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

For a polycyclic group $Λ$, $\text{rank} ( Λ)$ is defined as the number of $\mathbb{Z}$ factors in a polycyclic decomposition of $Λ$. For a finitely generated group $G$, $\text{rank} (G)$ is defined as the infimum of $ \text{rank} (Λ)$ among finite index polycyclic subgroups $Λ\leq G$. For a compact $\text{RCD} (K,N)$ space $(X,\mathsf{d}, \mathfrak{m})$ with $\text{diam} (X) \leq \varepsilon (K,N)$, the rank of $π_1(X)$ is at most $N$. We show that in case of equality, $X$ is homeomorphic to an infranilmanifold, generalizing a result by Kapovitch--Wilking to the non-smooth setting. We also fill a gap in the proof by Mondello--Mondino--Perales that if $π_1(X) = \mathbb{Z}^N$, then $X$ is bi-Hölder homeomorphic to a flat torus (diffeomorphic in the smooth case).