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. 22E40, 53C22

Abstract. We show that the non-arithmetic lattices in PO(n, 1) of Belolipetsky and Thomson [BT11], obtained as fundamental groups of closed hyperbolic manifolds with short systole, are quasi-arithmetic in the sense of Vinberg, and, by contrast, the well-known non-arithmetic lattices of Gromov and PiatetskiShapiro are not quasi-arithmetic. A corollary of this is that there are, for all n 2, non-arithmetic lattices in PO(n, 1) that are not commensurable with the Gromov-Piatetski-Shapiro lattices.

We compute invariants of quadratic forms associated to orthogonal hypergeometric groups of degree five. This allows us to determine some commensurabilities between these groups, as well as to say when some thin groups cannot be conjugate to each other.

No abstract