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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2021
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| Online Access: | https://arxiv.org/abs/2108.01275 |
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| _version_ | 1866909150270717952 |
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| author | Hong, Soonki Kwon, Sanghoon |
| author_facet | Hong, Soonki Kwon, Sanghoon |
| contents | We investigate the automorphic spectra of the natural weighted adjacency operator on the complex arising as a $PGL(3,\mathbb{F}_q[t])$ quotient of $\widetilde{A}_2$-type building. We prove that the set of non-trivial approximate eigenvalues $(λ^+,λ^-)$ of the weighted adjacency operators $A_w^\pm$ on the quotient induced from the colored adjacency operators $A^\pm$ on the building for $PGL_3$ contains the simultaneous spectrum of $A^\pm$ and another hypocycloid with three cusps. As a byproduct, we re-establish a proof of the fact that $PGL(3,\mathbb{F}_q[t])\backslash PGL(3,\mathbb{F}_q(\!(t^{-1})\!))/PGL(3,\mathbb{F}_q[\![t^{-1}]\!])$ is not a Ramanujan complex, from a combinatorial aspect. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2108_01275 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Spectrum of weighted adjacency operator on a non-uniform arithmetic quotient of $PGL_3$ Hong, Soonki Kwon, Sanghoon Number Theory Combinatorics Group Theory 20G25, 47A25, 05E45 We investigate the automorphic spectra of the natural weighted adjacency operator on the complex arising as a $PGL(3,\mathbb{F}_q[t])$ quotient of $\widetilde{A}_2$-type building. We prove that the set of non-trivial approximate eigenvalues $(λ^+,λ^-)$ of the weighted adjacency operators $A_w^\pm$ on the quotient induced from the colored adjacency operators $A^\pm$ on the building for $PGL_3$ contains the simultaneous spectrum of $A^\pm$ and another hypocycloid with three cusps. As a byproduct, we re-establish a proof of the fact that $PGL(3,\mathbb{F}_q[t])\backslash PGL(3,\mathbb{F}_q(\!(t^{-1})\!))/PGL(3,\mathbb{F}_q[\![t^{-1}]\!])$ is not a Ramanujan complex, from a combinatorial aspect. |
| title | Spectrum of weighted adjacency operator on a non-uniform arithmetic quotient of $PGL_3$ |
| topic | Number Theory Combinatorics Group Theory 20G25, 47A25, 05E45 |
| url | https://arxiv.org/abs/2108.01275 |