Title:

Short geodesics in hyperbolic manifolds

Given a closed Riemannian nmanifold M, its shortest closed geodesic is called its systole and the length of this geodesic is denoted syst_1(M). For any ε > 0 and any n at least 2 one may construct a closed hyperbolic nmanifold M with syst_1(M) at most equal to ε. Constructions are detailed herein. The volume of M is bounded from below, by A_n/syst_1(M)^(n−2) where A_n is a positive constant depending only on n. There also exist sequences of nmanifolds M_i with syst_1(M_i) → 0 as i → ∞, such that vol(M_i) may be bounded above by a polynomial in 1/syst_1(M_i). When ε is sufficiently small, the manifold M is nonarithmetic, so that its fundamental group is an example of a nonarithmetic lattice in PO(n,1). The lattices arising from this construction are also exhibited as examples of noncoherent groups in PO(n,1). Also presented herein is an overview of existing results in this vein, alongside the prerequisite theory for the constructions given.
