Title: Automorphisms of groups and a higher rank JSJ decomposition (continued)

  The JSJ decomposition encodes the automorphisms and the virtually cyclic splittings of a hyperbolic group. For general finitely presented groups, the JSJ decomposition encodes only their splittings. 

We generalize the structure and the construction of the JSJ decomposition, to study the automorphisms of hierarchically hyperbolic groups (HHG). Hierarchically hyperbolic groups generalize the structure of the mapping class groups of surfaces, and include many of the groups that act on non-positively curved cube complexes (a simple example  of an HHG is a group that acts properly and cocompactly on a product of hyperbolic spaces).

 The object that we construct can be viewed as a higher rank JSJ decomposition. Like the JSJ decomposition of a hyperbolic group it encodes information on the algebraic structure of the automorphism group and on the dynamics of individual automorphisms.

In the second talk I'll describe the higher JSJ decomposition. If time permits we will explain how the construction of the higher rank JSJ uses techniques from the study of the first order theory of groups (the solution to Tarski's problem), and why the techniques that we apply in this construction enable one to construct a canonical JSJ decomposition for (varieties over) semigroups and associative algebras.