Research

Published papers

Most papers can be downloaded at http://arxiv.org/a/kim_s_3

• 35. Inhyeok Choi, Sang-hyun Kim, Smoothing countable group actions on metrizable spaces, to appear in Advanced Studies in Pure Mathematics (Proceedings of Mathematical Society of Japan - Seasonal Institute).
  [Summary]

  Every topological action of a countable group on a metrizable space can be promoted to a bi-Lipschitz action with respect to a suitably chosen compatible metric. The same conclusion holds for closed subgroups of locally compact groups. This generalizes earlier work of Hamenstädt on finitely generated groups, and recovers a smoothing theorem of Deroin, Kleptsyn, and Navas for actions on one-manifolds.

  [Arxiv]

  [cite]

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  @misc{arxiv:2410.06077,
        title={Smoothing countable group actions on metrizable spaces}, 
        author={Inhyeok Choi and Sang-hyun Kim},
        year={2026},
        eprint={2410.06077},
        archivePrefix={arXiv},
        primaryClass={math.GR},
        url={https://arxiv.org/abs/2410.06077}
  }

• 34. Sang-hyun Kim, Nicolás Matte Bon, Mikael de la Salle and Michele Triestino, Subexponential growth and C¹ actions on one-manifolds, International Mathematics Research Notices, Volume 2025, Issue 13, July 2025, rnaf202.
  [Summary]

  For countable groups without finitely generated subgroups of exponential growth, every faithful topological action on a compact connected one-manifold can be blown up to a C¹ action. The proof relies on a combinatorial characterization of such groups and extends techniques from smooth dynamics. This identifies a precise algebraic condition under which topological rigidity can be upgraded to differentiable regularity.

  [Journal] [Arxiv]

  [cite]

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  @article {MR4929825,
      AUTHOR = {Kim, Sang-hyun and de la Salle, Mikael and Matte Bon,
                Nicol\'{a}s and Triestino, Michele},
       TITLE = {Subexponential growth and {$\rm C^1$} actions on
                one-manifolds},
     JOURNAL = {Int. Math. Res. Not. IMRN},
    FJOURNAL = {International Mathematics Research Notices. IMRN},
        YEAR = {2025},
      NUMBER = {13},
       PAGES = {Paper No. rnaf202, 12},
        ISSN = {1073-7928,1687-0247},
     MRCLASS = {57M60},
    MRNUMBER = {4929825},
  MRREVIEWER = {Sanghoon\ Kwon},
         DOI = {10.1093/imrn/rnaf202},
         URL = {https://doi.org/10.1093/imrn/rnaf202},
  }

• 33. Sang-hyun Kim, Thomas Koberda and Javier de la Nuez-Gonzalez. First order rigidity of homeomorphism groups of manifolds, Comm. Amer. Math. Soc. 5 (2025), 144-194.
  [Summary]

  A compact connected manifold of dimension at least two is uniquely determined, up to homeomorphism, by a single first-order sentence in the language of groups evaluated on its homeomorphism group. In particular, the first-order theory of the homeomorphism group is a complete invariant of the underlying manifold. Analogous rigidity results are established for measure-preserving homeomorphism groups and diffeomorphism groups, forging a direct link between the model theory of transformation groups and the topology of manifolds.

  [Journal] [Arxiv]

  [cite]

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  @article {MR4904300,
      AUTHOR = {Kim, Sang-hyun and Koberda, Thomas and de la Nuez
                Gonz\'{a}lez, J.},
       TITLE = {First order rigidity of homeomorphism groups of manifolds},
     JOURNAL = {Commun. Am. Math. Soc.},
    FJOURNAL = {Communications of the American Mathematical Society},
      VOLUME = {5},
        YEAR = {2025},
       PAGES = {144--194},
        ISSN = {2692-3688},
     MRCLASS = {03C07 (20A15 57M60 57S05 57S25)},
    MRNUMBER = {4904300},
         DOI = {10.1090/cams/47},
         URL = {https://doi.org/10.1090/cams/47},
  }

• 32. Nhat Minh Doan, Sang-hyun Kim, Mong Lung Lang, Ser Peow Tan. Optimal independent generating system for the congruence subgroups Γ₀(p) and Γ₀(pq), The Journal of Geometric Analysis (2025) 35:216.
  [Summary]

  For primes p and distinct primes p, q, an explicit independent generating system for the congruence subgroups Γ₀(p) and Γ₀(pq) of SL(2,ℤ) is constructed, where each generator has its (2,1)-entry equal to n or 0. This resolves a conjecture of Kulkarni for these subgroups. The proof is geometric, based on the action of Γ₀(n) on the Farey graph in the hyperbolic plane and the combinatorics of the resulting quotient.

  [Journal] [Arxiv]

  [cite]

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  @article {MR4917150,
      AUTHOR = {Doan, Nhat Minh and Kim, Sang-hyun and Lang, Mong Lung and
                Tan, Ser Peow},
       TITLE = {Optimal special polygons for the congruence subgroups {$\Gamma
                _0(p)$} and {$\Gamma _0(pq)$}},
     JOURNAL = {J. Geom. Anal.},
    FJOURNAL = {Journal of Geometric Analysis},
      VOLUME = {35},
        YEAR = {2025},
      NUMBER = {7},
       PAGES = {Paper No. 216, 28},
        ISSN = {1050-6926,1559-002X},
     MRCLASS = {11F06 (11B57 30F35)},
    MRNUMBER = {4917150},
  MRREVIEWER = {Michelle\ Chu},
         DOI = {10.1007/s12220-025-02051-y},
         URL = {https://doi.org/10.1007/s12220-025-02051-y},
  }

• 31. Sang-hyun Kim and Thomas Koberda. Non-freeness of groups generated by two parabolic elements with small rational parameters. Michigan Mathematical Journal (2022).
  [Summary]

  Given two parabolic elements in SL(2,ℂ) parameterized by a rational number s/r, the group they generate is shown to be non-free for all numerators s ≤ 27 (with a partial exception at s = 24). For larger numerators, density estimates are provided. The proofs combine algebraic arguments with computer-assisted verification, contributing to the broader question of when two-generator subgroups of SL(2,ℂ) fail to be free.

  [Journal] [Arxiv]

  [cite]

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  @article {MR4505367,
      AUTHOR = {Kim, Sang-hyun and Koberda, Thomas},
       TITLE = {Non-freeness of groups generated by two parabolic elements
                with small rational parameters},
     JOURNAL = {Michigan Math. J.},
    FJOURNAL = {Michigan Mathematical Journal},
      VOLUME = {71},
        YEAR = {2022},
      NUMBER = {4},
       PAGES = {809--833},
        ISSN = {0026-2285,1945-2365},
     MRCLASS = {20H10 (20-08 20F05)},
    MRNUMBER = {4505367},
  MRREVIEWER = {Jack\ O.\ Button},
         DOI = {10.1307/mmj/20205868},
         URL = {https://doi.org/10.1307/mmj/20205868},
  }

• 30. Sang-hyun Kim and Genevieve Walsh. Some groups with planar boundaries, Surveys in Differential Geometry (2022).
  [Summary]

  This survey illustrates Bowditch's canonical JSJ splitting of one-ended hyperbolic groups, with emphasis on how planarity of the Gromov boundary constrains the algebraic structure of the group. A central theme is the Kapovich–Kleiner conjecture, which predicts that a one-ended hyperbolic group with planar boundary acts geometrically on a 3-manifold with boundary.

