Edge zeta function and closed cycles in the standard non-uniform complex from $\operatorname{PGL}_3$

Soonki Hong; Sanghoon Kwon

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arXiv · 2411.15489

Edge zeta function and closed cycles in the standard non-uniform complex from $\operatorname{PGL}_3$

Soonki Hong·Sanghoon Kwon

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Abstract

In this paper, we define the edge zeta function of weighted complex. We also present the formula for the edge zeta function of the standard non-uniform complex $\operatorname{PGL}(3,\mathbb{F}_q[t])\backslash\operatorname{PGL}(3,\mathbb{F}_q(\!(t^{-1})\!))/\operatorname{PGL}(3,\mathbb{F}_q[\![t^{-1}]\!])$, arising from the group $\operatorname{PGL}_3$, as a rational function. Applying trunction in a specific direction is one of the main ingredient. As a result, we obtain the exact formula for the number of closed cycles coming from geodesics in the building.

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BibTeX RIS

Soonki Hong, Sanghoon Kwon. 2024-11-23. Edge zeta function and closed cycles in the standard non-uniform complex from $\operatorname{PGL}_3$. https://arxiv.org/abs/2411.15489

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