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arXiv · 2312.03504

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

Maxime Fortier Bourque·Émile Gruda-Mediavilla·Bram Petri·Mathieu Pineault
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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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Maxime Fortier Bourque, Émile Gruda-Mediavilla, Bram Petri, Mathieu Pineault. 2023-12-06. Three counterexamples to a conjecture of Colin de Verdi\`ere on multiplicity. https://arxiv.org/abs/2312.03504

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