Edge Zeta Functions and Eigenvalues for Buildings of Finite Groups of Lie Type

TL;DR

This paper investigates the edge zeta function of Tits buildings associated with finite groups of Lie type, revealing that nonzero edge eigenvalues are powers of q after a bounded exponent, using a uniform Hecke algebra approach.

Contribution

It extends previous results to all finite groups of Lie type and all edge-geodesic cycles using a uniform Hecke algebra method.

Findings

Nonzero edge eigenvalues become powers of q after raising to a bounded exponent.

The approach is uniform across all types of finite groups of Lie type.

Extends prior results from type A and oppositeness graphs to full edge-geodesic settings.

Abstract

For the Tits building B(G) of a finite group of Lie type G(Fq), we study the edge zeta function, which enumerates edge-geodesic cycles in the 1-skeleton. We show that every nonzero edge eigenvalue becomes a power of q after raising to a bounded exponent k depending on the type of G. The proof is uniform across types using a Hecke algebra approach. This extends previous results for type A and for oppositeness graphs to the full edge-geodesic setting and all finite groups of Lie type.