Gromov–Hausdorff limits of aspherical manifolds

Rong, Xiaochun

doi:10.2140/gt.2026.30.2685

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ISSN (electronic): 1364-0380

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Gromov–Hausdorff limits of aspherical manifolds

Xiaochun Rong

Geometry & Topology 30 (2026) 2685–2712

DOI: 10.2140/gt.2026.30.2685

Abstract

Let $X$ be a compact Gromov–Hausdorff limit space of a sequence of compact $n$ -manifolds, $M_{i}$ , of Ricci curvature ${Ric}_{M_{i}} \geq - (n - 1)$ , volume $vol (M_{i}) \to 0$ , and where all points in $M_{i}$ are $(δ, ρ)$ -local rewinding Reifenberg points, or the sectional curvature satisfies $\sec_{M_{i}} \geq - 1$ , respectively. We conjecture that if $M_{i}$ is an aspherical manifold of fundamental group satisfying a certain condition (e.g., a nilpotent group), then $X$ is a differentiable, or topological aspherical manifold, respectively. Our main result asserts that if $M_{i}$ is diffeomorphic or homeomorphic to a nilmanifold, then $X$ is diffeomorphic or homeomorphic to a nilmanifold, respectively.

Keywords

Gromov–Hausdorff limit, nilpotent manifolds, aspherical manifolds

Mathematical Subject Classification

Primary: 53C21, 53C23, 53C24

References

Publication

Received: 26 March 2025

Revised: 9 December 2025

Accepted: 18 March 2026

Published: 13 August 2026

Proposed: Aaron Naber

Seconded: Tobias H Colding, Gang Tian

Authors

Xiaochun Rong
School of Mathematics Sciences Capital Normal University Beijing China	Department of Mathematics Rutgers University Piscataway, NJ United States

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Volume 30, issue 7 (2026)