Topological rigidity of small RCD(K,N) spaces with maximal rank

For a polycyclic group Λ, rank(Λ ) is defined as the number of Z factors in a polycyclic decomposition of Λ. For a finitely generated group G, rank$(G)$ is defined as the infimum of rank(Λ ) among finite index polycyclic subgroups Λ ≤ G. For a compact RCD$(K,N)$ space (X,d, m ) with diam(X) ≤ ε (K,N), the rank of π₁(X) is at most N. In this note we show that in case of equality, X is homeomorphic to an infranilmanifold, generalizing to the non-smooth setting a result by Kapovitch--Wilking. We also fill a gap in the proof that if π₁(X) = ZN, then X is bi-H\"older homeomorphic to a flat torus (diffeomorphic in the smooth case).