Weekly Bulletin

On a contact manifold, Rumin's complex provides a sequence of differential operators that is intrinsic to the contact structure and computes de~Rham cohomology. This is a Rockland complex, the analogue of an elliptic complex in the Heisenberg calculus. Subelliptic analysis permits to define and study associated spectral invariants. In particular, the analytic torsion and the eta invariant of Rumin's contact complex have been compared to Riemannian an CR-invariants by Biquard-Herzlich-Rumin, Rumin-Seshadri, and Albin-Quan. While Rumin's complex can be defined for a broad class of filtered (Carnot) manifolds, one 5-dimensional geometry stands out because its Rumin complex is as well behaved as the contact analogue. This geometry is determined by a generic rank two distribution (i.e. 2-plane field) in dimension five, a.k.a. (2,3,5) distribution. Equivalently, it can be characterized as a particular Cartan geometry associated with the exceptional Lie group G2. In this talk we will recall the contact case and discuss recent results on the analytic torsion and the eta invariant of (2,3,5) distributions.