Areas of totally geodesic surfaces of hyperbolic 3-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.

Publisher

International Press Boston

Publication Date

6-5-2021

Publication Title

Pure and Applied Mathematics Quarterly

Department

Mathematics

Document Type

Article

DOI

https://dx.doi.org/10.4310/PAMQ.2021.v17.n1.a1

Keywords

Hyperbolic orbifolds, Totally geodesic surfaces

Language

English

Format

text

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