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Many cusped hyperbolic 3-manifolds do not bound geometrically


Authors: Alexander Kolpakov, Alan W. Reid and Stefano Riolo
Journal: Proc. Amer. Math. Soc. 148 (2020), 2233-2243
MSC (2010): Primary 57R90, 57M50, 20F55, 37F20
DOI: https://doi.org/10.1090/proc/14573
Published electronically: January 28, 2020
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Abstract: In this note we show that there exist cusped hyperbolic $ 3$-manifolds that embed geodesically but cannot bound geometrically. Thus, being a geometric boundary is a non-trivial property for such manifolds. Our result complements the work by Long and Reid on geometric boundaries of compact hyperbolic $ 4$-manifolds and by Kolpakov, Reid, and Slavich on embedding arithmetic hyperbolic manifolds.


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Additional Information

Alexander Kolpakov
Affiliation: Institut de mathématiques, Rue Emile-Argand 11, 2000 Neuchâtel, Switzerland
Email: kolpakov.alexander@gmail.com

Alan W. Reid
Affiliation: Department of Mathematics, Rice University, 6100 Main Street, Houston, Texas 77005
Email: alan.reid@rice.edu

Stefano Riolo
Affiliation: Institut de mathématiques, Rue Emile-Argand 11, 2000 Neuchâtel, Switzerland
Email: stefano.riolo@unine.ch

DOI: https://doi.org/10.1090/proc/14573
Keywords: $3$-manifold, $4$-manifold, hyperbolic geometry, cobordism, geometric boundary.
Received by editor(s): November 11, 2018
Received by editor(s) in revised form: November 12, 2018, and January 31, 2019
Published electronically: January 28, 2020
Additional Notes: The authors were supported by the Swiss National Science Foundation project no. PP00P2-170560 (first and third authors) and N.S.F. grant DMS-$1812397$ (second author)
Communicated by: David Futer
Article copyright: © Copyright 2020 Alexander Kolpakov, Alan W. Reid, and Stefano Riolo. We allow the use and reproduction of the whole or any part of this article in the public domain.