We show that for every n > 2 and any > 0 there exists a compact hyperbolic n-manifold with a closed geodesic of length less than .When is sufficiently small these manifolds are non-arithmetic, and they are obtained by a generalised inbreeding construction which was first suggested by Agol for n D 4. We also show that for n > 3 the volumes of these manifolds grow at least as 1= n 2 when ! 0.
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A 2011 study studied this question.