The Dirac operator on cusped hyperbolic manifolds

Bruno Martelli; Alan W. Reid

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arXiv · 2212.06811

The Dirac operator on cusped hyperbolic manifolds

Bruno Martelli·Alan W. Reid

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Abstract

We study how the spin structures on finite-volume hyperbolic n-manifolds restrict to cusps. When a cusp cross-section is a (n-1)-torus, there are essentially two possible behaviours: the spin structure is either bounding or Lie. We show that in every dimension n there are examples where at least one cusp is Lie, and in every dimension n <= 8 there are examples where all the cusps are bounding. By work of C. Bar, this implies that the spectrum of the Dirac operator is R in the first case, and discrete in the second. We therefore deduce that there are cusped hyperbolic manifolds whose spectrum of the Dirac operator is R in all dimensions, and whose spectrum is discrete in all dimensions n <= 8.

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Bruno Martelli, Alan W. Reid. 2022-12-13. The Dirac operator on cusped hyperbolic manifolds. https://arxiv.org/abs/2212.06811

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