Junior Topology and Group Theory Seminar

 Given a group acting on a metric space X, one is often interested in the kernel of the action, consisting of those elements that fix every point of X. From a coarse geometric perspective, however, this notion is unsatisfactory, as the kernel is generally not invariant under G-equivariant quasi-isometries. To address this, one can instead consider the coarse kernel, defined as the collection of group elements that move every point of X by a uniformly bounded amount. In this talk, we study this coarse kernel under various assumptions on the action. 

When the action is geometric, we give a purely algebraic characterisation of the coarse kernel as the FC-centre of the group. We then specialise to actions on CAT(0) spaces, where we investigate the coarse kernel via the curtain model, a hyperbolic space associated to a CAT(0) space introduced by Petyt, Spriano, and Zalloum. Along the way, we will meet centralisers, boundaries, and actions on hyperbolic spaces! This is based on my summer project supervised by Davide Spriano and Harry Petyt.