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   Christian Gaetz ; Sergi Elizalde ; Colin Defant and Nathan Williams ; Nicholas J. Williams ; Wenjie Fang ; Jesús A. De Loera and William J. Wesley ; Yasuhide Numata, Yusuke Takahashi ; Margaret Bayer, Mark Denker ; Christian Gaetz, Oliver Pechenik ; Jia Huang and Erkko Lehtonen ; et al ( April 2023 , Séminaire lotharingien de combinatoire)

   Krattenthaler, Christian ; Thibon, Jean-Yves (Ed.)
2. A relative version of the Turaev–Viro invariants and the volume of hyperbolic polyhedral 3‐manifolds

   https://doi.org/10.1112/topo.12300  

   Yang, Tian ( May 2023 , Journal of Topology)

   We define a relative version of the Turaev–Viro invariants for an ideally triangulated compact 3‐manifold with nonempty boundary and a coloring on the edges, generalizing the Turaev–Viro invariants [36] of the manifold. We also propose the volume conjecture for these invariants whose asymptotic behavior is related to the volume of the manifold in the hyperbolic polyhedral metric [22, 23] with singular locus of the edges and cone angles determined by the coloring, and prove the conjecture in the case that the cone angles are sufficiently small. This suggests an approach of solving the volume conjecture for the Turaev–Viro invariants proposed by Chen–Yang [8] for hyperbolic 3‐manifolds with totally geodesic boundary.

    

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3. Embedding closed hyperbolic 3–manifolds insmall volume hyperbolic 4–manifolds

   https://doi.org/10.2140/agt.2021.21.2627  

   Chu, Michelle ; Reid, Alan W ( January 2021 , Algebraic & Geometric Topology)
4. Hyperbolic rank rigidity for manifolds of -pinched negative curvature

   https://doi.org/10.1017/etds.2018.113  

   CONNELL, CHRIS ; NGUYEN, THANG ; SPATZIER, RALF ( May 2020 , Ergodic Theory and Dynamical Systems)

   A Riemannian manifold $M$ has higher hyperbolic rank if every geodesic has a perpendicular Jacobi field making sectional curvature $-1$ with the geodesic. If, in addition, the sectional curvatures of $M$ lie in the interval $[-1,-\frac{1}{4}]$ and $M$ is closed, we show that $M$ is a locally symmetric space of rank one. This partially extends work by Constantine using completely different methods. It is also a partial counterpart to Hamenstädt’s hyperbolic rank rigidity result for sectional curvatures $\leq -1$ , and complements well-known results on Euclidean and spherical rank rigidity. 

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5. Metric rigidity of Kahler manifolds with lower Ricci bounds and almost maximal volume.

   Ved Datar, Harish Seshadri ( January 2021 , Proceedings of the American Mathematical Society)