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Abstract
Given a hyperbolic surface with boundary, arc coordinates provide a parametrization of the Teichmüller space. They rely on the choice of a family of arcs which start and end at boundary components and are orthogonal to them. Higher rank Teichmüller theories are a generalization of classical Teichmüller theory and are concerned with the study of representations of the fundamental group of an oriented surface of negative Euler characteristic into simple real Lie groups G of higher rank. It is well known that maximal representations are a higher rank Teichmüller theory for G Hermitian. In this thesis we will discuss how to generalize arc coordinates for maximal representations, focusing on the case where the surface is a pair of pants and G is PSp(4,R). This will be possible by introducing geometric parameters on the space of right-angled hexagons in the Siegel space X, which lead to the visualization of a right-angled hexagon as a polygonal chain inside the hypervolic plane. We discuss geometric properties of reflections in X and introduce the notion of maximal representations of a reflection group W3. We give a parametrization of maximal representations of W3 into PSp(4,R), which allows us to parametrize a subset of maximal and Shilov hyperbolic representations into PSp(4,R).
Document type: | Dissertation |
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Supervisor: | Pozzetti, Prof. Dr. Maria Beatrice |
Place of Publication: | Heidelberg |
Date of thesis defense: | 1 March 2024 |
Date Deposited: | 13 Mar 2024 14:12 |
Date: | 2024 |
Faculties / Institutes: | The Faculty of Mathematics and Computer Science > Department of Computer Science The Faculty of Mathematics and Computer Science > Institut für Mathematik |
DDC-classification: | 510 Mathematics |