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1. Geodesic planes in geometrically finite acylindrical -manifolds

   https://doi.org/10.1017/etds.2021.19  

   BENOIST, YVES; OH, HEE (February 2022, Ergodic Theory and Dynamical Systems)

   Abstract Let M be a geometrically finite acylindrical hyperbolic $$3$$ -manifold and let $M^*$ denote the interior of the convex core of M . We show that any geodesic plane in $M^*$ is either closed or dense, and that there are only countably many closed geodesic planes in $M^*$ . These results were obtained by McMullen, Mohammadi and Oh [Geodesic planes in hyperbolic 3-manifolds. Invent. Math. 209 (2017), 425–461; Geodesic planes in the convex core of an acylindrical 3-manifold. Duke Math. J. , to appear, Preprint , 2018, arXiv:1802.03853] when M is convex cocompact. As a corollary, we obtain that when M covers an arithmetic hyperbolic $$3$$ -manifold $$M_0$$ , the topological behavior of a geodesic plane in $M^*$ is governed by that of the corresponding plane in $$M_0$$ . We construct a counterexample of this phenomenon when $$M_0$$ is non-arithmetic. 

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2. Shrinking targets for the geodesic flow on geometrically finite hyperbolic manifolds

   https://doi.org/10.3934/jmd.2021014  

   Kelmer, Dubi; Oh, Hee (January 2021, Journal of Modern Dynamics)

   Let \begin{document}$$ \mathscr{M} $$\end{document} be a geometrically finite hyperbolic manifold. We present a very general theorem on the shrinking target problem for the geodesic flow, using its exponential mixing. This includes a strengthening of Sullivan's logarithm law for the excursion rate of the geodesic flow. More generally, we prove logarithm laws for the first hitting time for shrinking cusp neighborhoods, shrinking tubular neighborhoods of a closed geodesic, and shrinking metric balls, as well as give quantitative estimates for the time a generic geodesic spends in such shrinking targets. 

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3. Geodesic Growth of Numbered Graph Products

   https://doi.org/10.46298/jgcc.2023.14.2.10019  

   Marjanski, Lindsay; Solon, Vincent; Zheng, Frank; Zopff, Kathleen (February 2023, journal of Groups, complexity, cryptology)

   In this paper, we study geodesic growth of numbered graph products; these area generalization of right-angled Coxeter groups, defined as graph products offinite cyclic groups. We first define a graph-theoretic condition calledlink-regularity, as well as a natural equivalence amongst link-regular numberedgraphs, and show that numbered graph products associated to link-regularnumbered graphs must have the same geodesic growth series. Next, we derive aformula for the geodesic growth of right-angled Coxeter groups associated tolink-regular graphs. Finally, we find a system of equations that can be used tosolve for the geodesic growth of numbered graph products corresponding tolink-regular numbered graphs that contain no triangles and have constant vertexnumbering. 

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4. IGG: Image Generation Informed by Geodesic Dynamics in Deformation Spaces

   Wu, Nian; Jayakumar, Nivetha; Xing, Jiarui; Zhang, Miaomiao (May 2025, Springer Cham)

   Generative models have recently gained increasing attention in image generation and editing tasks. However, they often lack a direct connection to object geometry, which is crucial in sensitive domains such as computational anatomy, biology, and robotics. This paper presents a novel framework for Image Generation informed by Geodesic dynamics (IGG) in deformation spaces. Our IGG model comprises two key components: (i) an efficient autoencoder that explicitly learns the geodesic path of image transformations in the latent space; and (ii) a latent geodesic diffusion model that captures the distribution of latent representations of geodesic deformations conditioned on text instructions. By leveraging geodesic paths, our method ensures smooth, topology-preserving, and interpretable deformations, capturing complex variations in image structures while maintaining geometric consistency. We validate the proposed IGG on plant growth data and brain magnetic resonance imaging (MRI). Experimental results show that IGG outperforms the state-of-the-art image generation/editing models with superior performance in generating realistic, high-quality images with preserved object topology and reduced artifacts. Our code is publicly available at https://github.com/nellie689/IGG. 

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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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