We will discuss stable commutator length on mapping class groups.

Given a family of rational maps or polynomials, a basic problem is to understand what happens (dynamically) as the family degenerates. In this talk, I will describe a limiting object as a "non-archimedean dynamical system" in terms of a rational function acting on a Berkovich space. In geometric terms, this is a dynamical system on an R-tree.

TBA

We will characterize convex cocompact subgroups of mapping class groups that arise as subgroups of specially embedded right-angled Artin groups. With this characterization, we construct convex cocompact subgroups of Mod(S) with "small translation length" on the curve complex. This is joint work with Johanna Mangahas.

The abstract commensurator of a group G is the group of all isomorphisms between finite index subgroups of G up to a natural equivalence relation. Commensurators of lattices in semisimple Lie groups are well understood, using strong rigidity results of Mostow, Prasad, and Margulis. We will describe commensurators of lattices in solvable groups, where strong rigidity fails. If time permits, we will extend these results to lattices in certain groups that are neither solvable nor semisimple.