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New York Journal of Mathematics
Volume 26 (2020), 149-183

  

Jeffrey S. Meyer, Christian Millichap, and Rolland Trapp

Arithmeticity and hidden symmetries of fully augmented pretzel link complements

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Published: February 12, 2020.
Keywords: hyperbolic link complement, arithmetic link, hidden symmetry.
Subject: Primary: 57M25; Secondary: 57M27, 57M50.

Abstract
This paper examines number theoretic and topological properties of fully augmented pretzel link complements. In particular, we determine exactly when these link complements are arithmetic and exactly which are commensurable with one another. We show these link complements realize infinitely many CM-fields as invariant trace fields, which we explicitly compute. Further, we construct two infinite families of non-arithmetic fully augmented link complements: one that has no hidden symmetries and the other where the number of hidden symmetries grows linearly with volume. This second family realizes the maximal growth rate for the number of hidden symmetries relative to volume for non-arithmetic hyperbolic 3-manifolds. Our work requires a careful analysis of the geometry of these link complements, including their cusp shapes and totally geodesic surfaces inside of these manifolds.

Acknowledgements

The authors acknowledge support from U.S. National Science Foundation grants DMS 1107452, 1107263, 1107367 "RNMS: Geometric Structures and Representation Varieties" (the GEAR Network). We would also like to thank Dave Futer for his helpful suggestions.


Author information

Jeffrey S. Meyer:
Department of Mathematics
California State University
San Bernardino, CA 92407, USA

jeffrey.meyer@csusb.edu

Christian Millichap:
Department of Mathematics
Furman University
Greenville, SC 29613, USA

Christian.Millichap@furman.edu

Rolland Trapp:
Department of Mathematics
California State University
San Bernardino, CA 92407, USA

rtrapp@csusb.edu