Harmonic spinors on closed hyperbolic manifolds 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.
 A: I'm happy to be able to answer my own question! John Ratcliffe, Steven Tschantz and I showed that the Dirac operator on the Davis manifold (a closed hyperbolic 4-manifold constructed by Mike Davis) has non-zero kernel. This is in the preprint Harmonic spinors on the Davis hyperbolic 4-manifold that we posted on the arxiv today.
The method is to find a symmetry of the manifold that respects a particular spin structure, and use the G-spin theorem to show that the G-spinor index is non-zero. This can only happen if the kernel of the Dirac operator is non-zero. The method also works in dimension 2 (where the result is of course not new) and in principle in any even dimension. It doesn't (to my knowledge) work in odd dimensions, although Misha's suggestions in the comments above seem promising there.
A: According to Rula Tabbash in this paper, all negatively curved closed manifolds admit harmonic spinors (and no positively curved ones do). I am not 100% certain, since this is a physics paper, and so it is not clear if he is assuming something unstated...
