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Three counterexamples to a conjecture of Colin de Verdi\`ere on multiplicity

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arxiv 2312.03504 v2 pith:WQUDRZ7B submitted 2023-12-06 math.SP math.GT

Maxime Fortier Bourque , Émile Gruda-Mediavilla , Bram Petri , Mathieu Pineault This is my paper · ORCID
classification math.SPmath.GT
keywords colingroupsmultiplicitysurfacesverdiapplyclosedconjecture
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abstract

We exhibit closed hyperbolic surfaces of genus $10$, $17$, and $37$ such that the multiplicity of the first nonzero eigenvalue of their Laplacian is larger than the maximum conjectured by Yves Colin de Verdi\`ere in 1986. In order to determine these multiplicities, we apply the twisted Selberg trace formula to the representations induced by the isometry groups of these surfaces on corresponding triangle groups.

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  1. Limit points of uniform arithmetic bass notes

    math.SP 2024-12 conditional novelty 7.0 of 10

    For closed arithmetic hyperbolic surfaces, the possible values of the first nonzero Laplace eigenvalue form a dense subset of [0, 1/4].

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