Paulo Tirao

The Cohomology of Lattices in SL(2, ℂ)

This paper contains both theoretical results and experimental data on the behavior of the dimensions of the cohomology spaces H 1(Γ,E n ), where Γ is a lattice in SL(2,ℂ) and , 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 present a large amount of experimental data for this case, as well as for some geometrically constructed and mostly nonarithmetic groups. The computations for SL(2,O) lead us to discover two instances with nonlifted 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.

On the homology of free 2-step nilpotent Lie algebras

Abstract We find an explicit formula for the total dimension of the homology of a free 2-step nilpotent Lie algebra. We analyse the asymptotics of this formula and use it to find an improved lower bound on the total dimension of the homology of any 2-step nilpotent Lie algebra.

The cohomology of the cotangent bundle of Heisenberg groups

Abstract Given a parabolic subalgebra g1×n of a semisimple Lie algebra, Kostant (Ann. Math. 1963) and Griffiths (Acta Math. 1963) independently computed the g1 invariants in the cohomology group of n with exterior adjoint coefficients. By a theorem of Bott (Ann. Math. 1957), this is the cohomology of the associated compact homogeneous space with coefficients in the sheaf of local holomorphic forms. In this paper we determine explicitly the full module structure, over the symplectic group, of the cohomology group of the Heisenberg Lie algebra with exterior adjoint coefficients. This is the cohomology of the cotangent bundle of the Heisenberg group.

Five-Dimensional Bieberbach Groups with Holonomy Group Z2⊕Z2

In this paper we determine all five-dimensional compact flat Riemannian manifolds with holonomy group Z2⊕Z2. The classification is achieved by classifying their fundamental groups up to isomorphism. The Betti numbers of all these manifolds are also computed.

FILIFORM LIE ALGEBRAS OF DIMENSION 8 AS DEGENERATIONS

For each complex 8-dimensional filiform Lie algebra we find another non isomorphic Lie algebra that degenerates to it. Since this is already known for nilpotent Lie algebras of rank

Comparison morphisms and the Hochschild cohomology ring of truncated quiver algebras

\ge 1

Compact symplectic nilmanifolds associated with graphs

, only the caracteristically nilpotent ones should be considered.