Explicit Constructions of Ramanujan Complexes of Type $\tilde{A}_d$

TL;DR

This paper provides explicit methods to construct Ramanujan complexes of type , generalizing previous work to higher dimensions and all local fields of positive characteristic, with potential applications in mathematics and computer science.

Contribution

The authors introduce explicit constructions of Ramanujan complexes for all dimensions  and local fields of positive characteristic, expanding the class of known Ramanujan complexes.

Findings

Explicit constructions for all  and local fields of positive characteristic.

Ramanujan complexes as quotients of Bruhat-Tits buildings.

Potential applications in combinatorics and theoretical computer science.

Abstract

In this paper we present for every $d \geq 2$ and every local field $F$ of positive characteristic, explicit constructions of Ramanujan complexes which are quotients of the Bruhat-Tits building $\B_d(F)$ associated with $\operatorname{PGL}_d(F)$.