Coarse Geometry of Groups and Spaces

Abstract: Given two metric spaces X and Y it is natural to ask how faithfully, from the point of view of the metric, one can embed X into Y. One way of making this precise is asking whether there exists a coarse embedding of X into Y.

Positive results are plentiful and diverse, from Assouad's embedding theorem for doubling metric spaces to the elementary fact that any finitely generated subgroup of a finitely generated group is coarsely embedded with respect to word metrics. Moreover, the consequences of admitting a coarse embedding into a sufficiently nice space can be very strong. By contrast, there are few invariants which provide obstructions to coarse embeddings, leaving many elementary geometric questions open. I will present new families of invariants which resolve some of these questions. In particular I will show that the Baumslag-Solitar group BS(m,n) coarsely embeds into some hyperbolic group if and only if |m|=|n|=1.

algebraic topologydifferential geometrygeneral topologygroup theorygeometric topology

Audience: researchers in the topic

Ohio State Topology and Geometric Group Theory Seminar

Series comments: https://sites.google.com/view/topoandggt