Short Geodesics in Hyperbolic Manifolds
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879KbAbstractGiven a closed Riemannian n manifold M, its shortest closed geodesic is called its systole and the length of this geodesic is denoted syst_1(M). For any > 0 and any n at least 2 one may construct a closed hyperbolic n manifold M with syst_1(M) at most equal to. Constructions are detailed herein. The replica hermes black berkin
volume of M is bounded from below, by A_n/syst_1(M)^(n2) where A_n is a positive constant depending only on n. There also exist sequences of n manifolds 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 is sufficiently small, the manifold M is non arithmetic, so that its fundamental group is an example of a non arithmetic lattice in PO(n,1). The lattices arising from this construction are also exhibited as examples of non coherent 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.