  [Journal] [Arxiv]

  [cite]

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  @incollection {MR4479754,
      AUTHOR = {Kim, Sang-hyun and Walsh, Genevieve},
       TITLE = {Some groups with planar boundaries},
   BOOKTITLE = {Surveys in differential geometry 2020. {S}urveys in 3-manifold
                topology and geometry},
      SERIES = {Surv. Differ. Geom.},
      VOLUME = {25},
       PAGES = {253--280},
   PUBLISHER = {Int. Press, Boston, MA},
        YEAR = {[2022] \copyright 2022},
        ISBN = {978-1-57146-419-4},
     MRCLASS = {20F67 (20F65 57Sxx)},
    MRNUMBER = {4479754},
  }

• 29. Sang-hyun Kim, Thomas Koberda and Cristóbal Rivas. Virtual critical regularity of mapping class group actions on the circle, Transformation Groups (2022).
  [Summary]

  The free product (G × H) ∗ ℤ, where G and H are non-solvable, admits no embedding into the group of orientation-preserving C^{1+τ} diffeomorphisms of the circle for any τ > 0. As a consequence, no finite-index subgroup of the mapping class group of a closed surface of genus at least two can act faithfully on the circle by C^{1+τ} diffeomorphisms. This pins down the virtual critical regularity of mapping class group actions at exactly C¹.

  [Journal] [Arxiv]

  [cite]

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  @article {MR4788024,
      AUTHOR = {Kim, Sang-hyun and Koberda, Thomas and Rivas, Crist\'{o}bal},
       TITLE = {Virtual critical regularity of mapping class group actions on
                the circle},
     JOURNAL = {Transform. Groups},
    FJOURNAL = {Transformation Groups},
      VOLUME = {29},
        YEAR = {2024},
      NUMBER = {3},
       PAGES = {1105--1114},
        ISSN = {1083-4362,1531-586X},
     MRCLASS = {57M60 (20F14 20F36 37C05 37C85 57K20)},
    MRNUMBER = {4788024},
  MRREVIEWER = {Marja\ K.\ Kankaanrinta},
         DOI = {10.1007/s00031-022-09698-9},
         URL = {https://doi.org/10.1007/s00031-022-09698-9},
  }

• 28. Sang-hyun Kim and Thomas Koberda. Structure and regularity of group actions on one-manifolds, Springer Monographs in Mathematics (2021).
  [Summary]

  This monograph presents a systematic study of the interplay between algebraic structure of groups and their actions on one-manifolds by diffeomorphisms of varying regularity. The theory of critical regularity is developed, showing how the optimal smoothness of a group action is determined by algebraic constraints such as the presence of direct products or free products. Key topics include right-angled Artin group actions, mapping class group dynamics, and phenomena related to Denjoy's theorem, bringing together classical results and recent developments at the interface of geometric group theory and smooth dynamics.

  [Journal] [Arxiv]

  [cite]

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  @book {MR4381312,
      AUTHOR = {Kim, Sang-hyun and Koberda, Thomas},
       TITLE = {Structure and regularity of group actions on one-manifolds},
      SERIES = {Springer Monographs in Mathematics},
   PUBLISHER = {Springer, Cham},
        YEAR = {[2021] \copyright 2021},
       PAGES = {xiv+323},
        ISBN = {978-3-030-89005-6; 978-3-030-89006-3},
     MRCLASS = {57M60 (20F65 37C85)},
    MRNUMBER = {4381312},
         DOI = {10.1007/978-3-030-89006-3},
         URL = {https://doi.org/10.1007/978-3-030-89006-3},
  }

• 27. Sang-hyun Kim, Thomas Koberda and Cristóbal Rivas. Direct products, overlapping actions, and critical regularity, Journal of Modern Dynamics, Vol. 17 (June 2021), pp. 285-304.
  [Summary]

  The free product (G × H) ∗ ℤ, where G and H are non-solvable, does not embed into the group of orientation-preserving C^{1+τ} diffeomorphisms of a compact interval for any τ > 0. The proof introduces the technique of analyzing "overlapping" actions of direct product subgroups to detect regularity obstructions. This provides a counterpart on the interval to parallel results on the circle, further constraining how algebraic complexity limits differentiable regularity.

  [Journal] [Arxiv]

  [cite]

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  @article {MR4288175,
      AUTHOR = {Kim, Sang-hyun and Koberda, Thomas and Rivas, Crist\'{o}bal},
       TITLE = {Direct products, overlapping actions, and critical regularity},
     JOURNAL = {J. Mod. Dyn.},
    FJOURNAL = {Journal of Modern Dynamics},
      VOLUME = {17},
        YEAR = {2021},
       PAGES = {285--304},
        ISSN = {1930-5311,1930-532X},
     MRCLASS = {57M60 (20F14 20F36 20F60 37C85 37E05)},
    MRNUMBER = {4288175},
  MRREVIEWER = {Andrzej\ W.\ Bi\'{s}},
         DOI = {10.3934/jmd.2021009},
         URL = {https://doi.org/10.3934/jmd.2021009},
  }

• 26. Sang-hyun Kim, Thomas Koberda, Jaejeong Lee, Ken'ichi Ohshika and Ser Peow Tan; appendix by Xinghua Gao. Shapes of hyperbolic triangles and once-punctured torus groups, Mathematische Zeitschrift (2021).
  [Summary]

  The paper studies the moduli of once-punctured torus groups parameterized by the shape of a hyperbolic ideal triangle. For irrational cone angles, the precise regions in parameter space where the monodromy representation yields a free group are determined. The analysis uses continued fraction expansions and the combinatorics of the Farey graph to detect when discreteness and faithfulness hold.

  [Journal] [Arxiv]

  [cite]

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  @article {MR4329281,
      AUTHOR = {Kim, Sang-hyun and Koberda, Thomas and Lee, Jaejeong and
                Ohshika, Ken'ichi and Tan, Ser Peow and Gao, Xinghua},
       TITLE = {Shapes of hyperbolic triangles and once-punctured torus
                groups},
     JOURNAL = {Math. Z.},
    FJOURNAL = {Mathematische Zeitschrift},
      VOLUME = {299},
        YEAR = {2021},
      NUMBER = {3-4},
       PAGES = {2103--2130},
        ISSN = {0025-5874,1432-1823},
     MRCLASS = {30F35 (11J70 20E05 20H10 30F40)},
    MRNUMBER = {4329281},
  MRREVIEWER = {Michelle\ Chu},
         DOI = {10.1007/s00209-021-02745-3},
         URL = {https://doi.org/10.1007/s00209-021-02745-3},
  }

• 25. Sang-hyun Kim and Thomas Koberda. Integrability of moduli and regularity of Denjoy counterexamples. Discrete and Continuous Dynamical Systems-A, October 2020, 40(10): 6061-6088.
  [Summary]

  For a concave modulus of continuity α whose reciprocal is integrable near zero, there exists a C^{1,α} diffeomorphism of the circle possessing an exceptional minimal set (a Cantor set). This result extends to actions of groups with polynomial growth, giving a near-optimal condition for constructing Denjoy-type counterexamples. A partial converse is also established, delineating the threshold between smooth rigidity and the existence of exceptional minimal sets.

