We study the group of polynomial automorphisms of C^3 generated by affine maps and all (x,y,z)->(x+P(y,z),y,z). We exhibit a contractible hyperbolic 2-complex on which that group acs. We also find a loxodromic weakly properly discontinuous element. This is joint work with Stephane Lamy

A hierarchically hyperbolic structure gives a way of reducing the study of a given metric space to the study of a specified family of hyperbolic spaces. Spaces with hierarchically hyperbolic structures include CAT(0) cube complexes admitting a proper and cocompact isometric action, mapping class groups, Teichmuller spaces with either the Teichmuller or the Weil-Petersson metric and many 3-manifold groups.

I will outline what a hierarchically hyperbolic structure is and how to give one to a CAT(0) cube complex admitting a factor system, which is a "large enough" locally finite collection of convex subcomplexes. Finally, I will give applications, in particular one regarding acylindrical actions.
Based on joint works with J. Behrstock and M. Hagen

I will discuss a collection of hyperbolic graphs associated to a CAT(0) cube complex and explain how the geometry of the cube complex can be recovered -- up to quasi-isometry -- from its shadows on these graphs.  I will explain how this mirrors the Masur-Minsky theory enabling the study of the mapping class group of a surface via projections to curve graphs of subsurfaces.  I'll then define "hierarchical hyperbolicity", which is a common generalisation of these two classes of examples, and discuss some applications.  This is based on joint work with J. Behrstock and A. Sisto

One of the most prominent class of CAT(0) spaces is the class of Affine Buildings.

In dimension 1, an affine building is nothing but a tree. In dimension 3 and higher (irreducible) affine buildings are always classical, that is they are the Bruhat-Tits buildings of algebraic groups over valued fields. In dimension 2 there are loads of exotic (ie, non-classical) buildings. Some are intimately related with some sporadic finite simple groups. Many have a cocompact group of isometries.

A rank-1 isometry of an irreducible CAT(0) space is an isometry that exhibits hyperbolic-type behavior regardless of whether the ambient space is indeed hyperbolic. A regular isometry of an (essential) CAT(0) cube complex is an isometry that is rank-1 in each irreducible factor. In a joint work with Lécureux and Mathéus, we study random walks and deduce that regular isometries are plentiful, provided the action is nonelementary. This generalizes previous results of Caprace-Sageev and Caprace-Zadnik (where it is assumed that the acting group has lattice-type properties).

In Gromov's density model for random groups at low densities groups can be cubulated (Ollivier-Wise) whereas at higher densities the groups have Property (T) (Zuk, Kotowski-Kotowski), but there's a gap where things are unclear.  I'll discuss two results, with Piotr Przytycki and Cornelia Drutu respectively, about this problem.

Bowditch's JSJ tree is a quasi-isometry invariant for one-ended hyperbolic groups, which uses the local cut point structure of their visual boundary.  We compute this tree for a large family of hyperbolic right-angled Coxeter groups, and identify a subfamily for which this tree is a complete quasi-isometry invariant.  We then investigate the commensurability classification of groups in this subfamily.  For our work on commensurability, a key step is proving that these Coxeter groups are virtually geometric amalgams of surfaces.  This is joint work with Pallavi Dani (Louisiana State University) and Emily Stark (University of Haifa).