A CAT(0) alternative for amenable groups and a Kazhdan-type rigidity principle

We prove that finitely generated amenable groups acting on CAT(0) spaces satisfy the following alternative: either every action on a geodesically complete CAT(0) space with bounded geometry (or finite dimension) has a global fixed point, or the group admits a fixed-point-free action on $\mathbb{R}^n$. As a consequence, finitely generated amenable torsion groups and finitely generated virtually simple amenable groups cannot act nontrivially on geodesically complete CAT(0) spaces with bounded geometry or on finite-dimensional complete CAT(0) spaces. The proof relies on a Kazhdan-type rigidity theorem for groups with the Euclidean fixed point property: if such a group acts on a geodesically complete CAT(0) space of bounded geometry with almost fixed points, then it has a genuine fixed point. This yields several further corollaries, including a rigidity dichotomy for drift and that any finitely generated torsion group acting on a geodesically complete visibility CAT(0) space with bounded geometry must have a global fixed point. These results make substantial progress on the longstanding problem of understanding actions of torsion groups on CAT(0) spaces.