New York Journal of Mathematics
Volume 21 (2015) 955-972

  

Benjamin Linowitz, Jeffrey S. Meyer, and Paul Pollack

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

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Published: September 21, 2015
Keywords: Hyperbolic manifolds, length spectrum, totally geodesic surfaces
Subject: Primary 53C22; secondary 57M50

Abstract
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 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?

Acknowledgements

The first author was partially supported by NSF RTG grant DMS-1045119 and an NSF Mathematical Sciences Postdoctoral Fellowship. The third author was partially supported by NSF grant DMS-1402268.


Author information

Benjamin Linowitz:
Department of Mathematics, University of Michigan, Ann Arbor, MI 48109
linowitz@umich.edu

Jeffrey S. Meyer:
Department of Mathematics, University of Oklahoma, Norman, OK 73019
jmeyer@math.ou.edu

Paul Pollack:
Department of Mathematics, University of Georgia, Athens, GA 30602
pollack@uga.edu