The multiplicity of the smallest positive eigenvalue of the Laplacian on the Klein Quartic

This thesis discusses various ideas related to the proof of the statement the multiplicity $m_1(K)$ of the smallest positive eigenvalue $\lambda_1(K)$ of the Laplacian on the hyperbolic surface called the Klein Quartic is equal to 8. First, we take a look at how the Selberg Trace Formula can be used to obtain bounds for $m_1(K)$. Second, we show the existence of a representation whose decomposition into irreducible summands implies an integer equation for the multiplicity $m_1(K)$ of $\lambda_1(K)$. Third, we investigate the conditions under which a polyhedron and its reflection group provides a tessellation of hyperbolic space. Finally, we study the macro structure of the proof of the statement $m_1(K) = 8$ and how the different strategies are combined to prove the desired result. This work contains no original results.