One boundary, infinitely geometries

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.