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This is closed related to the question asked here. I wonder if there is any progress on Problem 13 from the "Problem Section" in Schoen and Yau, page 281, problem 13, which asks: Let $M_1$ and $M_2$ be compact Einstein manifolds with negative curvature. Suppose that $\pi_1(M_1)=\pi_1(M_2)$ and $\dim M_1\geq 3$. Is $M_1$ isometric to $M_2$?

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Paul
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    To my knowledge, there has been no progress. Topology of Einstein manifolds is largely a mystery and one can ask many simple sounding questions with no answers in sight. – Igor Belegradek Oct 21 '14 at 18:00

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