Closed geodesics and Fr{\o}yshov invariants of hyperbolic three-manifolds

TL;DR

This paper explores the relationship between Froyshov invariants and hyperbolic geometry in minimal L-spaces, providing bounds, algorithms for computation, and concrete examples involving hyperbolic three-manifolds.

Contribution

It establishes a connection between Froyshov invariants and hyperbolic geometry, offering new bounds and an algorithm for their computation in minimal L-spaces.

Findings

Derived effective upper bounds for Froyshov invariants based on volume and injectivity radius.

Developed an algorithm to compute Froyshov invariants from hyperbolic geometric data.

Computed Froyshov invariants for all spin^c structures on the Seifert-Weber space.

Abstract

Froyshov 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 three and four-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 Froyshov invariants and closed geodesics using ideas from analytic number theory. We discuss two main applications of our approach. First, we derive effective upper bounds for the Froyshov invariants of minimal hyperbolic $L$-spaces purely in terms of volume and injectivity radius. Second, we describe an algorithm to compute Froyshov invariants of minimal $L$-spaces in terms of data arising from hyperbolic geometry. As a concrete example of our method, we compute the Froyshov 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.