  [Journal] [Arxiv]

  [cite]

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  @article {MR4128339,
      AUTHOR = {Kim, Sang-Hyun and Koberda, Thomas},
       TITLE = {Integrability of moduli and regularity of {D}enjoy
                counterexamples},
     JOURNAL = {Discrete Contin. Dyn. Syst.},
    FJOURNAL = {Discrete and Continuous Dynamical Systems},
      VOLUME = {40},
        YEAR = {2020},
      NUMBER = {10},
       PAGES = {6061--6088},
        ISSN = {1078-0947,1553-5231},
     MRCLASS = {37E10 (37C05 37C15 37C85 37E45)},
    MRNUMBER = {4128339},
  MRREVIEWER = {Puneet\ Sharma},
         DOI = {10.3934/dcds.2020259},
         URL = {https://doi.org/10.3934/dcds.2020259},
  }

• 24. Christian Bonatti, Sang-hyun Kim, Thomas Koberda and Michele Triestino. Small C¹ actions of semidirect products on compact manifolds, Algebraic & Geometric Topology 20 (2020) 3183–3203.
  [Summary]

  For a finitely generated semidirect product G = H ⋋ ⟨t⟩ whose conjugation action on the first cohomology of H is hyperbolic, every C¹ action on a compact manifold sufficiently close to the trivial action factors through an abelian group. This applies, for instance, to fundamental groups of fibered hyperbolic 3-manifolds. The result reveals that local rigidity phenomena for smooth group actions are controlled by the spectral properties of the monodromy.

  [Journal] [Arxiv]

  [cite]

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  @article {MR4185939,
      AUTHOR = {Bonatti, Christian and Kim, Sang-hyun and Koberda, Thomas and
                Triestino, Michele},
       TITLE = {Small {$C^1$} actions of semidirect products on compact
                manifolds},
     JOURNAL = {Algebr. Geom. Topol.},
    FJOURNAL = {Algebraic \& Geometric Topology},
      VOLUME = {20},
        YEAR = {2020},
      NUMBER = {6},
       PAGES = {3183--3203},
        ISSN = {1472-2747,1472-2739},
     MRCLASS = {37C85 (20E22 37D30 57M50 57M60 57R35)},
    MRNUMBER = {4185939},
  MRREVIEWER = {Shigenori\ Matsumoto},
         DOI = {10.2140/agt.2020.20.3183},
         URL = {https://doi.org/10.2140/agt.2020.20.3183},
  }

• 23. Sang-hyun Kim and Thomas Koberda. Diffeomorphism groups of critical regularity. Inventiones mathematicae, 221(2), 421-501 (2020).
  [Summary]

  For each real number α ≥ 1, finitely generated groups are constructed that embed into the group of C^α diffeomorphisms of the circle but admit no embedding at any higher regularity. Complementary groups are also built that embed at every regularity below α but not at α itself. These groups, and their commutator subgroups (which are simple), witness the fact that the regularity spectrum of diffeomorphism groups is fully realized, confirming that critical regularity is a genuine and sharp phenomenon.

  [Journal] [Arxiv]

  [cite]

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  @article {MR4121156,
      AUTHOR = {Kim, Sang-hyun and Koberda, Thomas},
       TITLE = {Diffeomorphism groups of critical regularity},
     JOURNAL = {Invent. Math.},
    FJOURNAL = {Inventiones Mathematicae},
      VOLUME = {221},
        YEAR = {2020},
      NUMBER = {2},
       PAGES = {421--501},
        ISSN = {0020-9910,1432-1297},
     MRCLASS = {57M60 (20F36 37C05 37C85 57S05)},
    MRNUMBER = {4121156},
  MRREVIEWER = {Shigenori\ Matsumoto},
         DOI = {10.1007/s00222-020-00953-y},
         URL = {https://doi.org/10.1007/s00222-020-00953-y},
  }

• 22. Sang-hyun Kim, Thomas Koberda and Mahan Mj, Flexibility of Group Actions on the Circle, Springer Lecture Notes in Mathematics 2231 (2019).
  [Summary]

  The paper characterizes finitely generated groups admitting uncountably many pairwise non-conjugate faithful representations into PSL(2,ℝ). A combination theorem for such groups is established, encompassing free products and HNN extensions amalgamated over maximal cyclic subgroups. This shows that flexibility of circle actions, in the sense of having a large moduli space of representations, is widespread among groups acting on the circle.

  [Journal] [Arxiv]

  [cite]

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  @book {MR3887602,
      AUTHOR = {Kim, Sang-hyun and Koberda, Thomas and Mj, Mahan},
       TITLE = {Flexibility of group actions on the circle},
      SERIES = {Lecture Notes in Mathematics},
      VOLUME = {2231},
   PUBLISHER = {Springer, Cham},
        YEAR = {2019},
       PAGES = {x+134},
        ISBN = {978-3-030-02854-1; 978-3-030-02855-8},
     MRCLASS = {37-02 (20F34 20F65 37C85 37D40 37F30 57M60)},
    MRNUMBER = {3887602},
  MRREVIEWER = {Jo\~{a}o\ Paulo\ Almeida},
         DOI = {10.1007/978-3-030-02855-8},
         URL = {https://doi.org/10.1007/978-3-030-02855-8},
  }

• 21. Sang-hyun Kim, Thomas Koberda and Yash Lodha, Chain groups of homeomorphisms of the interval and the circle, Annales Scientifiques de l'École Normale Supérieure, Series 4, Volume (52), 2019, 797-820.
  [Summary]

  Subgroups of the homeomorphism group of the real line generated by finitely many homeomorphisms, each supported on a single bounded interval, are studied systematically. The group-theoretic structure of these "chain groups" is determined: they are characterized in terms of simplicity, finite generation, and left-orderability. As the main application, uncountably many pairwise non-isomorphic finitely generated simple left-orderable groups are constructed.

  [Journal] [Arxiv]

  [cite]

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  @article {MR4038452,
      AUTHOR = {Kim, Sang-hyun and Koberda, Thomas and Lodha, Yash},
       TITLE = {Chain groups of homeomorphisms of the interval},
     JOURNAL = {Ann. Sci. \'{E}c. Norm. Sup\'{e}r. (4)},
    FJOURNAL = {Annales Scientifiques de l'\'{E}cole Normale Sup\'{e}rieure.
                Quatri\`eme S\'{e}rie},
      VOLUME = {52},
        YEAR = {2019},
      NUMBER = {4},
       PAGES = {797--820},
        ISSN = {0012-9593,1873-2151},
     MRCLASS = {57S05 (20F65 37C85)},
    MRNUMBER = {4038452},
  MRREVIEWER = {Michele\ Triestino},
         DOI = {10.24033/asens.2397},
         URL = {https://doi.org/10.24033/asens.2397},
  }

• 20. Hyungryul Baik, Sang-hyun Kim and Thomas Koberda. Unsmoothable group actions on compact one-manifolds, Journal of European Mathematical Society, Volume 21, Issue 8, 2019, pp. 2333–2353.
  [Summary]

  Mapping class groups of surfaces that are not virtually free admit no faithful C^{1+bv} actions on the circle, even after passing to finite-index subgroups. The proof exploits algebraic constraints arising from Dehn twist relations in the mapping class group. This establishes that the topological actions of mapping class groups on the circle are inherently unsmoothable beyond Lipschitz regularity.

