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Oct 12, 2016 at 12:32 comment added Igor Belegradek You probably meant to assume that $X$ is closed or finite volume. In the arithmetic case a lot is known (even though I doubt that complete cohomology ring is ever computed). I would start from the seminal paper [Millson, John J. On the first Betti number of a constant negatively curved manifold. Ann. of Math. (1976)] and use MathSciNet to trace references to the modern times.
Oct 12, 2016 at 5:37 history edited YCor
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Oct 12, 2016 at 4:30 history asked David Treumann CC BY-SA 3.0