Finite quotients of Bruhat-Tits buildings as geometric expanders

TL;DR

This paper extends the construction of finite quotients of Bruhat-Tits buildings as geometric expanders from specific cases to more general high rank buildings, enhancing understanding of their combinatorial and geometric properties.

Contribution

It generalizes the explicit construction of geometric expanders as finite quotients of Bruhat-Tits buildings to all high rank cases.

Findings

Finite quotients of high rank Bruhat-Tits buildings are geometric expanders.

Extension of previous constructions to more general high rank cases.

Supports the use of these complexes in combinatorial and geometric applications.

Abstract

In \cite{FGLNP}, Fox, Gromov, Lafforgue, Naor and Pach, in a respond to a question of Gromov \cite{G}, constructed bounded degree geometric expanders, namely, simplical complexes having the affine overlapping property. Their explicit constructions are finite quotients of $\tilde{A_d}$-buildings, for $d\geq 2$, over local fields. In this paper, this result is extended to general high rank Bruhat-Tits buildings.