The genus spectrum of a hyperbolic 3-manifold

TL;DR

This paper investigates the spectrum of totally geodesic surfaces in hyperbolic 3-manifolds, revealing that for arithmetic cases it determines the manifold's commensurability class, and explores spectral properties of finite covers.

Contribution

It establishes that the spectrum of totally geodesic surfaces uniquely determines the commensurability class in arithmetic hyperbolic 3-manifolds and constructs examples of non-isometric covers with identical spectra.

Findings

Spectrum determines commensurability class in arithmetic cases

Existence of non-isometric covers with identical spectra

Unbounded volume ratios in spectral pairs

Abstract

In this article we study the spectrum of totally geodesic surfaces of a finite volume hyperbolic 3-manifold. We show that for arithmetic hyperbolic 3-manifolds that contain a totally geodesic surface, this spectrum determines the commensurability class. In addition, we show that any finite volume hyperbolic 3-manifold has many pairs of non-isometric finite covers with identical spectra. Forgetting multiplicities, we can also construct pairs where the volume ratio is unbounded.