On finite arithmetic simplicial complexes

TL;DR

This paper calculates the Euler-Poincaré characteristic of certain quotient complexes derived from Bruhat-Tits buildings, providing insights into their structure under arithmetic group actions over function fields.

Contribution

It introduces a method to compute the Euler-Poincaré characteristic for quotients of Bruhat-Tits buildings associated with PGL(n) over function fields.

Findings

Explicit formulas for Euler-Poincaré characteristic of quotient complexes

Structural descriptions of quotient simplicial complexes in specific cases

Application of these results to understanding arithmetic group actions

Abstract

We compute the Euler-Poincar\'e characteristic of quotients of the Bruhat-Tits building of PGL(n) under the action of arithmetic groups arising from central division algebras over rational function fields of positive characteristic. We use this result to determine the structure of the quotient simplicial complex in certain cases.