International Young Seminar on
Bounded Cohomology and Simplicial Volume 

Abstract:

There's a conjecture that a compact Riemannian manifold with almost nonnegative scalar curvature, with respect to a normalized volume, has vanishing simplicial volume.  I'll explain the background to this conjecture, how it motivates problems about the existence of finite volume Riemannian metrics with positive scalar curvature on noncompact manifolds, and some results in the spin case.

Abstract:

Stable commutator length is a measure of homological complexity of group elements, which is known to take large values in the presence of various notions of negative curvature. We will present a new geometric proof of a theorem of Heuer on sharp lower bounds for scl in right-angled Artin groups. Our proof relates letter-quasimorphisms (which are analogues of real-valued quasimorphisms with image in free groups) to negatively curved angle structures for surfaces estimating scl.

Abstract:

The space of subgroups of a countable group G is a compact topological space equipped with a continuous action of G which encodes many of the properties of its non-free actions. I will discuss some approaches to studying the Cantor-Bendixson decomposition of this space in the context of hyperbolic groups and groups which act (nicely) on trees, and give some conditions under which the conjugation action on the perfect kernel of this space is highly topologically transitive. I will then explain how this information can be used to find many new examples of groups which admit a faithful transitive amenable action, including for instance all virtually compact special groups.

This is joint work with Damien Gaboriau.