Closed geodesics and Frøyshov invariants of hyperbolic three-manifolds

Abstract

Frøyshov invariants are numerical invariants of rational homology three-spheres derived from gradings in monopole Floer homology. In the past few years, they have been employed to solve a wide range of problems in 3- and 4-dimensional topology. In this paper, we look at connections with hyperbolic geometry for the class of minimal $L$-spaces. In particular, we study relations between Frøyshov invariants and closed geodesics using ideas from analytic number theory. We discuss two main applications of our approach. First, we derive explicit upper bounds for the Frøyshov invariants of minimal hyperbolic $L$-spaces purely in terms of volume and injectivity radius. Second, we describe an algorithm to compute Frøyshov invariants of minimal $L$-spaces in terms of data arising from hyperbolic geometry. As a concrete example of our method, we compute the Frøyshov invariants for all spin${}^c$ structures on the Seifert–Weber dodecahedral space. Along the way, we also prove several results about the eta invariants of the odd signature and Dirac operators on hyperbolic three-manifolds which might be of independent interest.