In this paper, we construct for the first time, Witten genus and elliptic genera on noncompact manifolds with a proper cocompact action by an almost connected Lie group prove vanishing rigidity results that generalise known compact actions manifolds. We also compute our some interesting examples, study non-spinc case in appendix.

We explain simple methods to establish the property of Rapid Decay for a number of groups arising geometrically. Those lead to new examples of groups with the property of Rapid Decay, notably including non-cocompact lattices in rank one Lie groups.

We show that, under some very weak assumption of effective variation for the magnetic field, a periodic Schrödinger operator with magnetic wells on a noncompact Riemannian manifold M such that H(M, R) = 0 equipped with a properly disconnected, cocompact action of a finitely generated, discrete group of isometries has an arbitrarily large number of spectral gaps in the semi-classical limit.

Let M be a non-elementary convex cocompact hyperbolic 3-manifold and δ be the critical exponent of its fundamental group. We prove that a one-dimensional unipotent flow for the frame bundle of M is ergodic for the Burger-Roblin measure if and only if δ > 1.

For a finitely generated discrete group $\Gamma$ acting properly on spin manifold $M$, we formulate new topological obstructions to $\Gamma$-invariant metrics of positive scalar curvature $M$ that take into account the cohomology classifying space $\underline{B}\Gamma$ for proper actions. In cocompact case, this leads natural generalisation Gromov-Lawson's notion higher $\hat{A}$-genera setting...