subjects: Factorization (Mathematics) Riemann surfaces