Research

Published papers

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

• 34. 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). [arXiv]
  ▶ Abstract

  We prove that every topological action of a countable group on a metrizable space can be realized as a bi-Lipschitz action with respect to some compatible metric. We also establish an analogous result for closed subgroups of locally compact groups. Our results are based on an earlier result of U. Hamenst\"{a}dt regarding finitely, and recover a theorem by Deroin, Kleptsyn and Navas regarding one-manifolds.

• 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. [arXiv] [Journal]
  ▶ Abstract

  We prove that if a countable group does not contain a finitely generated subgroup of exponential growth, then every topological action of the group on a compact connected one-manifold can be blown-up to a C 1 action. The proof is based on a functional characterisation of such groups.

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

  For a rational number q = s / r in (-4, 4), consider two 2 x 2 matrices a = ( (1, 0), (1, 1) ) and b = ( (1, q), (0, 1) ). We prove that the group < a , b > is non-free if s ≤ 27 and s ≠ 24. If s = 24, we prove the same statements for almost all r in the sense of natural density. We give an estimate for such a density when s > 27. Some of the proofs are computer-assisted, the output of which is given below.

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

  We give an illustration of Bowditch's canonical splitting of one-ended hyperbolic groups, as well as a survey on the planarity conjecture regarding the Gromov boundary of a hyperbolic group.

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

  We prove that if G and H are nonsolvable groups then (G x H) * Z does not embed into the group of the orientation preserving C^{1+t} diffeomorphisms of the circle for any t > 0. It follows that no finite index subgroup of the mapping class group of a closed hyperbolic surface can faithfully act on the circle by C^{1+t} diffeomorphisms, for any t > 0.

• 28. Sang-hyun Kim and Thomas Koberda. Structure and regularity of group actions on one-manifolds, 285 pages, Springer Monographs in Mathematics (November 21, 2021). [pdf]
• 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. [pdf]
  ▶ Abstract

  We prove that if G and H are nonsolvable groups then (G x H) * Z does not embed into the group of the orientation preserving C^{1+t} diffeomorphisms of the interval for any t > 0.

• 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) [pdf]
  ▶ Abstract

  We study the question of how often the monodromy image of a once-punctured torus with a fixed irrational cone angle becomes a free group.

• 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. [arXiv]
  ▶ Abstract

  We prove that if α(x)​ ​ is a concave modulus such that 1/ α(x)​ is integrable near zero, then there exists a C 1,α ​ ​ diffeomorphism of the circle with an exceptional minimal set. We also generalize this to groups with polynomial growth, and deduce a partial converse.

• 24. Chritian 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. [pdf]
  ▶ Abstract

  For a compact Riemannian manifold M and for a finitely generated group G = H ⧕ < t > with a hyperbolic linear action of t on H^1(H, ℤ), every C¹--action of G on M that is sufficiently close to the trivial action has an abelian image. As an example, we deduce the same conclusion when G is a fibered 3-manifold group.

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

  We prove that for each real numbers a ≥ 1, there exists a finitely generated subgroup G a of Diff a (S 1 ) with the property that G a admits no injective homomorphisms into Diff b (S 1 ) for all b > a. We also prove that there exists another fg group H a that embeds into Diff b (S 1 ) for all b < a, but not into Diff a (S 1 ). One can further require the same properties are inherited to all finite index subgroups and to the commutator groups of G a and of H a . The commutator groups will be simple.

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

  We study which finitely generated groups admit uncountably many pairwise-non-conjugate embeddings into PSL(2,R). In particular, we obtain a combination theorem of such a class of groups, encompassing free product and HNN extensions, both amalgamated over maximal abelian subgroups.

• 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. [pdf]
  ▶ Abstract

  We study subgroups of Homeo(R) generated by finitely many homeomorphisms each of which is supported on a single interval. As a consequence, we construct uncountably many non-pairwise isomorphic countable simple orderable groups.

• 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. [arXiv]
  ▶ Abstract

  Let G be the mapping class group of a surface (possibly with punctures or boundary), such that G is not virtually free. We prove that G never admits, even virtually, an embedding into the C 1+bv diffeomorphism group of the circle.