  [Journal] [Arxiv]

  [cite]

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  @article {MR4035847,
      AUTHOR = {Baik, Hyungryul and Kim, Sang-hyun and Koberda, Thomas},
       TITLE = {Unsmoothable group actions on compact one-manifolds},
     JOURNAL = {J. Eur. Math. Soc. (JEMS)},
    FJOURNAL = {Journal of the European Mathematical Society (JEMS)},
      VOLUME = {21},
        YEAR = {2019},
      NUMBER = {8},
       PAGES = {2333--2353},
        ISSN = {1435-9855,1435-9863},
     MRCLASS = {20F36 (20F65 37E10 57M60)},
    MRNUMBER = {4035847},
  MRREVIEWER = {Jens\ Harlander},
         DOI = {10.4171/jems/886},
         URL = {https://doi.org/10.4171/jems/886},
  }

• 19. Sang-hyun Kim and Thomas Koberda. Free products and algebraic structures of diffeomorphism groups, Journal of Topology, Volume 11, Issue 4, December 2018, Pages 1053-1075.
  [Summary]

  If a finitely generated group G is not virtually abelian, then the free product (G × ℤ) ∗ ℤ does not embed into the C^{1+bv} diffeomorphism group of a compact one-manifold. This yields a complete classification of right-angled Artin groups that embed into the diffeomorphism group of the circle at each regularity 0 ≤ r ≤ ω, extending earlier results that covered only the cases r ≤ 1 and r = ω.

  [Journal] [Arxiv]

  [cite]

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  @article {MR3989437,
      AUTHOR = {Kim, Sang-Hyun and Koberda, Thomas},
       TITLE = {Free products and the algebraic structure of diffeomorphism
                groups},
     JOURNAL = {J. Topol.},
    FJOURNAL = {Journal of Topology},
      VOLUME = {11},
        YEAR = {2018},
      NUMBER = {4},
       PAGES = {1054--1076},
        ISSN = {1753-8416,1753-8424},
     MRCLASS = {57M60 (20F36 37C05 37C85 57S05)},
    MRNUMBER = {3989437},
  MRREVIEWER = {Ji-Young\ Ham},
         DOI = {10.1112/topo.12079},
         URL = {https://doi.org/10.1112/topo.12079},
  }

• 18. Sang-hyun Kim. Surface subgroups of word-hyperbolic groups (survey), Handbook of group actions. Vol. III, 89-102.
  [Summary]

  This survey examines Gromov's question of whether every one-ended word-hyperbolic group contains a surface subgroup. Known positive results are collected, including cases arising from doubles of free groups and groups acting on CAT(0) cube complexes. The connections to Cannon's conjecture and the structure of hyperbolic group boundaries are also discussed.

  [cite]

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  @incollection {MR3888617,
      AUTHOR = {Kim, Sang-hyun},
       TITLE = {Surface subgroups of word-hyperbolic groups},
   BOOKTITLE = {Handbook of group actions. {V}ol. {III}},
      SERIES = {Adv. Lect. Math. (ALM)},
      VOLUME = {40},
       PAGES = {87--100},
   PUBLISHER = {Int. Press, Somerville, MA},
        YEAR = {2018},
        ISBN = {978-1-57146-364-7},
     MRCLASS = {20F36 (05C25 20F65)},
    MRNUMBER = {3888617},
  MRREVIEWER = {Jordi\ Delgado},
  }

• 17. Sang-hyun Kim and Thomas Koberda. RAAGs in Diffeos (survey), Advanced Studies in Pure Mathematics: Volume 73, 215-224 (2017).
  [Summary]

  This survey article describes embeddings of right-angled Artin groups (RAAGs) into diffeomorphism groups of manifolds. The interplay between the defining graph of a RAAG and the regularity of the resulting diffeomorphism action is discussed, highlighting how combinatorial graph-theoretic properties translate into dynamical constraints.

  [Journal] [Arxiv]

  [cite]

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  @incollection {MR3728499,
      AUTHOR = {Kim, Sang-hyun and Koberda, Thomas},
       TITLE = {R{AAG}s in diffeos},
   BOOKTITLE = {Hyperbolic geometry and geometric group theory},
      SERIES = {Adv. Stud. Pure Math.},
      VOLUME = {73},
       PAGES = {215--224},
   PUBLISHER = {Math. Soc. Japan, Tokyo},
        YEAR = {2017},
        ISBN = {978-4-86497-042-6},
     MRCLASS = {20F36 (57M60)},
    MRNUMBER = {3728499},
  MRREVIEWER = {Mustafa\ Korkmaz},
         DOI = {10.2969/aspm/07310215},
         URL = {https://doi.org/10.2969/aspm/07310215},
  }

• 16. Cheol-Hyun Cho, Hansol Hong, Sang-hyun Kim and Siu-Cheong Lau. Lagrangian Floer potential of orbifold spheres, Advances in Mathematics, Volume 306, 14 January 2017, Pages 344-426.
  [Summary]

  Lagrangian Floer potentials are computed for Seidel Lagrangians on hyperbolic 2-orbifold spheres—orbifold projective lines with three orbifold points and negative orbifold Euler characteristic. The computation involves counting pseudo-holomorphic polygon contributions within the framework of A-infinity categories. This provides explicit formulas connecting Lagrangian Floer theory to the geometry of orbifold surfaces.

  [Journal] [Arxiv]

  [cite]

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  @article {MR3581306,
      AUTHOR = {Cho, Cheol-Hyun and Hong, Hansol and Kim, Sang-hyun and Lau,
                Siu-Cheong},
       TITLE = {Lagrangian {F}loer potential of orbifold spheres},
     JOURNAL = {Adv. Math.},
    FJOURNAL = {Advances in Mathematics},
      VOLUME = {306},
        YEAR = {2017},
       PAGES = {344--426},
        ISSN = {0001-8708,1090-2082},
     MRCLASS = {14N35 (14J33)},
    MRNUMBER = {3581306},
  MRREVIEWER = {Guangbo\ Xu},
         DOI = {10.1016/j.aim.2016.10.017},
         URL = {https://doi.org/10.1016/j.aim.2016.10.017},
  }

• 15. Hyungryul Baik, Sang-hyun Kim and Thomas Koberda. Right-angled Artin groups in the C^∞ diffeomorphism group of the real line, Israel Journal of Mathematics, June 2016, Volume 213, Issue 1, pp 175-182.
  [Summary]

  Every right-angled Artin group (RAAG) is shown to embed into the group of C∞ diffeomorphisms of the real line. The proof constructs explicit smooth bump functions that realize the required group action. This settles the embeddability question for RAAGs into smooth diffeomorphism groups in the non-compact setting.

  [Journal] [Arxiv]

  [cite]

  Copy

  @article {MR3509472,
       AUTHOR = {Baik, Hyungryul and Kim, Sang-hyun and Koberda, Thomas},
        TITLE = {Right-angled {A}rtin groups in the {$C^\infty$} diffeomorphism
                 group of the real line},
      JOURNAL = {Israel J. Math.},
     FJOURNAL = {Israel Journal of Mathematics},
       VOLUME = {213},
         YEAR = {2016},
       NUMBER = {1},
        PAGES = {175--182},
         ISSN = {0021-2172,1565-8511},
      MRCLASS = {20F36 (03D35 20E26 58D05)},
     MRNUMBER = {3509472},
   MRREVIEWER = {Valeriy\ G.\ Bardakov},
          DOI = {10.1007/s11856-016-1307-8},
          URL = {https://doi.org/10.1007/s11856-016-1307-8},
  }

• 14. Sang-hyun Kim and Thomas Koberda. Right-angled Artin groups and finite subgraphs of curve graphs, Osaka Journal of Mathematics, Volume 53, Number 3 (2016), 705-716.
  [Summary]

  For surfaces S with complexity ξ(S) < 3, a RAAG A(Γ) embeds into the mapping class group Mod(S) only if Γ appears as an induced subgraph of the curve graph C(S). For surfaces of higher complexity, explicit counterexamples show that this criterion fails. The result clarifies the extent to which the combinatorics of the curve graph governs RAAG embeddings into mapping class groups.

