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1. Surfaces in The Tesseract

   Estévez, Manuel; Roldán, Érika; Segerman, Henry (July 2023, Bridges 2023 Conference Proceedings)

   Holdener, Judy; Torrence, Eve; Fong, Chamberlain (Ed.)

   How can we visualize all the surfaces that can be made from the faces of the tesseract? In recent work, Aveni, Govc, and Roldán showed that the torus and the sphere are the only closed surfaces that can be realized by a subset of two-dimensional faces of the tesseract. They also gave an exhaustive list of all the isomorphic types of embedings of these two surfaces. Here, we generate 3D models of all these surfaces. We also exhibit, with the help of some hyper-ants, the minimum realization of the Möbius strip on the tesseract. 

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2. Algorithms for Contractibility of Compressed Curves on 3-Manifold Boundaries

   https://doi.org/10.1007/s00454-022-00411-x  

   Chambers, Erin Wolf; Lazarus, Francis; de Mesmay, Arnaud; Parsa, Salman (September 2023, Discrete & Computational Geometry)

   In this paper we prove that the problem of deciding contractibility of an arbitrary closed curve on the boundary of a 3-manifold is in NP. We emphasize that the manifold and the curve are both inputs to the problem. Moreover, our algorithm also works if the curve is given as a compressed word. Previously, such an algorithm was known for simple (non-compressed) curves, and, in very limited cases, for curves with self-intersections. Furthermore, our algorithm is fixed-parameter tractable in the size of the input 3-manifold. As part of our proof, we obtain new polynomial-time algorithms for compressed curves on surfaces, which we believe are of independent interest. We provide a polynomial-time algorithm which, given an orientable surface and a compressed loop on the surface, computes a canonical form for the loop as a compressed word. In particular, contractibility of compressed curves on surfaces can be decided in polynomial time; prior published work considered only constant genus surfaces. More generally, we solve the following normal subgroup membership problem in polynomial time: given an arbitrary orientable surface, a compressed closed curve, and a collection of disjoint normal curves, there is a polynomial-time algorithm to decide if the curve lies in the normal subgroup generated by components of the normal curves in the fundamental group of the surface after attaching the curves to a basepoint. 

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3. Extensions of multicurve stabilizers are hierarchically hyperbolic

   Russell, Jacob (December 2024, ArXiv)

   For a closed and orientable surface of genus at least 2, we prove the surface group extensions of the stabilizers of multicurves are hierarchically hyperbolic groups. This answers a question of Durham, Dowdall, Leininger, and Sisto. We also include an appendix that employs work of Charney, Cordes, and Sisto to characterize the Morse boundaries of hierarchically hyperbolic groups whose largest acylindrical action on a hyperbolic space is on a quasi-tree. 

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4. Diameter of Compact Riemann Surfaces

   https://doi.org/10.1007/s40315-024-00546-3  

   Stepanyants, Huck; Beardon, Alan; Paton, Jeremy; Krioukov, Dmitri (June 2024, Computational Methods and Function Theory)

   Diameter is one of the most basic properties of a geometric object, while Riemann surfaces are one of the most basic geometric objects. Surprisingly, the diameter of compact Riemann surfaces is known exactly only for the sphere and the torus. For higher genuses, only very general but loose upper and lower bounds are available. The problem of calculating the diameter exactly has been intractable since there is no simple expression for the distance between a pair of points on a high-genus surface. Here we prove that the diameters of a class of simple Riemann surfaces known as generalized Bolza surfaces of any genus greater than 1 are equal to the radii of their fundamental polygons. This is the first exact result for the diameter of a compact hyperbolic manifold. 

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5. Cusp volumes of alternating knots on surfaces

   https://doi.org/10.2140/agt.2022.22.2467  

   Bavier, B. (November 2022, Algebraic geometric topology)

   We study the geometry of hyperbolic knots that admit alternating projections on embedded surfaces in closed 3–manifolds. We show that, under mild hypothesis, their cusp area admits two-sided bounds in terms of the twist number of the alternating projection and the genus of the projection surface. As a result, we derive diagrammatic estimates of slope lengths and give applications to Dehn surgery. These generalize results of Lackenby and Purcell about alternating knots in the 3–sphere. Using a result of Kalfagianni and Purcell, we point out that alternating knots on surfaces of higher genus can have arbitrarily small cusp density, in contrast to alternating knots on spheres whose cusp densities are bounded away from zero due to Lackenby and Purcell. 

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