Does anyone know an example of a closed spin hyperbolic manifold of dimension 3 or greater such that the kernel of the Dirac operator is non-trivial?

I'm mainly interested in the 3-dimensional case but would be happy to hear about higher dimensions as well. Note that Hitchin showed that for a particular choice of spin structure on a surface, the Dirac operator necessarily has kernel. There are strong results of Bär (The Dirac Operator on Hyperbolic Manifolds of Finite Volume, J. Diff. Geom. 54 (2000), 439--488) for finite-volume hyperbolic manifolds but I haven't found anything for closed hyperbolic manifolds.

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