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arXiv · 2610.15406

One boundary, infinitely geometries

Jiming Ma
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Abstract

Let $N$ be a 3-dimensional integral homology sphere. We prove the existence of a sequence of 4-dimensional topologically negatively curved manifolds $M_k$, $k=1,2,\dots$, satisfying the following: (1) all aspherical manifolds $M_k$ have the same universal cover $U_N$ depending only on $N$; (2) all fundamental groups $π_1(M_k)$ are hyperbolic and share the same Gromov boundary $X_N$ depending only on $N$; (3) the groups $π_1(M_k)$ represent infinitely many distinct quasi-isometry classes. The last conclusion, which represents the new feature, shows that negatively curved geometries in dimension 4 exhibit substantially richer structure than those in dimension 3.

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Jiming Ma. 2026-10-11. One boundary, infinitely geometries. https://arxiv.org/abs/2610.15406

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