We discuss Agol's construction of real hyperbolic lattices with arbitrarily small systole. Agol proposed his strategy in dimension 4, where the problem had theretofore been open, but Belolipetsky–Thomson and Bergeron–Haglund–Wise independently showed that this strategy goes through in all dimensions. As a byproduct, one obtains nonarithmetic real hyperbolic lattices in each dimension, and genuinely different examples from those of Gromov and Piatetski-Shapiro.
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Emmanuel Ullmo
