Non-arithmetic hyperbolic manifolds via short closed geodesics

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.