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

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