Areas of totally geodesic surfaces of hyperbolic 3-orbifolds

TL;DR

This paper investigates the totally geodesic area sets of hyperbolic 3-orbifolds, showing how these sets can overlap significantly among non-commensurable arithmetic examples, thus providing insights into their geometric and topological properties.

Contribution

It introduces the totally geodesic area set as a new invariant and quantifies its overlap among non-commensurable arithmetic hyperbolic 3-orbifolds.

Findings

Non-commensurable arithmetic hyperbolic 3-orbifolds can have arbitrarily large overlaps in their totally geodesic area sets.

The totally geodesic area set is a coarser invariant than the geometric genus spectrum.

Results relate the geometric properties of surfaces to the arithmetic structure of orbifolds.

Abstract

The geodesic length spectrum of a complete, finite volume, hyperbolic 3-orbifold M is a fundamental invariant of the topology of M via Mostow-Prasad Rigidity. Motivated by this, the second author and Reid defined a two-dimensional analogue of the geodesic length spectrum given by the multiset of isometry types of totally geodesic, immersed, finite-area surfaces of M called the geometric genus spectrum. They showed that if $M$ is arithmetic and contains a totally geodesic surface, then the geometric genus spectrum of M determines its commensurability class. In this paper we define a coarser invariant called the totally geodesic area set given by the set of areas of surfaces in the geometric genus spectrum. We prove a number of results quantifying the extent to which non-commensurable arithmetic hyperbolic 3-orbifolds can have arbitrarily large overlaps in their totally geodesic area sets.