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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Abelian spiders and real cyclotomic integers

Authors: Frank Calegari and Zoey Guo
Journal: Trans. Amer. Math. Soc. 370 (2018), 6515-6533
MSC (2010): Primary 11R18, 46L37
Published electronically: February 26, 2018
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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.

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Additional Information

Frank Calegari
Affiliation: Department of Mathematics, The University of Chicago, 5734 S University Avenue, Chicago, Illinois 60637

Zoey Guo
Affiliation: Department of Mathematics, Marlboro College, 2582 South Road, Marlboro, Vermont 05344

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.
Article copyright: © Copyright 2018 American Mathematical Society

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