  [Journal] [Arxiv]

  [cite]

  Copy

  @article {MR3533464,
       AUTHOR = {Kim, Sang-Hyun and Koberda, Thomas},
        TITLE = {Right-angled {A}rtin groups and finite subgraphs of curve
                 graphs},
      JOURNAL = {Osaka J. Math.},
     FJOURNAL = {Osaka Journal of Mathematics},
       VOLUME = {53},
         YEAR = {2016},
       NUMBER = {3},
        PAGES = {705--716},
         ISSN = {0030-6126},
      MRCLASS = {20F36 (20F34 20F65 57M15)},
     MRNUMBER = {3533464},
   MRREVIEWER = {Daniel\ Matei},
          URL = {http://projecteuclid.org/euclid.ojm/1470413985},
  }

• 13. Sang-hyun Kim and Thomas Koberda. Anti-trees and right-angled Artin subgroups of braid groups, Geometry & Topology 19-6 (2015), 3289–3306.
  [Summary]

  Every RAAG embeds into a RAAG whose defining graph is an anti-tree. Since anti-tree RAAGs are known to embed into braid groups, it follows that every RAAG embeds into some braid group. As a further consequence, every RAAG embeds quasi-isometrically into the area-preserving diffeomorphism group of the 2-sphere with respect to the L^p metric for p > 2.

  [Journal] [Arxiv]

  [cite]

  Copy

  @article {MR3447104,
      AUTHOR = {Kim, Sang-hyun and Koberda, Thomas},
       TITLE = {Anti-trees and right-angled {A}rtin subgroups of braid groups},
     JOURNAL = {Geom. Topol.},
    FJOURNAL = {Geometry \& Topology},
      VOLUME = {19},
        YEAR = {2015},
      NUMBER = {6},
       PAGES = {3289--3306},
        ISSN = {1465-3060,1364-0380},
     MRCLASS = {20F36 (03D35 20F10 20F67 57M07)},
    MRNUMBER = {3447104},
  MRREVIEWER = {Volker\ Gebhardt},
         DOI = {10.2140/gt.2015.19.3289},
         URL = {https://doi.org/10.2140/gt.2015.19.3289},
  }

• 12. Sang-hyun Kim and Genevieve Walsh. Coxeter groups, hyperbolic cubes, and acute triangulations, Journal of Topology (2016) 9 (1): 117-142.
  [Summary]

  A combinatorial triangulation of the 2-sphere can be realized as an acute geodesic triangulation if and only if it contains no separating 3- or 4-cycle. The proof leverages the theory of hyperbolic Coxeter groups and the geometry of right-angled polytopes in hyperbolic space. This gives a complete combinatorial characterization of acute triangulability for the sphere.

  [Journal] [Arxiv]

  [cite]

  Copy

  @article {MR3465843,
      AUTHOR = {Kim, S.-h. and Walsh, G. S.},
       TITLE = {Coxeter groups, hyperbolic cubes and acute triangulations},
     JOURNAL = {J. Topol.},
    FJOURNAL = {Journal of Topology},
      VOLUME = {9},
        YEAR = {2016},
      NUMBER = {1},
       PAGES = {117--142},
        ISSN = {1753-8416,1753-8424},
     MRCLASS = {57M07 (20F65)},
    MRNUMBER = {3465843},
  MRREVIEWER = {Bruno\ P.\ Zimmermann},
         DOI = {10.1112/jtopol/jtv038},
         URL = {https://doi.org/10.1112/jtopol/jtv038},
  }

• 11. Sang-hyun Kim and Thomas Koberda. The geometry of the curve complex of a right-angled Artin group, International Journal of Algebra and Computation (2014) 24 (2) 121-169.
  [Summary]

  A theory of right-angled Artin group actions on their extension graphs is developed, paralleling the classical theory of mapping class group actions on curve graphs. In particular, these actions are shown to be acylindrical, and explicit acylindricity constants are computed. The framework establishes extension graphs as a natural geometric tool for studying the internal structure and quasi-isometric rigidity of RAAGs.

  [Journal] [Arxiv]

  [cite]

  Copy

  @article {MR3192368,
      AUTHOR = {Kim, Sang-Hyun and Koberda, Thomas},
       TITLE = {The geometry of the curve graph of a right-angled {A}rtin
                group},
     JOURNAL = {Internat. J. Algebra Comput.},
    FJOURNAL = {International Journal of Algebra and Computation},
      VOLUME = {24},
        YEAR = {2014},
      NUMBER = {2},
       PAGES = {121--169},
        ISSN = {0218-1967,1793-6500},
     MRCLASS = {20F36 (05E45)},
    MRNUMBER = {3192368},
  MRREVIEWER = {Thomas\ Haettel},
         DOI = {10.1142/S021819671450009X},
         URL = {https://doi.org/10.1142/S021819671450009X},
  }

• 10. Sang-hyun Kim and Sang-il Oum. Hyperbolic Surface subgroups of one-ended doubles of free groups, Journal of Topology (2014) 7 (4): 927–947.
  [Summary]

  The double of a rank-two free group along a one-ended amalgamation is shown to either split as a nontrivial free product or contain a closed hyperbolic surface subgroup. This provides evidence for Gromov's conjecture that every one-ended word-hyperbolic group contains a surface subgroup. The proof uses a combination of combinatorial group theory and the geometry of van Kampen diagrams.

  [Journal] [Arxiv]

  [cite]

  Copy

  @article {MR3286893,
      AUTHOR = {Kim, Sang-hyun and Oum, Sang-il},
       TITLE = {Hyperbolic surface subgroups of one-ended doubles of free
                groups},
     JOURNAL = {J. Topol.},
    FJOURNAL = {Journal of Topology},
      VOLUME = {7},
        YEAR = {2014},
      NUMBER = {4},
       PAGES = {927--947},
        ISSN = {1753-8416,1753-8424},
     MRCLASS = {20F65 (05C70 57M20)},
    MRNUMBER = {3286893},
  MRREVIEWER = {David\ Hume},
         DOI = {10.1112/jtopol/jtu004},
         URL = {https://doi.org/10.1112/jtopol/jtu004},
  }

• 9. Sang-hyun Kim and Thomas Koberda. An obstruction to embedding right-angled Artin groups in mapping class groups, International Mathematics Research Notices (2014) #2014 (14): 3912–3918.
  [Summary]

  The chromatic number of the defining graph of a RAAG provides an obstruction to embedding the RAAG into a mapping class group. More precisely, if A(Γ) embeds into the mapping class group of a surface S, then the chromatic number of Γ is bounded by a function of the complexity of S. This gives a simple graph-theoretic necessary condition for RAAG embeddings into mapping class groups.

