Coarse Kernels of Group Actions

TL;DR

This paper investigates the algebraic structure of the coarse kernel in group actions, especially on CAT(0) spaces, providing a complete characterization and exploring its properties in various geometric contexts.

Contribution

It offers a complete algebraic description of the coarse kernel in group actions and characterizes its structure on CAT(0) spaces, including conditions for finiteness and cyclicity.

Findings

Coarse kernel is virtually abelian on CAT(0) spaces.

Characterization of when the coarse kernel is finite or cyclic.

Relation between coarse kernels of actions on CAT(0) spaces and curtain models.

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.