We show that for every and any there exists a compact hyperbolic –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 . We also show that for the volumes of these manifolds grow at least as when .
"Systoles of hyperbolic manifolds." Algebr. Geom. Topol. 11 (3) 1455 - 1469, 2011. https://doi.org/10.2140/agt.2011.11.1455