 

Benjamin Linowitz, Jeffrey S. Meyer, and Paul Pollack
The length spectra of arithmetic hyperbolic 3manifolds and their totally geodesic surfaces view print


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 3orbifold 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 3orbifold determined by the lengths of closed geodesics lying on totally geodesic surfaces?, Do there exist arithmetic hyperbolic 3orbifolds whose "short'' geodesics do not lie on any totally geodesic surfaces?, and Do there exist arithmetic hyperbolic 3orbifolds whose "short'' geodesics come from distinct totally geodesic surfaces?


Acknowledgements
The first author was partially supported by NSF RTG grant DMS1045119 and an NSF Mathematical Sciences Postdoctoral Fellowship. The third author was partially supported by NSF grant DMS1402268.


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

