On the kernel of actions on asymptotic cones

Any finitely generated group G acts on its asymptotic cones. The purpose of this paper is to calculate the kernel of such actions. First, we show that when G is acylindrically hyperbolic, the following subgroups all coincide: (1) the kernel of the natural action on asymptotic cones, (2) the kernel of the group action on the Gromov boundary ∂ X of a δ-hyperbolic space X on which G acts non-elementarily and acylindrically, (3) the set FC(G) of elements with finite conjugacy classes, (4) the unique maximal finite normal subgroup $U(G)$ of G, and (5) the amenable radical A(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 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 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. One direct corollary of these results is that the kernels of actions on asymptotic cones are invariant under the choice of the ultrafilter and non-decreasing sequence used to define the asymptotic cone. As an application, we show that the cardinality of the kernel can determine whether the group admits non-elementary action under some mild assumptions.