Browsing by Author "Reid, Alan W"

Now showing 1 - 4 of 4
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    Cusps and commensurability classes of hyperbolic 4-manifolds
    (2023-04-21) Sell, Connor; Reid, Alan W
    It is well-known that the cusp cross-sections of finite-volume, cusped hyperbolic n-manifolds are flat, compact (n-1)-manifolds. In 2002, Long and Reid proved that each of the finitely many homeomorphism classes of flat, compact (n-1)-manifolds occur as the cusp cross-section of some arithmetic hyperbolic n-orbifold; the orbifold was upgraded to a manifold by McReynolds in 2004. There are six orientable, compact, flat 3-manifolds that can occur as cusp cross-sections of hyperbolic 4-manifolds. This thesis provides criteria for exactly when a given commensurability class of arithmetic hyperbolic 4-manifolds contains a representative with a given cusp type. In particular, for three of the six cusp types, we provide infinitely many examples of commensurability classes that contain no manifolds with cusps of the given type; no such examples were previously known for any cusp type in any dimension. Further, we extend this result to find commensurability classes of hyperbolic 5-manifolds that avoid some compact, flat 4-manifolds as cusp cross-sections, and classes of non-arithmetic manifolds in both dimensions that avoid some cusp types.
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    Filling links and minimal surfaces in 3-manifolds
    (2022-04-22) Stagner, William; Reid, Alan W
    This thesis studies this existence of filling links 3-manifolds. A link L in a 3-manifold M is filling in M if, for any spine G of M disjoint from L, π_1(G) injects into π_1(M-L). Conceptually, a filling link cuts through all of the topology 3-manifold. These links were first studied by Freedman-Krushkal in the concrete case of the 3-torus M = T^3, but they leave open the question of whether a filling link actually exists in T^3. We answer this question affirmatively by proving in fact that every closed, orientable 3-manifold M with fundamental group of rank 3 contains a filling link.
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    Finite Quotients of Hyperbolic 3-Manifold Groups
    (2023-04-24) Cheetham-West, Tamunonye; Reid, Alan W
    This thesis provides further evidence of the seemingly very close relationship between the geometry of a finite-volume hyperbolic 3-manifold and the profinite completion of its fundamental group.
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    Thin Surface Subgroups in Non-uniform Arithmetic Lattices in SO+(n,1)
    (2024-04-17) Edelman-Munoz, Sara; Reid, Alan W
    In this thesis I show that the fundamental groups of all non-compact, arithmetic, hyperbolic, n-manifolds for n $\geq$ 4 have thin surface subgroups. As a consequence of the proof of this theorem I show that the fundamental groups of the doubles of cusped, arithmetic, hyperbolic manifolds are GFERF.