  [Journal] [Arxiv]

  [cite]

  Copy

  @article {MR3239092,
      AUTHOR = {Kim, Sang-Hyun and Koberda, Thomas},
       TITLE = {An obstruction to embedding right-angled {A}rtin groups in
                mapping class groups},
     JOURNAL = {Int. Math. Res. Not. IMRN},
    FJOURNAL = {International Mathematics Research Notices. IMRN},
        YEAR = {2014},
      NUMBER = {14},
       PAGES = {3912--3918},
        ISSN = {1073-7928,1687-0247},
     MRCLASS = {20F65},
    MRNUMBER = {3239092},
  MRREVIEWER = {Kevin\ D.\ Schreve},
         DOI = {10.1093/imrn/rnt064},
         URL = {https://doi.org/10.1093/imrn/rnt064},
  }

• 8. Sang-hyun Kim and Thomas Koberda. Embeddability between right-angled Artin groups, Geometry & Topology 17 (2013) 493–530.
  [Summary]

  The notion of an "extension graph" associated to a RAAG is introduced and shown to provide a systematic tool for studying embeddings between RAAGs. Specifically, A(Γ) embeds into A(Λ) whenever Γ appears as an induced subgraph of the extension graph of A(Λ). This unifies and extends previous ad hoc constructions, yielding many new embedding results and establishing a conceptual framework for the embeddability problem.

  [Journal] [Arxiv]

  [cite]

  Copy

  @article {MR3039768,
      AUTHOR = {Kim, Sang-hyun and Koberda, Thomas},
       TITLE = {Embedability between right-angled {A}rtin groups},
     JOURNAL = {Geom. Topol.},
    FJOURNAL = {Geometry \& Topology},
      VOLUME = {17},
        YEAR = {2013},
      NUMBER = {1},
       PAGES = {493--530},
        ISSN = {1465-3060,1364-0380},
     MRCLASS = {20F36 (05C25 20E05 20F05)},
    MRNUMBER = {3039768},
  MRREVIEWER = {Yago\ Antol\'{\i}n},
         DOI = {10.2140/gt.2013.17.493},
         URL = {https://doi.org/10.2140/gt.2013.17.493},
  }

• 7. Sang-hyun Kim and Henry Wilton. Polygonal words in free groups, Quarterly Journal of Mathematics (2012) 63(2), 399–421.
  [Summary]

  The notion of "polygonal" words in free groups is introduced, capturing words whose associated van Kampen diagrams over any double of the free group admit a CAT(−1) metric. These words are shown to be plentiful and are used to construct surface subgroups in doubles of free groups. This provides strong evidence for Gromov's question on the ubiquity of surface subgroups in word-hyperbolic groups.

  [Journal] [Arxiv]

  [cite]

  Copy

  @article {MR2925298,
      AUTHOR = {Kim, Sang-Hyun and Wilton, Henry},
       TITLE = {Polygonal words in free groups},
     JOURNAL = {Q. J. Math.},
    FJOURNAL = {The Quarterly Journal of Mathematics},
      VOLUME = {63},
        YEAR = {2012},
      NUMBER = {2},
       PAGES = {399--421},
        ISSN = {0033-5606,1464-3847},
     MRCLASS = {20F10 (03D40 20E05 20F67 68R15)},
    MRNUMBER = {2925298},
  MRREVIEWER = {Serge\ Perrine},
         DOI = {10.1093/qmath/haq045},
         URL = {https://doi.org/10.1093/qmath/haq045},
  }

• 6. Sang-hyun Kim. Surface subgroups of graph products of groups, International Journal of Algebra and Computation (2012) 22 (8).
  [Summary]

  For a graph product of groups, the kernel of the natural map to the direct product of vertex groups is shown to embed into a RAAG. When each vertex group is finite or cyclic, the graph product is virtually special in the sense of Haglund and Wise. As a consequence, for graphs on at most seven vertices, right-angled Coxeter groups contain hyperbolic surface subgroups if and only if the underlying graph is weakly chordal.

  [Journal] [Arxiv]

  [cite]

  Copy

  @article {MR3010817,
      AUTHOR = {Kim, Sang-Hyun},
       TITLE = {Surface subgroups of graph products of groups},
     JOURNAL = {Internat. J. Algebra Comput.},
    FJOURNAL = {International Journal of Algebra and Computation},
      VOLUME = {22},
        YEAR = {2012},
      NUMBER = {8},
       PAGES = {1240003, 20},
        ISSN = {0218-1967,1793-6500},
     MRCLASS = {20F65 (20F36)},
    MRNUMBER = {3010817},
         DOI = {10.1142/S0218196712400036},
         URL = {https://doi.org/10.1142/S0218196712400036},
  }

• 5. Sang-hyun Kim. Geometricity and polygonality in free groups, International Journal of Algebra and Computation 21(1–2) (2011) 235–256.
  [Summary]

  "Geometric" words in free groups, as defined by Gordon and Wilton, are shown to be "polygonal" in the sense of the companion work on polygonal words. This connects two independent approaches to Gromov's question on surface subgroups: geometric words arising from immersed surfaces and polygonal words arising from CAT(−1) van Kampen diagrams.

  [Journal] [Arxiv]

  [cite]

  Copy

  @article {MR2787460,
      AUTHOR = {Kim, Sang-Hyun},
       TITLE = {Geometricity and polygonality in free groups},
     JOURNAL = {Internat. J. Algebra Comput.},
    FJOURNAL = {International Journal of Algebra and Computation},
      VOLUME = {21},
        YEAR = {2011},
      NUMBER = {1-2},
       PAGES = {235--256},
        ISSN = {0218-1967,1793-6500},
     MRCLASS = {20F65 (20E06 20F67)},
    MRNUMBER = {2787460},
  MRREVIEWER = {Jos\'{e}\ Burillo},
         DOI = {10.1142/S0218196711006157},
         URL = {https://doi.org/10.1142/S0218196711006157},
  }

• 4. Sang-hyun Kim. On right-angled Artin groups without surface subgroups, Groups, Geometry, and Dynamics 4(2) (2010) 275–307.
  [Summary]

  A combination theorem is proved for the class of right-angled Artin groups that do not contain closed hyperbolic surface subgroups: certain amalgamations and HNN extensions preserve this property. The proof relies on the theory of special cube complexes. This provides a structural characterization of surface-subgroup-free RAAGs and contributes to the understanding of which RAAGs arise as fundamental groups of non-positively curved spaces.

  [Journal] [Arxiv]

  [cite]

  Copy

  @article {MR2595093,
      AUTHOR = {Kim, Sang-hyun},
       TITLE = {On right-angled {A}rtin groups without surface subgroups},
     JOURNAL = {Groups Geom. Dyn.},
    FJOURNAL = {Groups, Geometry, and Dynamics},
      VOLUME = {4},
        YEAR = {2010},
      NUMBER = {2},
       PAGES = {275--307},
        ISSN = {1661-7207,1661-7215},
     MRCLASS = {20F36 (05C25 20F65 57M07)},
    MRNUMBER = {2595093},
  MRREVIEWER = {Matthew\ B.\ Day},
         DOI = {10.4171/GGD/84},
         URL = {https://doi.org/10.4171/GGD/84},
  }

• 3. Chan-Byoung Chae, Sang-hyun Kim and Robert W. Heath Jr., Linear network coordinated beamforming for cell-boundary users, Proc. of IEEE Workshop on Signal Processing Advances in Wireless Communications (SPAWC) (2009).
  [Summary]

  A linear coordinated beamforming strategy is proposed for multi-cell wireless networks to mitigate inter-cell interference affecting cell-boundary users. Adjacent base stations cooperatively optimize their transmitted beams to steer nulls toward vulnerable cell-edge users in neighboring cells. Simulations confirm that this coordination significantly improves signal quality and overall downlink throughput compared to uncoordinated methods.

