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

For a polycyclic group $\mathrm{\Lambda }$, $\text{rank}(\mathrm{\Lambda })$ is defined as the number of $\mathbb{Z}$ factors in a polycyclic decomposition of $\mathrm{\Lambda }$. For a finitely generated group $G$, $\text{rank}(G)$ is defined as the infimum of $\text{rank}(\mathrm{\Lambda })$ among finite index polycyclic subgroups $\mathrm{\Lambda }\le G$. For a compact $\text{RCD}(K,N)$ space $(X,\mathsf{d},\mathfrak{m})$ with $\text{diam}(X)\le \epsilon (K,N)$, the rank of ${\pi }_1(X)$ 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.