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

Ballantine, Cristina; Ciubotaru, Dan

doi:10.1090/S0002-9939-2011-10856-6

Ramanujan bigraphs associated with $SU(3)$ over a $p$-adic field
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by Cristina Ballantine and Dan Ciubotaru PDF

Proc. Amer. Math. Soc. 139 (2011), 1939-1953 Request permission

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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