  [Journal]

  [cite]

  Copy

  @inproceedings{spawc2009chae,
    author={Chae, Chan-Byoung and Kim, Sang-hyun and Heath, Robert W.},
    booktitle={2009 IEEE 10th Workshop on Signal Processing Advances in Wireless Communications (SPAWC)}, 
    title={Linear network coordinated beamforming for cell-boundary users}, 
    year={2009},
    pages={534-538},
    doi={10.1109/SPAWC.2009.5161842}
  }

• 2. Chan-Byoung Chae, Sang-hyun Kim and Robert W. Heath Jr., Network coordinated beamforming for cell-boundary users: linear and non-linear approaches, IEEE Journal of Selected Topics in Signal Processing (J-STSP), vol. 3, no. 6 (2009) 1094–1105.
  [Summary]

  Both linear and non-linear network coordinated beamforming algorithms are developed for multi-cell downlink environments with multi-antenna users at cell boundaries. The algorithms transmit independent data streams while suppressing mutual interference through cooperative coordination among base stations. The non-linear approach further optimizes spatial resources by decoupling the interacting channels, yielding substantial throughput and reliability gains over uncoordinated methods.

  [Journal]

  [cite]

  Copy

  @article{jstsp2009chae,
    author={Chae, Chan-Byoung and Kim, Sang-hyun and Heath, Robert W.},
    journal={IEEE Journal of Selected Topics in Signal Processing}, 
    title={Network coordinated beamforming for cell-boundary users: linear and non-linear approaches}, 
    year={2009},
    volume={3},
    number={6},
    pages={1094-1105},
    doi={10.1109/JSTSP.2009.2035857}
  }

• 1. Sang-hyun Kim. Co-contractions of graphs and right-angled Artin groups, Algebraic and Geometric Topology 8 (2008) 849–868.
  [Summary]

  A natural map between right-angled Artin groups induced by a "co-contraction" of the underlying defining graphs is shown to be injective. The family of "contraction words" in free groups that yield such graph co-contractions is explicitly characterized. This provides a systematic method for constructing embeddings between RAAGs from combinatorial operations on graphs.

  [Journal] [Arxiv]

  [cite]

  Copy

  @article {MR2443098,
      AUTHOR = {Kim, Sang-hyun},
       TITLE = {Co-contractions of graphs and right-angled {A}rtin groups},
     JOURNAL = {Algebr. Geom. Topol.},
    FJOURNAL = {Algebraic \& Geometric Topology},
      VOLUME = {8},
        YEAR = {2008},
      NUMBER = {2},
       PAGES = {849--868},
        ISSN = {1472-2747,1472-2739},
     MRCLASS = {20F65 (20F36)},
    MRNUMBER = {2443098},
         DOI = {10.2140/agt.2008.8.849},
         URL = {https://doi.org/10.2140/agt.2008.8.849},
  }

Submitted or Accepted Papers

• 9. Leonardo Dinamarca, Maximiliano Escayola, Sang-hyun Kim, Thomas Koberda. Hyperbolicity and obstructions to PL actions of the circle (2026).
  [Summary]

  We investigate acylindrically hyperbolic groups acting on the circle by piecewise linear homeomorphisms. We prove that a finitely generated acylindrically hyperbolic group acting faithfully by piecewise linear homeomorphisms of the circle is virtually free, is virtually a closed hyperbolic surface group, or virtually splits over a two-ended subgroup; as a consequence, we prove a general structural result for Gromov hyperbolic groups of piecewise linear homeomorphisms. Along the way, we show that a right-angled Artin subgroup of piecewise linear homeomorphisms of the circle is either free or abelian, generalizing a result of Bleak and Salazar-Diaz. We also show that the fundamental group of a finite volume hyperbolic n-manifold cannot act faithfully on the circle by piecewise linear homeomorphisms, provided that n≥3, complementing 2-dimensional examples of Ghys and Minakawa.

  [Arxiv]

  [cite]

  Copy

  @misc{arxiv:2608.12168,
        title={Hyperbolicity and obstructions to PL actions of the circle}, 
        author={Leonardo Dinamarca and Maximiliano Escayola and Sang-hyun Kim and Thomas Koberda},
        year={2026},
        eprint={2608.12168},
        archivePrefix={arXiv},
        primaryClass={math.GR},
        url={https://arxiv.org/abs/2608.12168}
  }

• 8. Koji Fujiwara, Sang-hyun Kim, Ryokichi Tanaka. Small growth rates of free groups (2026).
  [Summary]

  We introduce the notion of a Magnus marking for a finite generating set of a group and prove a certain expansion property. Using this property, we determine the second and third smallest growth rates of the rank-d free group for d≥2. We also give a new lower bound for the smallest growth rate of the genus-g surface group for g≥2, as well as a lower bound for the growth rate associated with a one-relator presentation.

  [Arxiv]
• 7. TBA on Samurai (2026).
  [Summary]

  Details of this paper will be posted as soon as the preprint becomes available.
• 6. Aletheia tackles FirstProof autonomously (2026).
  [Summary]

  This report describes the performance of Aletheia, an autonomous mathematics research agent powered by Gemini, on the FirstProof challenge—a set of 10 research-level open problems in mathematics. Aletheia solved 6 of the 10 problems autonomously within the challenge timeframe, with solutions verified by expert mathematicians. The agent architecture, prompts, and detailed inference traces are open-sourced to enable community verification and further development.

  [Arxiv]

  [cite]

  Copy

  @misc{arxiv:2602.21201,
        title={Aletheia tackles FirstProof autonomously}, 
        author={Tony Feng and Junehyuk Jung and Sang-hyun Kim and Carlo Pagano and Sergei Gukov and Chiang-Chiang Tsai and David Woodruff and Adel Javanmard and Aryan Mokhtari and Dawsen Hwang and Yuri Chervonyi and Jonathan N. Lee and Garrett Bingham and Trieu H. Trinh and Vahab Mirrokni and Quoc V. Le and Thang Luong},
        year={2026},
        eprint={2602.21201},
        archivePrefix={arXiv},
        primaryClass={math.GR},
        url={https://arxiv.org/abs/2602.21201}
  }

• 5. Towards Autonomous Mathematics Research (2026).
  [Summary]

  This paper introduces Aletheia, a research agent designed for autonomous mathematical exploration and proof generation. Aletheia integrates large language model capabilities with symbolic verification and literature search to conduct multi-step mathematical reasoning over extended horizons. The system is evaluated on open problems in combinatorics, number theory, and algebra, demonstrating progress toward AI-assisted autonomous mathematical discovery.

