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Ideal points of character varieties, algebraic non-integral representations, and undetected closed essential surfaces in 3-manifolds


Authors: Alex Casella, Charles Katerba and Stephan Tillmann
Journal: Proc. Amer. Math. Soc. 148 (2020), 2257-2271
MSC (2010): Primary 57M27
DOI: https://doi.org/10.1090/proc/14684
Published electronically: February 12, 2020
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Abstract: Closed essential surfaces in a 3-manifold can be detected by ideal points of the character variety or by algebraic non-integral representations. We give examples of closed essential surfaces not detected in either of these ways. For ideal points, we use Chesebro's module-theoretic interpretation of Culler-Shalen theory. As a corollary, we construct an infinite family of closed hyperbolic Haken 3-manifolds with no algebraic non-integral representation into $ \textrm {PSL}_2 (\mathbb{C})$, resolving a question of Schanuel and Zhang.


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Additional Information

Alex Casella
Affiliation: Department of Mathematics, Florida State University, Tallahassee, Florida 32304
Email: acasella@fsu.edu

Charles Katerba
Affiliation: Department of Mathematical Sciences, Montana State University, Bozeman, Montana 59717
Email: ckaterba@fvcc.edu

Stephan Tillmann
Affiliation: School of Mathematics and Statistics F07, The University of Sydney, New South Wales 2006, Australia
Email: tillmann@maths.usyd.edu.au

DOI: https://doi.org/10.1090/proc/14684
Keywords: Character variety, essential surface, algebraic non-integral representation
Received by editor(s): August 14, 2018
Received by editor(s) in revised form: April 8, 2019
Published electronically: February 12, 2020
Additional Notes: The first author acknowledges support by the Commonwealth of Australia.
The second author acknowledges support by an NSF-EAPSI Fellowship (project number 1713920).
The research of the third author was supported by an Australian Research Council Future Fellowship (project number FT170100316).
Communicated by: David Futer
Article copyright: © Copyright 2020 American Mathematical Society