This paper contains both theoretical results and experimental data on the behavior of the dimensions of the cohomology spaces H 1 (Γ, En), where Γ is a lattice in SL(2, C) and En = Sym n ⊗Sym n , n ∈ N ∪ {0}, is one of the standard self-dual modules. In the case Γ = SL(2, O) for the ring of integers O in an imaginary quadratic number field, we make the theory of lifting explicit and obtain lower bounds linear in n. We have accumulated a large amount of experimental data in this case, as well as for some geometrically constructed and mostly non-arithmetic groups. The computations for SL(2, O) lead us to discover two instances with non-lifted classes in the cohomology. We also derive an upper bound of size O(n 2 / log n) for any fixed lattice Γ in the general case. We discuss a number of new questions and conjectures suggested by our results and our experimental data.

A main contribution of this paper is the explicit construction of comparison morphisms between the standard bar resolution and Bardzell's minimal resolution for truncated quiver algebras over arbitrary fields (TQA's). As a direct application we describe explicitly the Yoneda product and derive several results on the structure of the cohomology ring of TQA's over a field of characteristic zero. For instance, we show that the product of odd degree cohomology classes is always zero. We prove that TQA's associated with quivers with no cycles or with neither sinks nor sources have trivial cohomology rings. On the other side we exhibit a fundamental example of a TQA with nontrivial cohomology ring. Finally, for truncated polynomial algebras in one variable, we construct explicit cohomology classes in the bar resolution and give a full description of their cohomology ring.

a b s t r a c tIn this paper we consider 2-step nilpotent Lie algebras, Lie groups and nilmanifolds associated with graphs. We present a combinatorial construction of the second cohomology group for these Lie algebras. This enables us to characterize those graphs giving rise to symplectic or contact nilmanifolds.In this paper we consider finite-dimensional real 2-step nilpotent Lie algebras and nilmanifolds associated with finite graphs. This is a small class of Lie algebras with a nice combinatorial structure. In fact they are only finitely many in a given dimension, while there are infinitely many (non-isomorphic) 2-step nilpotent Lie algebras already in dimension 9 [1]. However five out of the seven six-dimensional symplectic 2-step Lie algebras [2] are associated with graphs, and it is not clear to us how the symplectic 2-step Lie algebras are related in general to the ones considered here.Our interest in this class of Lie algebras and the corresponding nilmanifolds is twofold.On the one hand we are interested in the explicit construction of cohomology classes for nilpotent Lie algebras, which is in turn related to the Toral Rank Conjecture for nilpotent Lie algebras. We look for general constructions such as the one in [3]. The combinatorial construction we present in this paper can be seen as the first step in a more general approach to the construction of cohomology classes for certain classes of 2-step nilpotent Lie algebras. At this stage it constructs all of the (first and) second cohomology groups for 2-step algebras associated with graphs. Note that the 2-step free ones are included.On the other hand, we are interested in the geometry of nilmanifolds and in particular in the construction of geometric structures such as symplectic and contact forms.We show that the first and second cohomology groups of the Lie algebras considered here can be easily read off the graph directly. This gives us an easy criterion to decide, just by inspection, whether the Lie algebra associated with a given graph admits a symplectic or contact form. It turns out that the three-dimensional Heisenberg Lie algebra is the only one of this class with a contact form while those admitting a symplectic form are exactly those corresponding to a graph where each connected component has at least as many vertices as arrows. Moreover, the proof of the criterion yields an algorithm to construct such a form.

Abstract. It is known that the total (co)-homoloy of a 2-step nilpotent Lie algebra g is at least 2 |z| , where z is the center of g. We improve this result by showing that a better lower bound is 2 t , where t = |z| + |v|+1 2and v is a complement of z in g. Furthermore, we provide evidence that this is the best possible bound of the form 2 t .