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

For a polycyclic group, rank () is defined as the number of Z factors in a polycyclic decomposition of. For a finitely generated group G, rank (G) is defined as the infimum of rank () among finite index polycyclic subgroups G. For a compact RCD (K, N) space (X, d, m) with diam (X) (K, N), the rank of ₁ (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 ₁ (X) = ZN, then X is bi-H\"older homeomorphic to a flat torus (diffeomorphic in the smooth case).