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

TL;DR

This paper proves that compact RCD(K,N) spaces with maximal fundamental group rank are topologically infranilmanifolds, extending classical rigidity results to non-smooth metric measure spaces.

Contribution

It establishes a topological rigidity result for RCD(K,N) spaces with maximal fundamental group rank, generalizing smooth manifold theorems to the non-smooth setting.

Findings

Spaces with maximal rank are homeomorphic to infranilmanifolds.

The fundamental group rank is bounded above by N.

Equality case characterizes the space as an infranilmanifold.

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 $\Lambda \leq G$. For a compact $ \text{RCD} (K,N)$ space $(X,\mathsf{d}, \mathfrak{m})$ with $ \text{diam} (X) \leq \varepsilon (K,N)$, the rank of $\pi_1(X)$ 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.