Seminar

The study of median spaces is really exciting since it provides a comprehensive framework to study a wide variety of spaces and phenomenon. The study of what types of actions a group admits on a median space encompasses the study of Property (T) and the Haagerup Property (sometimes also called a-(T)-menability). CAT(0) cube complexes admit a (simplicial) median structure, and R-trees provide a family of non-simplicial examples. Furthermore, the asymptotic cones of mapping class groups admit bi-Lipschitz embedding into a finite product of R-trees. Since products of median spaces are again median, this provides another interesting example and points to possible applications.