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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. Equivariant hyperbolization of 3-manifolds via homology cobordisms

   Dave Auckly, Hee Jung (January 2023, Topology and its applications)

   The main result of this paper is that any 3-dimensional manifold with a finite group action is equivariantly invertibly homology cobordant to a hyperbolic manifold; this result holds with suitable twisted coefficients as well. The following two consequences motivated this work. First, there are hyperbolic equivariant corks (as defined in previous work of the authors) for a wide class of finite groups. Second, any finite group that acts on a homology 3-sphere also acts on a hyperbolic homology 3-sphere. The theorem has other corollaries, including the existence of infinitely many hyperbolic homology spheres that support free Zp-actions that do not extend over any contractible manifolds, and (from the non-equivariant version of the theorem) infinitely many that bound homology balls but do not bound contractible manifolds. In passing, it is shown that the invertible homology cobordism relation on 3-manifolds is antisymmetric, and thus a partial order. 

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4. Relative cubulation of relative strict hyperbolization

   https://doi.org/10.1112/jlms.70093  

   Lafont, Jean‐François; Ruffoni, Lorenzo (April 2025, Journal of the London Mathematical Society)

   Abstract We prove that many relatively hyperbolic groups obtained by relative strict hyperbolization admit a cocompact action on a cubical complex. Under suitable assumptions on the peripheral subgroups, these groups are residually finite and even virtually special. We include some applications to the theory of manifolds, such as the construction of new non‐positively curved Riemannian manifolds with residually finite fundamental group, and the existence of non‐triangulable aspherical manifolds with virtually special fundamental group. 

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5. Asymptotics of the Poisson kernel and Green's functions of the fractional conformal Laplacian

   https://doi.org/10.3934/dcds.2022085  

   Mayer, Martin; Ndiaye, Cheikh Birahim (January 2022, Discrete and Continuous Dynamical Systems)

   We study the asymptotics of the Poisson kernel and Green's functions of the fractional conformal Laplacian for conformal infinities of asymptotically hyperbolic manifolds. We derive sharp expansions of the Poisson kernel and Green's functions of the conformal Laplacian near their singularities. Our expansions of the Green's functions answer the first part of the conjecture of Kim-Musso-Wei[21] in the case of locally flat conformal infinities of Poincare-Einstein manifolds and together with the Poisson kernel asymptotic is used also in our paper [25] to show solvability of the fractional Yamabe problem in that case. Our asymptotics of the Green's functions on the general case of conformal infinities of asymptotically hyperbolic space is used also in [29] to show solvability of the fractional Yamabe problem for conformal infinities of dimension \begin{document}$ 3 $$\end{document} and fractional parameter in \begin{document}$$ (\frac{1}{2}, 1) $$\end{document}$ corresponding to a global case left by previous works. 

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