Coarse Kernels of Group Actions

Abstract

In this paper, we study the coarse kernel of a group action, namely the normal subgroup of elements that translate every point by a uniformly bounded amount. We give a complete algebraic characterization of this object. We specialize to $\mathrm{CAT}(0)$ spaces and show that the coarse kernel must be virtually abelian, characterizing when it is finite or cyclic in terms of the curtain model. As an application, we characterize the relation between the coarse kernels of the action on a $\mathrm{CAT}(0)$ space and the induced action on its curtain model. Along the way, we study weakly acylindrical actions on quasi-lines.