Abdul Zalloum

Hyperplanes all over the place

(Joint with Petyt and Spriano) A revolutionary work of Sageev shows that the entire structure of a CAT(0) cube complex is encoded in the combinatorics of its hyperplanes. I will describe a very natural way in which hyperplanes exist beyond the world of CAT(0) cube complexes including CAT(0) spaces/groups, hierarchically hyperbolic groups, and coarse median spaces that satisfy an extra condition. Using such hyperplanes, we build a family of hyperbolic spaces in the same fashion the contact graph of Hagen and the separation graph of Genevious are built. For instance, in the mapping class group case, the "contact graph" we build coincides (up to quasi-isometry) with the curve graph, and in RAAGS, the contact graph we build coincides with the contact graph of Hagen, up to quasi-isometry. A great deal of the well-known theorems regarding actions on such graphs almost effortlessly carry through from the world of cubulated groups to our setting. The main new class of groups our work provides a ``contact graph" for is the class of CAT(0) groups.