We recall the theory of non-abelian first cohomology as presented in [Se97] I 5 and emphasize the role played by conjugacy classes of sections and by equivariant torsors. The difference between two sections can be described by either a cocycle or an equivariant torsor.The analogue in the non-abelian setup of the familiar abelian long exact sequences for short exact sequences of coefficients or of low degree terms in the Hochschild-Serre spectral sequence turn out to exist but in a restricted form depending on the amount of commutativity that one is willing to spend as an assumption.

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 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).

For an abelian variety A over a number field k we discuss the divisibility in H 1 (k, A) of elements of the subgroup X(A/k). The results are most complete for elliptic curves over Q.

A Seifert-Van Kampen theorem describes the fundamental group of a space in terms of the fundamental groups of the constituents of a covering and the configuration of connected components of the covering. Here we provide the combinatorial part of such a theorem for the most general sort of coverings. Thus a Seifert-Van Kampen theorem is reduced to a purely geometric statement of effective descent.