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

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.

Keywords

hyperbolic orbifolds, totally geodesic surfaces

2010 Mathematics Subject Classification

Primary 57M50. Secondary 53C42.

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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