Dirac spectral flow and Floer theory of hyperbolic three-manifolds

We study the interplay between hyperbolic geometry and monopole Floer homology for a closed oriented three-manifold $Y$ with $b_1=1$ equipped with a torsion spin$^c$ structure $\mathfrak{s}$. We show that, under favorable circumstances, one can completely describe the Floer theory of $(Y,\mathfrak{s})$ purely in terms of geometric data such as the lengths and holonomies of closed geodesics. In particular, we perform the first computations of monopole Floer chain complexes with non-trivial homology for hyperbolic three-manifolds. The examples we consider admit no irreducible solutions to the Seiberg-Witten equations, and the non-triviality of the Floer homology groups is a consequence of the geometry of the $1$-parameter family of Dirac operators associated to flat spin$^c$ connections. The main technical challenge is to understand explicitly how the Dirac eigenvalues with small absolute value cross the value zero in this family; we tackle this using Fourier analytic tools via the corresponding $1$-parameter family of odd Selberg trace formulas and its derivative.