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We construct and classify triangle presentations associated to the smallest generalized quadrangle. We use these constructions to find groups acting cocompactly on hyperbolic buildings with n-sided chambers for arbitrary n ≥ 3.

We construct an infinite series of simply transitive irreducible lattices in PGL 2 (F q ((t))) × PGL 2 (F q ((t))) by means of a quaternion algebra over F q (t). The lattices depend on an odd prime power q = p r and a parameter τ ∈ F * q , τ = 1, and are the fundamental group of a square complex with just one vertex and universal covering T q+1 × T q+1 , a product of trees with constant valency q + 1.Our lattices give rise via non-archimedian uniformization to smooth projective surfaces of general type over F q ((t)) with ample canonical class, Chern ratio c 2 1 / c 2 = 2, trivial Albanese variety and non-reduced Picard scheme. For q = 3, the Zariski-Euler characteristic attains its minimal value χ = 1: the surface is a non-classical fake quadric.

We construct an infinite family of triples (G k , H k , T k ), where G k are 2-groups of increasing order, H k are index-2 subgroups of G k , and T k are pairs of generators of H k . We show that the triples u k = (G k , H k , T k ) are mixed Beauville structures if k is not a power of 2. This is the first known infinite family of 2-groups admitting mixed Beauville structures. Moreover, the associated Beauville surface S(u 3 ) is real and, for k > 3 not a power of 2, the Beauville surface S(u k ) is not biholomorphic to S(u k ).

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).