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Ramanujan bigraphs associated with $ SU(3)$ over a $ p$-adic field

Authors: Cristina Ballantine and Dan Ciubotaru
Journal: Proc. Amer. Math. Soc. 139 (2011), 1939-1953
MSC (2010): Primary 11F70, 22E50
Published electronically: January 6, 2011
MathSciNet review: 2775370
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Abstract: We use the representation theory of the quasisplit form $ G$ of $ SU(3)$ over a $ p$-adic field to investigate whether certain quotients of the Bruhat-Tits tree associated to this form are Ramanujan bigraphs. We show that a quotient of the tree associated with $ G$ (which is a biregular bigraph) is Ramanujan if and only if $ G$ satisfies a Ramanujan type conjecture. This result is analogous to the seminal case of $ PGL_2(\mathbb{Q}_p)$ considered by Lubotzky, Phillips, and Sarnak. As a consequence, the classification of the automorphic spectrum of the unitary group in three variables by Rogawski implies the existence of certain infinite families of Ramanujan bigraphs.

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Additional Information

Cristina Ballantine
Affiliation: Department of Mathematics and Computer Science, College of the Holy Cross, Worcester, Massachusetts 01610

Dan Ciubotaru
Affiliation: Department of Mathematics, University of Utah, Salt Lake City, Utah 84112

Received by editor(s): June 1, 2010
Published electronically: January 6, 2011
Additional Notes: The second author was supported in part by NSA-AMS 081022.
Communicated by: Matthew A. Papanikolas
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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