Kernel vs. amenable radical inclusion for actions on asymptotic cones

Ascertain whether, for every finitely generated group G and for all choices of ultrafilter ω and scaling sequence d_n, the inclusion ker(G acting naturally on the asymptotic cone Cone_ω(G,d_n)) ⊆ A(G) (the amenable radical of G) always holds; alternatively, construct a finitely generated group for which ker(G acting naturally on Cone_ω(G,d_n)) strictly contains A(G).

Background

The paper studies the kernel of the natural action of a finitely generated group G on its asymptotic cones and compares it to several canonical subgroups, including the amenable radical A(G), the FC-center, and the maximal finite normal subgroup K(G). For acylindrically hyperbolic groups, these subgroups coincide, but for general groups, only certain inclusions are established.

The authors exhibit examples where ker(G acting on Cone_ω(G,d_n)) is a proper subset of A(G) (e.g., for Baumslag–Solitar groups), and note they cannot find examples showing the reverse inclusion. They therefore pose whether the containment ker ⊆ A(G) is universal or whether there exist finitely generated groups with ker strictly larger than A(G).