Infinite series of quaternionic 1-vertex cube complexes, the doubling construction, and explicit cubical Ramanujan complexes

TL;DR

This paper constructs high-dimensional, explicit cubical Ramanujan complexes using quaternionic lattices and the doubling construction, extending previous work on Ramanujan graphs to higher dimensions.

Contribution

It introduces a method to build vertex transitive lattices on products of trees in any dimension using quaternion algebras, and constructs explicit cubical Ramanujan complexes.

Findings

Construction of non-residually finite quaternionic lattices in higher dimensions

Development of a generalized doubling construction for cubical sets

Explicit examples of cubical Ramanujan complexes with optimal expansion properties

Abstract

We construct vertex transitive lattices on products of trees of arbitrary dimension $d \geq 1$ based on quaternion algebras over global fields with exactly two ramified places. Starting from arithmetic examples, we find non-residually finite groups generalizing earlier results of Wise, Burger and Mozes to higher dimension. We make effective use of the combinatorial language of cubical sets and the doubling construction generalized to arbitrary dimension. Congruence subgroups of these quaternion lattices yield explicit cubical Ramanujan complexes, a higher dimensional cubical version of Ramanujan graphs (optimal expanders).