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arXiv · 2206.00567

Weak Ramanujan property of the standard non-uniform arithmetic quotient of $PGL_4$

Soonki Hong·Sanghoon Kwon
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Abstract

Let $F$ be a field of formal series over a finite field and $\mathcal{B}_d$ be the affine building associated to $PGL_d(F)$. Given a lattice $\Gamma$ in $PGL_d(F)$, the complex arising as a quotient $\Gamma\backslash \mathcal{B}_d$ is called weakly Ramanujan if every non-tivial discrete simultaneous spectrum of the colored adjacency operators $A_1,A_2,\ldots,A_{d-1}$ acting on $L^2(\Gamma\backslash \mathcal{B}_d)$ is contained in the simultaneous spectrum of those operators acting on $L^2(\mathcal{B})$. In this paper, we prove that the standard non-uniform arithmetic quotient $PGL_4(\mathbb{F}_q[t])\backslash \mathcal{B}_4$ of $PGL_4(F)$ is weakly Ramanujan.

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Soonki Hong, Sanghoon Kwon. 2022-06-01. Weak Ramanujan property of the standard non-uniform arithmetic quotient of $PGL_4$. https://arxiv.org/abs/2206.00567

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