We show that the group cohomology of simple connected Lie groups, and consequently their lattices and symmetric spaces, with L^p-coefficients vanishes below the rank. This confirms a conjecture of Gromov in the special case of the coefficient module L^p(G). We relate the vanishing of L^1-cohomology to a fixed-point theorem and confirm a conjecture of Farb. Farb conjectured a fixed point property for actions of lattices in such groups on CAT(0) cell complexes of dimension lower than the rank.
In this talk, I will give a short introduction to group cohomology and discuss ideas from the proofs of both results. This talk is based on joint work with Saar Bader, Uri Bader, and Roman Sauer.

Riemannian manifolds whose holonomy group lies inside the exceptional Lie group G_2 are called G_2-manifolds. These manifolds have several interesting properties (they are Ricci-flat) and are of interest in geometric analysis as well as in mathematical physics and other fields. In this seminar, we will give an introduction to the theory of G_2-manifolds and discuss an associated non-linear Laplacian-type operator whose kernel essentially determines whether a compact manifold is G_2. We will present uniqueness and existence results for the Poisson’s equation of this operator on homogeneous and cohomogeneity-one manifolds.