I will describe a construction of closed hyperbolic manifolds with arbitrarily short closed geodesics, and explain why, if the closed geodesic is sufficiently short in a particular sense, then these manifolds are non-arithmetic. The construction starts from arithmetic lattices in PO(n,1) and is a generalisation (by Belolipetsky-T.) of a construction of I. Agol.
Links:
[1] https://www.mpim-bonn.mpg.de/taxonomy/term/39
[2] https://www.mpim-bonn.mpg.de/node/3444
[3] https://www.mpim-bonn.mpg.de/node/3050