• 19. Sang-hyun Kim and Thomas Koberda. Free products and algebraic structures of diffeomorphism groups, Journal of Topology, Volume11, Issue4, December 2018, Pages 1053-1075. [arXiv]
  ▶ Abstract

  We prove that if a finitely generated group G is not virtually abelian, then (G x Z) * Z never admits an embedding into the C 1+bv diffeomorphism group of a compact one-manifold. As a consequence, we have a complete classification of RAAGs that embed into Diff r (S 1 ) for each 0 ≤ r ≤ ω. (The cases r ≤ 1 and r = ω were previously known)

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

  We survey known results on Gromov's question regarding surface subgroups of word-hyperbolic groups.

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

  We give an exposition on embeddability results related to RAAGs (right-angled Artin groups) and various automorphism groups of manifolds.

• 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. [pdf]
  ▶ Abstract

  We compute Lagrangian Floer potentials for Seidel Lagrangians on hyperbolic 2--orbifold spheres.

• 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. [arXiv]
  ▶ Abstract

  We prove that every RAAG embeds into the C ∞ diffeomorphism group of the real line.

• 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. [arXiv]
  ▶ Abstract

  Let S be a surface with the complexity xi(S) < 3. We prove that if a RAAG A(X) embeds into Mod(S), then X must appear in the curve graph C(S) as an induced subgraph. We also give counterexamples for xi(S)>3.

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

  We prove that every RAAG (right-angled Artin group) embeds into some RAAG defined by an anti-tree. As a consequence, every RAAG embeds into some braid group, and also into Symp(S 2 ) by a quasi-isometry with word-- or L p --metric for p>2; this strengthens M. Kapovich's result.

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

  We prove that a combinatorial triangulation L of S 2 can be realized as an acute geodesic triangulation if and only if L does not have a separating three- or four-cycle.

• 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. [pdf]
  ▶ Abstract

  We develop a theory of right-angled Artin group actions on extension graphs, which parallels mapping class group actions on curve graphs. In particular, we concretely compute an acylindricity constant of the action.

• 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. [pdf]
  ▶ Abstract

  We prove that the double of a rank-two free group either splits as a nontrivial free product or contains a closed hyperbolic surface subgroup.

• 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. [pdf]
  ▶ Abstract

  We prove that a large chromatic number of the defning graph is an obstruction for a RAAG to embed into a given mapping class group.

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

  We propose that a notion of ``extension graph can be used for a systematic study of embedability between two RAAGs.

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

  We define a combinatorial group theoretic notion ``polygonality , and show that this notion can be used to find surface subgroups in many (conjecturally, all) doubles of free groups.

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

  For a graph product G of groups {G i }, we study the kernel K of the map G -> ∏ i G i . We show K embeds into some RAAG. When each G i is finite or cyclic, then G is virtually special. We deduce that when X is a graph with up to seven vertices, then the right-angled Coxeter groups on X contains a hyperbolic surface subgroup if and only if X is weakly chordal.

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

  We prove that ``geometric words (defined by Gordon--Wilton) are polygonal, as defined in [5].

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

  We prove a combination theorem for the family of RAAGs that do not admit ``relative embedding of surface groups.

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

  We prove the injectivity of a map between RAAGs, which comes from a graph operation called co-contraction . A family of words that yield such embeddings, called contraction words , is also described.

Submitted or Accepted Papers

• 7. TBA on Samurai (2026), [arXiv]
• 6. Aletheia tackles FirstProof autonomously (2026), [arXiv]
• 5. Towards Autonomous Mathematics Research (2026), [arXiv]
• 4. Semi-Autonomous Mathematics Discovery with Gemini: A Case Study on the Erdős Problems (2026), [arXiv]
  ▶ Abstract

  We present a case study in semi-autonomous mathematics discovery, using a custom built agent based on Gemini to systematically evaluate 700 conjectures labeled `Open' in Bloom's Erd ő s Problems database. We employ a hybrid methodology: AI-driven natural language verification to narrow the search space, followed by human expert evaluation to gauge correctness and novelty. We address 13 problems that were marked `Open' in the database: 5 through seemingly novel autonomous solutions, and 8 through identification of previous solutions in the existing literature.

• 3. Irrationality of rapidly converging series: a problem of Erdős and Graham (2026), [arXiv]
• 2. Elementary equivalence and diffeomorphism groups of smooth manifolds. [arXiv]
• 1. Algebraic Structure of Diffeomorphism Groups of One--Manifolds [arXiv]

Unpublished notes

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

  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.

• 1. (With Thomas Koberda and Juyoung Lee), Finite subgraphs of extension graphs [arXiv]
  ▶ Abstract

  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.

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