Quasi-isometric embeddings of Ramanujan complexes

TL;DR

This paper investigates the large-scale geometric properties of Ramanujan complexes, showing that complexes associated with different primes do not quasi-isometrically embed into each other, using rigidity theorems.

Contribution

It establishes that Ramanujan complexes from different primes are not quasi-isometrically embeddable into one another, revealing their distinct large-scale geometries.

Findings

Ramanujan complexes associated with different primes are not quasi-isometrically embeddable.

The study applies box space rigidity and Euclidean building rigidity theorems.

Different prime-based complexes exhibit fundamentally different large-scale geometries.

Abstract

Ramanujan complexes were defined as high dimensional analogues of the optimal expanders, Ramanujan graphs. They were constructed as quotients of the Euclidean building (also called the affine building and the Bruhat-Tits building) of $\mathrm{PGL}_d(\mathbb{F}_p((y)))$ by certain cocompact lattices by Lubotzky-Samuels-Vishne. We distinguish the Ramanujan complexes up to large-scale geometry. More precisely, we show that if $p$ and $q$ are distinct primes, then the associated Ramanujan complexes do not quasi-isometrically embed into one another. The main tools are the box space rigidity of Khukhro-Valette and the Euclidean building rigidity of Kleiner-Leeb and Fisher-Whyte.