There is a pleasing correspondence between splittings of a free group over finitely generated subgroups and systems of surfaces in a doubled handlebody. One can use this to describe a family of hyperbolic complexes on which Out(F_n) acts. This is joint work with Camille Horbez.

Two subsets A and B of R^n are equidecomposable if it is possible to partition A into pieces and rearrange them via isometries to form a partition of B. Motivated by what is nowadays known as Banach-Tarski paradox, Tarski asked if the unit square and the disc of unit area in R^2 are equidecomposable. 65 years later Laczkovich showed that they are, at least when the pieces are allowed to be non-measurable sets. I will talk about a joint work with A. Mathe and O. Pikhurko which implies in particular the existence of a measurable equidecomposition of circle and square in R^2.

Divergence, thickness, and relative hyperbolicity are three geometric properties which determine aspects of the quasi-isometric geometry of a finitely generated group. We will discuss the basic properties of these notions and some of the relations between them. We will then then survey how these properties manifest in right-angled Coxeter groups and detail various ways to classify Coxeter groups using them.

This is joint work with Hagen and Sisto.

The notion of prime decomposition will be defined and illustrated for
manifolds. Two proofs of existence will be given, including Kneser's
classical proof using normal surface theory.