On the kernel of actions on asymptotic cones

Abstract

Any finitely generated group $G$ acts on its asymptotic cones in natural ways. The purpose of this paper is to calculate the kernel of such actions. First, we show that when $G$ is acylindrically hyperbolic, the kernel of the natural action on every asymptotic cone coincides with the unique maximal finite normal subgroup $K(G)$ of $G$. Secondly, we use this equivalence to interpret the kernel of the actions on asymptotic cones as the kernel of the actions on many spaces at "infinity". For instance, if $G \curvearrowright M$ is a non-elementary convergence group, then we show that the kernel of actions on the limit set $L(G)$ coincides with the kernel of the action on asymptotic cones. Similar results can also be established for the non-trivial Floyd boundary and the $\mathrm{CAT}(0)$ groups with the visual boundary, contracting boundary, and sublinearly Morse boundary. Additionally, the results are extended to another action on asymptotic cones, called Paulin's construction. In the last section, we calculate the kernel on asymptotic cones for various groups, and as an application, we show that the cardinality of the kernel can determine whether the group admits non-elementary action under some mild assumptions.