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

Abstract

We construct vertex transitive lattices on products of trees of arbitrary dimension d ≥ 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).

Higher dimensional generalizations of the Thompson groups

“…Our construction consists of two steps: first, we construct a family of cube complexes with two vertices, covered by products of k trees, and second, we explain how to get a k-graph from each complex. For background on cube complexes covered by products of k trees see [26] and references in the paper.…”

Parameterized (Modular) Counting and Cayley Graph Expanders

“…The set of relations is described by explicit algebraic equations in the field F p 2 . In [62] these groups were realized by mapping the generators a k , b j to explicit generalized quaternions, leading ultimately to an explicit injective group homomorphism…”

“…The generators a k , b j from the construction of Γ p;α,β map to symmetric generating sets T i of Γ i , i.e., to the set of cosets a k N i , b j N i when Γ i = Γ p;α,β /N i is considered as a factor group. Using results from [62], we know that the Cayley graphs G i = C(Γ i , T i ) associated to the congruence quotient groups Γ i with respect to the generating sets T i are expanders. This argument is worked out in [62] by Rungtanapirom and two of the authors, and it is based on a similar approach in the classical papers by Lubotzky, Phillips and Sarnak [48] and by Morgenstern [54].…”

“…Using results from [62], we know that the Cayley graphs G i = C(Γ i , T i ) associated to the congruence quotient groups Γ i with respect to the generating sets T i are expanders. This argument is worked out in [62] by Rungtanapirom and two of the authors, and it is based on a similar approach in the classical papers by Lubotzky, Phillips and Sarnak [48] and by Morgenstern [54]. We note here that the results of [62] ultimately rely on deep number theoretic results, namely a translation of the spectrum of the adjacency operator into Satake parameters of an associated automorphic representation and most crucially on work of Drinfeld on the geometric Langlands programme for GL 2 .…”

“…In this section we recall the construction of certain discrete groups from [65,62] that are lattices in generalized quaternion algebras. Their group theory is controlled by representations with values in power series rings with coefficients in F p .…”