  [Arxiv]

  [cite]

  Copy

  @misc{arxiv:2602.10177,
        title={Towards Autonomous Mathematics Research}, 
        author={Tony Feng and Trieu H. Trinh and Garrett Bingham and Dawsen Hwang and Yuri Chervonyi and Junehyuk Jung and Joonkyung Lee and Carlo Pagano and Sang-hyun Kim and Federico Pasqualotto and Sergei Gukov and Jonathan N. Lee and Junsu Kim and Kaiying Hou and Golnaz Ghiasi and Yi Tay and YaGuang Li and Chenkai Kuang and Yuan Liu and Hanzhao Lin and Evan Zheran Liu and Nigamaa Nayakanti and Xiaomeng Yang and Heng-Tze Cheng and Demis Hassabis and Koray Kavukcuoglu and Quoc V. Le and Thang Luong},
        year={2026},
        eprint={2602.10177},
        archivePrefix={arXiv},
        primaryClass={math.GR},
        url={https://arxiv.org/abs/2602.10177}
  }

• 4. Semi-Autonomous Mathematics Discovery with Gemini: A Case Study on the Erdős Problems (2026).
  [Summary]

  A Gemini-based AI agent is used to systematically evaluate 700 open conjectures from Bloom's Erdős Problems database. Using AI-driven verification to narrow the search space followed by human expert evaluation, 13 problems are addressed: 5 through novel autonomous solutions and 8 through identification of existing solutions in the literature. The study demonstrates a viable methodology for semi-autonomous mathematical discovery at scale.

  [Arxiv]

  [cite]

  Copy

  @misc{arxiv:2601.22401,
        title={Semi-Autonomous Mathematics Discovery with Gemini: A Case Study on the Erdős Problems}, 
        author={Tony Feng and Trieu Trinh and Garrett Bingham and Jiwon Kang and Shengtong Zhang and Sang-hyun Kim and Kevin Barreto and Carl Schildkraut and Junehyuk Jung and Jaehyeon Seo and Carlo Pagano and Yuri Chervonyi and Dawsen Hwang and Kaiying Hou and Sergei Gukov and Cheng-Chiang Tsai and Hyunwoo Choi and Youngbeom Jin and Wei-Yuan Li and Hao-An Wu and Ruey-An Shiu and Yu-Sheng Shih and Quoc V. Le and Thang Luong},
        year={2026},
        eprint={2601.22401},
        archivePrefix={arXiv},
        primaryClass={math.GR},
        url={https://arxiv.org/abs/2601.22401}
  }

• 3. Kevin Barreto, Jiwon Kang, Sang-hyun Kim, Vjekoslav Kovač, Shengtong Zhang. Irrationality of rapidly converging series: a problem of Erdős and Graham, accepted in B. Lond. Math. Soc. (2026).
  [Summary]

  An open problem of Erdős and Graham on the irrationality of rapidly converging series is resolved. It is shown that if a sequence of positive integers grows at least as fast as the golden ratio raised to a double exponential, then the sum ∑ 1/(a_n a_{n+1}) is irrational. The result is further generalized to sums of the form ∑ 1/(a_n a_{n+k}), extending the scope of classical irrationality criteria for series defined by fast-growing integer sequences.

  [Arxiv]

  [cite]

  Copy

  @misc{arxiv:2601.21442,
        title={Irrationality of rapidly converging series: a problem of Erdős and Graham}, 
        author={Kevin Barreto and Jiwon Kang and Sang-hyun Kim and Vjekoslav Kovač and Shengtong Zhang},
        year={2026},
        eprint={2601.21442},
        archivePrefix={arXiv},
        primaryClass={math.GR},
        url={https://arxiv.org/abs/2601.21442}
  }

• 2. Elementary equivalence and diffeomorphism groups of smooth manifolds (2025).
  [Summary]

  If the diffeomorphism groups Diff^r(M) and Diff^s(N) of two smooth manifolds are elementarily equivalent as groups in first-order logic, then M and N are diffeomorphic and r = s. This extends classical reconstruction theorems of Takens and Filipkiewicz from group isomorphisms to elementary equivalence, a strictly weaker condition. The result demonstrates that first-order logic alone is sufficient to recover both the smooth topology of a manifold and the regularity of its diffeomorphism group.

  [Arxiv]

  [cite]

  Copy

  @misc{arxiv:2507.07427,
        title={Elementary equivalence and diffeomorphism groups of smooth manifolds}, 
        author={Sang-hyun Kim and Thomas Koberda and J. de la Nuez González},
        year={2026},
        eprint={2507.07427},
        archivePrefix={arXiv},
        primaryClass={math.GR},
        url={https://arxiv.org/abs/2507.07427}
  }

• 1. Jaewon Chang, Sang-hyun Kim and Thomas Koberda. Algebraic Structure of Diffeomorphism Groups of One-Manifolds, accepted, Panoramas et synthèses - Société mathématique de France.
  [Summary]

  This expository article gives a detailed account of Mather's theorem on the simplicity of diffeomorphism groups, specialized to one-manifolds. By restricting to dimension one, the exposition avoids higher-dimensional technicalities and isolates the core geometric and dynamical mechanisms underlying Mather's proof. The smoothing techniques and fragmentation lemmas that form the backbone of the argument are carefully developed, providing an accessible entry point for researchers studying the algebraic structure of transformation groups.

  [Journal] [Arxiv]

  [cite]

  Copy

  @misc{arxiv:1904.08793,
        title={Algebraic Structure of Diffeomorphism Groups of One-Manifolds}, 
        author={Jaewon Chang and Sang-hyun Kim and Thomas Koberda},
        year={2026},
        eprint={1904.08793},
        archivePrefix={arXiv},
        primaryClass={math.GR},
        url={https://arxiv.org/abs/1904.08793}
  }

Unpublished

• 3. A simplicial-norm lower bound for some one-relator group (2026).
  [Summary]

  For the word $w = AABabacbCCBc \in F(\{a, b, c\})$, we establish the bounds $\frac{222}{17} \le V(w^3) \le \frac{670}{51}$. Since $\mathrm{scl}(w) = 5/4$, this yields $\|\alpha_3\|_1 \ge \frac{222}{17} > 13 = 4\,\mathrm{scl}(w^3) - 2$, answering a question of Heuer–Löh negatively. The lower bound comes from an exact parity certificate for every reduced ribbon surface, including signed boundary degrees.

  [cite]

  Copy

  @misc{kim2026simplicial,
        title={A simplicial-norm lower bound for some one-relator group}, 
        author={Sang-hyun Kim},
        year={2026},
        note={Preprint}
  }

• 2. The Arens space $S_2$ is a paracomplex: A note on Nagata's Problem 3.
  [Summary]

  In 1974, J. Nagata posed the problem of finding a paracomplex that is not the closed continuous image of any metric space. We prove that Arens space $S_2$ is such an example.

  [Journal]
• 1. Sang-hyun Kim, Thomas Koberda and Juyoung Lee. Finite subgraphs of extension graphs.
  [Summary]

  A strengthening and also a very detailed proof of Lemma 3.1 that originally appeared in Embedability between right-angled Artin groups , Geometry & Topology 17 (2013) 493--530.

  [Arxiv]

  [cite]

  Copy

  @misc{arxiv:1708.02088,
        title={Finite subgraphs of an extension graph}, 
        author={Sang-hyun Kim and Thomas Koberda and Juyoung Lee},
        year={2026},
        eprint={1708.02088},
        archivePrefix={arXiv},
        primaryClass={math.GR},
        url={https://arxiv.org/abs/1708.02088}
  }

Computes

• Computes — Computations in geometric group theory, hyperbolic geometry, and the mapping class group.
  [Link]

Slides and videos

• A problem set for my two-hour introductory lecture to hyperbolic geometry
• H. Short's proof of Howson's property for free groups (2012 KAIST graduate class)
• Free products of finite groups are virtually free (2012 KAIST graduate class)
• Nielsen generating set of Aut(Fn) (2012 KAIST graduate class)
• Finitely generated residually finite groups are Hopfian (2012 KAIST graduate class)

Talks & Panel