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1. A Hyperbolic Counterpart to Rokhlin’s Cobordism Theorem

   https://doi.org/10.1093/imrn/rnaa158  

   Chu, Michelle; Kolpakov, Alexander (July 2020, International Mathematics Research Notices)

   null (Ed.)

   Abstract The purpose of the present paper is to prove existence of super-exponentially many compact orientable hyperbolic arithmetic $$n$$-manifolds that are geometric boundaries of compact orientable hyperbolic $(n+1)$-manifolds, for any $$n \geq 2$$, thereby establishing that these classes of manifolds have the same growth rate with respect to volume as all compact orientable hyperbolic arithmetic $$n$$-manifolds. An analogous result holds for non-compact orientable hyperbolic arithmetic $$n$$-manifolds of finite volume that are geometric boundaries for $$n \geq 2$$. 

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2. Cusps and commensurability classes ofhyperbolic 4–manifolds

   https://doi.org/10.2140/agt.2023.23.3805  

   Sell, Connor (January 2023, Algebraic & Geometric Topology)

   There are six orientable compact flat 3–manifolds that can occur as cusp cross-sections of hyperbolic 4–manifolds. We provide 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. 

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3. Extremality and rigidity for scalar curvature in dimension four

   https://doi.org/10.1007/s00029-023-00892-5  

   Bettiol, Renato G.; Goodman, McFeely Jackson (February 2024, Selecta Mathematica)

   Following Gromov, a Riemannian manifold is called area-extremal if any modification that increases scalar curvature must decrease the area of some tangent 2-plane. We prove that large classes of compact 4-manifolds, with or without boundary, with nonnegative sectional curvature are area-extremal. We also show that all regions of positive sectional curvature on 4-manifolds are locally area-extremal. These results are obtained analyzing sections in the kernel of a twisted Dirac operator constructed from pairs of metrics, and using the Finsler–Thorpe trick for sectional curvature bounds in dimension 4. 

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4. Partial hyperbolicity and pseudo-Anosov dynamics

   https://doi.org/10.1007/s00039-024-00670-1  

   Fenley, Sergio; Potrie, Rafael (January 2024, Geometric and functional analysis)

   We study partial hyperbolic (PH) diffeomorphisms in 3-manifolds satisfying a commuting property. This is mainly applicable when the manifold is either hyperbolic or Seifert. As a consequence we prove that if a hyperbolic 3-manifold admits a partially hyperbolic diffeomorphism, then it also admits an Anosov flow. 

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5. Totally Geodesic Surfaces in Hyperbolic 3-Manifolds: Algorithms and Examples

   https://doi.org/10.4230/LIPIcs.SoCG.2024.14  

   Basilio, Brannon; Lee, Chaeryn; Malionek, Joseph (January 2024, Schloss Dagstuhl – Leibniz-Zentrum für Informatik)

   Mulzer, Wolfgang; Phillips, Jeff M (Ed.)

   Finding a totally geodesic surface, an embedded surface where the geodesics in the surface are also geodesics in the surrounding manifold, has been a problem of interest in the study of 3-manifolds. This has especially been of interest in hyperbolic 3-manifolds and knot complements, complements of piecewise-linearly embedded circles in the 3-sphere. This is due to Menasco-Reid’s conjecture stating that hyperbolic knot complements do not contain such surfaces. Here, we present an algorithm that determines whether a given surface is totally geodesic and an algorithm that checks whether a given 3-manifold contains a totally geodesic surface. We applied our algorithm on over 150,000 3-manifolds and discovered nine 3-manifolds with totally geodesic surfaces. Additionally, we verified Menasco-Reid’s conjecture for knots up to 12 crossings. 

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