Superexpanders from group actions on compact manifolds

It is known that the expanders arising as increasing sequences of level sets of warped cones, as introduced by the second-named author, do not coarsely embed into a Banach space as soon as the corresponding warped cone does not coarsely embed into this Banach space. Combining this with non-embeddability results for warped cones by Nowak and Sawicki, which relate the non-embeddability of a warped cone to a spectral gap property of the underlying action, we provide new examples of expanders that do not coarsely embed into any Banach space with nontrivial type. Moreover, we prove that these expanders are not coarsely equivalent to a Lafforgue expander. In particular, we provide infinitely many coarsely distinct superexpanders that are not Lafforgue expanders. In addition, we prove a quasi-isometric rigidity result for warped cones.Comment: 16 pages, to appear in Geometriae Dedicat

Mini-Workshop: Superexpanders and Their Coarse Geometry

It is a deep open problem whether all expanders are superexpanders. In fact, it was already a major challenge to prove the mere existence of superexpanders. However, by now, some classes of examples are known: Lafforgue’s expanders constructed as sequences of finite quotients of groups with strong Banach property (T), the examples coming from zigzag products due to Mendel and Naor, and the recent examples coming from group actions on compact manifolds. The methods which are used to construct superexpanders are typically functional analytic in nature, but also rely on arguments from geometry and combinatorics. Another important aspect of the study of superexpanders is their (coarse) geometry, in particular in order to distinguish them from each other. The aim of this workshop was to bring together researchers working on superexpanders and their coarse geometry from different perspectives, with the aim of sharing expertise and stimulating new research

Non-stable groups

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

Noncommutative Geometry

These reports contain an account of 2015’s meeting on noncommutative geometry. Noncommutative geometry has developed itself over the years to a completely new branch of mathematics shedding light on many other areas as number theory, differential geometry and operator algebras. A connection that was highlighted in particular in this meeting was the connection with the theory of $II_1$-factors and geometric group theory