Algorithmic Construction of Fundamental Polygons for Certain Fuchsian Groups
Independent thesis Advanced level (degree of Master (Two Years)), 20 credits / 30 HE creditsStudent thesis
The work of mathematical giants, such as Lobachevsky, Gauss, Riemann, Klein and Poincaré, to name a few, lies at the foundation of the study of the highly structured Riemann surfaces, which allow definition of holomorphic maps, corresponding to analytic maps in the theory of complex analysis. A topological result of Poincaré states that every path-connected Riemann surface can be realised by a construction of identifying congruent points in the complex plane, the Riemann sphere or the hyperbolic plane; just three simply connected surfaces that cover the underlying Riemann surface. This requires the discontinuous action of a discrete subgroup of the automorphisms of the corresponding space. In the hyperbolic plane, which is the richest source for Riemann surfaces, these groups are called Fuchsian, and there are several ways to study the action of such groups geometrically by computing fundamental domains. What is accomplished in this thesis is a combination of the methods found by Reidemeister & Schreier, Singerman and Voight, and thus provides a unified way of finding Dirichlet domains for subgroups of cofinite groups with a given index. Several examples are considered in-depth.
Place, publisher, year, edition, pages
2015. , 91 p.
Compact Riemann surfaces and uniformization; Fuchsian groups and automorphic functions; Hyperbolic geometry; Topological groups; Permutation groups; Computational methods; Covering spaces, branched coverings; Generators, relations, and presentations
IdentifiersURN: urn:nbn:se:liu:diva-119916ISRN: LiTH-MAT-EX--2015/05--SEOAI: oai:DiVA.org:liu-119916DiVA: diva2:833650
Subject / course
Costa F., Antonio
Izquierdo Barros, Milagros