Thomson, Scott Andrew (2012) Short Geodesics in Hyperbolic Manifolds. Doctoral thesis, Durham University.
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Given 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 ε > 0 and any n at least 2 one may construct a closed hyperbolic n-manifold 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 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 ε 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.
|Item Type:||Thesis (Doctoral)|
|Award:||Doctor of Philosophy|
|Faculty and Department:||Faculty of Science > Mathematical Sciences, Department of|
|Copyright:||Copyright of this thesis is held by the author|
|Deposited On:||01 Jun 2012 10:11|