Pure and Applied Mathematics Quarterly

Volume 17 (2021)

Number 1

Areas of totally geodesic surfaces of hyperbolic $3$-orbifolds

Pages: 1 – 25

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


Benjamin Linowitz (Department of Mathematics, Oberlin College, Oberlin, Ohio, U.S.A.)

D. B. McReynolds (Department of Mathematics, Purdue University, West Lafayette, Indiana, U.S.A.)

Nicholas Miller (Department of Mathematics, University of California, Berkeley, Calif., U.S.A.)


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.


hyperbolic orbifolds, totally geodesic surfaces

2010 Mathematics Subject Classification

Primary 57M50. Secondary 53C42.

B. Linowitz was partially supported by a Simons Collaboration Grant and by NSF Grant Number DMS-1905437.

D. B. McReynolds was partially supported by NSF grants DMS-1408458 and DMS-1812153.

N. Miller was partially supported by a Bilsland dissertation fellowship and by NSF grant DMS-2005438.

Received 14 July 2017

Accepted 2 February 2021

Published 11 April 2021