Non-stable groups

Abstract: In this article we discuss cohomological obstructions to two kinds of group stability. In the first part, we show that residually finite groups $\Gamma$ which arise as fundamental groups of compact Riemannian manifolds with strictly negative sectional curvature are not uniform-to-local stable with respect to the operator norm if their even Betti numbers $b_{2i}(\Gamma)$ do not vanish. In the second part, we show that non-vanishing of Betti numbers $b_{i}(\Gamma)$ in dimension $i>1$ obstructs $C^*$-algebra stability for groups approximable by unitary matrices that admit a coarse embedding in a Hilbert space.