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Spectrum of the weighted adjacency operator on a nonuniform arithmetic quotient of $\operatorname{PGL}_3$

Soonki Hong and Sanghoon Kwon

Vol. 13 (2024), No. 2, 103–122
Abstract

We investigate the automorphic spectra of the natural weighted adjacency operator on the complex arising as a PGL (3, 𝔽q[t]) quotient of an Ã2-type building. We prove that the set of nontrivial approximate eigenvalues (λ+,λ) of the weighted adjacency operators Aw± on the quotient induced from the colored adjacency operators A± on the building for PGL 3 contains the simultaneous spectrum of A± and another hypocycloid with three cusps. As a byproduct, we reestablish a proof of the fact that PGL (3, 𝔽q[t])PGL (3, 𝔽q((t1)))PGL (3, 𝔽q[[t1]]) is not a Ramanujan complex, from a combinatorial aspect.

Keywords
Bruhat–Tits building, spectrum, arithmetic lattice
Mathematical Subject Classification
Primary: 20E42, 20G25
Secondary: 47A25
Milestones
Received: 4 October 2023
Revised: 1 February 2024
Accepted: 16 February 2024
Published: 22 March 2024
Authors
Soonki Hong
Department of Mathematics
Postech
Pohang
South Korea
Sanghoon Kwon
Department of Mathematical Education
Catholic Kwandong University
Gangneung
South Korea