For every n≥4 there are infinitely many closed n-manifolds admitting negatively curved Einstein metrics but no locally symmetric metric.
Arithmetic groups
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Lecture notes on an introductory course on arithmetic lattices (EPFL 2014).
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Negatively curved Einstein metrics on Gromov-Thurston manifolds
For every n≥4 there are infinitely many closed n-manifolds admitting negatively curved Einstein metrics but no locally symmetric metric.