Embedding of some finite geometries into Riemann surfaces

Abstract

In this thesis we provide examples of a new approach to the field of finite geometries, namely by considering the conformal embedding of a finite geometry into a Riemann surface.  A finite geometry is a particular arrangement of a finite number of points and lines satisfying some well known axioms.  We will cover the first two examples of the family of Hadamard designs, which are the Fano plane and the 3-biplane.

Riemann surfaces and dessins are introduced and explained in chapter one.  We explore their common relationship to cocompact Fuchsian groups and display some results regarding the calculation of their automorphisms groups.  We also describe the three most common geometric representations of a dessin:  those by Cori, James and Walsh.

Chapter two is divided into two different parts.  In the first one we cover the family of finite groups PSL(2,p) where p is a prime number, particularly for the cases where p ∈ {5,7,11}.  In the second part of the chapter we introduce the family of Hecke groups H^q and their special congruence subgroups, with special regards to the cases where q = 3 and q = 5.

In chapter three we cover finite geometries and their properties.  Projective planes and biplanes are studied in different sections paying special attention to the Fano plane as our chosen representative for the projective planes and the 3-biplane.

Finally in chapter four we make extensive use of all the preliminar material by finding and describing several conformal embeddings of the Fano plane and the 3-biplane into Riemann surfaces, especially into those Riemann surfaces with automorphism group isomorphic to PSL(2,p) and that can be uniformized by a special congruence subgroup of H^q.

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