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

Abstract

For a polycyclic group $\Lambda$, $\text{rank}(\Lambda )$ is defined as the number of $\mathbb{Z}$ factors in a polycyclic decomposition of $\Lambda$. For a finitely generated group $G$, $\text{rank}(G)$ is defined as the infimum of $\text{rank}(\Lambda )$ among finite index polycyclic subgroups $\Lambda \le G$. 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.

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