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# Loughborough University Institutional Repository

 Please use this identifier to cite or link to this item: https://dspace.lboro.ac.uk/2134/36294

 Title: Properties of eigenvalues on Riemann surfaces with large symmetry groups Authors: Cook, Joseph Keywords: Riemann surfaceKlein quarticBolza surfaceRepresentation theorySymmetrySpectral theoryAnalysisHyperbolic geometry Issue Date: 2018 Publisher: © Joseph Cook Abstract: On compact Riemann surfaces, the Laplacian $\Delta$ has a discrete, non-negative spectrum of eigenvalues $\{\lambda_{i}\}$ of finite multiplicity. The spectrum is intrinsically linked to the geometry of the surface. In this work, we consider surfaces of constant negative curvature with a large symmetry group. It is not possible to explicitly calculate the eigenvalues for surfaces in this class, so we combine group theoretic and analytical methods to derive results about the spectrum. In particular, we focus on the Bolza surface and the Klein quartic. These have the highest order symmetry groups among compact Riemann surfaces of genera 2 and 3 respectively. The full automorphism group of the Bolza surface is isomorphic to $\mathrm{GL}_{2}(\mathbb{Z}_{3})\rtimes\mathbb{Z}_{2}. We analyze the irreducible representations of this group and prove that the multiplicity of$\lambda_{1}$is 3, building on the work of Jenni, and identify the irreducible representation that corresponds to this eigenspace. This proof relies on a certain conjecture, for which we give substantial numerical evidence and a hopeful method for proving. We go on to show that$\lambda_{2}\$ has multiplicity 4. Description: A Doctoral Thesis. Submitted in partial fulfilment of the requirements for the award of Doctor of Philosophy of Loughborough University.This Thesis has been redacted for reasons relating to the law of copyright. For more information please contact the author. Sponsor: EPSRC. URI: https://dspace.lboro.ac.uk/2134/36294 Appears in Collections: PhD Theses (Maths)

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