The length spectra of arithmetic hyperbolic 3-manifolds and their totally geodesic surfaces

TL;DR

This paper investigates how the length spectrum of arithmetic hyperbolic 3-orbifolds relates to their totally geodesic surfaces, exploring whether geodesic lengths determine the orbifold's geometry and classification.

Contribution

It provides new insights into the extent to which the geometry and commensurability class of arithmetic hyperbolic 3-orbifolds are determined by geodesics on totally geodesic surfaces, using analytic number theory techniques.

Findings

The commensurability class may not be fully determined by geodesic lengths on totally geodesic surfaces.

Existence of orbifolds with short geodesics not lying on any totally geodesic surface.

Existence of orbifolds with short geodesics from different totally geodesic surfaces.

Abstract

In this paper we examine the relationship between the length spectrum and the geometric genus spectrum of an arithmetic hyperbolic 3-orbifold M. In particular we analyze the extent to which the geometry of M is determined by the closed geodesics coming from finite area totally geodesic surfaces. Using a variety of techniques from analytic number theory, we address the following problems: Is the commensurability class of an arithmetic hyperbolic 3-orbifold determined by the lengths of closed geodesics lying on totally geodesic surfaces?, Do there exist arithmetic hyperbolic 3-orbifolds whose "short" geodesics do not lie on any totally geodesic surfaces?, and Do there exist arithmetic hyperbolic 3-orbifolds whose "short" geodesics come from distinct totally geodesic surfaces?