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

TL;DR

This paper presents explicit counterexamples of hyperbolic surfaces where the multiplicity of the first nonzero Laplacian eigenvalue exceeds the bounds conjectured by Colin de Verdière, using advanced trace formula techniques.

Contribution

It provides the first known counterexamples to Colin de Verdière's conjecture on eigenvalue multiplicities for hyperbolic surfaces.

Findings

Counterexamples for genus 10 and 17 surfaces.

Eigenvalue multiplicity exceeds conjectured bounds.

Application of twisted Selberg trace formula to determine multiplicities.

Abstract

We exhibit closed hyperbolic surfaces of genus $10$ and $17$ 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.