Abelian spiders and real cyclotomic integers
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- by Frank Calegari and Zoey Guo PDF
- Trans. Amer. Math. Soc. 370 (2018), 6515-6533 Request permission
Abstract:
If $\Gamma$ is a finite graph, then the largest eigenvalue $\lambda$ of the adjacency matrix of $\Gamma$ is a totally real algebraic integer ($\lambda$ is the Perron-Frobenius eigenvalue of $\Gamma$). We say that $\Gamma$ is abelian if the field generated by $\lambda ^2$ is abelian. Given a fixed graph $\Gamma$ and a fixed set of vertices of $\Gamma$, we define a spider graph to be a graph obtained by attaching to each of the chosen vertices of $\Gamma$ some $2$-valent trees of finite length. The main result is that only finitely many of the corresponding spider graphs are both abelian and not Dynkin diagrams, and that all such spiders can be effectively enumerated; this generalizes a previous result of Calegari, Morrison, and Snyder. The main theorem has applications to the classification of finite index subfactors. We also prove that the set of Salem numbers of “abelian type” is discrete.References
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Additional Information
- Frank Calegari
- Affiliation: Department of Mathematics, The University of Chicago, 5734 S University Avenue, Chicago, Illinois 60637
- MR Author ID: 678536
- Email: fcale@math.uchicago.edu
- Zoey Guo
- Affiliation: Department of Mathematics, Marlboro College, 2582 South Road, Marlboro, Vermont 05344
- Email: zoeyguo@gmail.com
- Received by editor(s): July 23, 2016
- Received by editor(s) in revised form: December 11, 2016, and December 15, 2016
- Published electronically: February 26, 2018
- Additional Notes: The authors were supported in part by NSF Grant DMS-1648702.
- © Copyright 2018 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 370 (2018), 6515-6533
- MSC (2010): Primary 11R18, 46L37
- DOI: https://doi.org/10.1090/tran/7237
- MathSciNet review: 3814339