Building lattices and zeta functions

TL;DR

This paper introduces a new concept of building lattices, generalizes existing zeta function theories to higher dimensions, and provides formulas connecting these structures, advancing the understanding of geometric and algebraic properties of buildings.

Contribution

It generalizes tree lattice concepts to building lattices and extends Bass's approach to higher-dimensional zeta functions.

Findings

Established a Lefschetz formula for building lattices

Generalized Ihara zeta functions to higher-dimensional buildings

Connected geometric zeta functions with algebraic structures

Abstract

We introduce the notion of a building lattice generalizing tree lattices. We give a Lefschetz formula and apply it to geometric zeta functions. We further generalize Bass's approach to Ihara zeta functions to the higher dimensional case of a building.