# Writings

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

## Published

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]

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.

Ancillary file of the above paper for the computer--assisted proofs

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

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]

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 mapping class groups of genus at least two do not faithfully act on the circle by C^{1+t} diffeomorphisms.

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] 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. [pdf] https://doi.org/10.3934/jmd.2021009

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] https://doi.org/10.1007/s00209-021-02745-3

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. doi: 10.3934/dcds.2020259

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. DOI: 10.2140/agt.2020.20.3183 [pdf]

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] [journal page]

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). [journal page] [Amazon page]

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. doi:10.24033/asens.2397

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, doi: 10.4171/JEMS/886, Journal of European Mathematical Society, Volume 21, Issue 8, 2019, pp. 2333–2353

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. https://doi.org/10.1112/topo.12079

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, Adv. Lect. Math. (ALM), 40, Higher Education Press and International Press, Beijing-Boston. Book chapter

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).

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. Published

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. Published

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.

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. DOI 10.2140/gt.2015.19.3289 Published

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. doi: 10.1112/jtopol/jtv038 Published

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. Published

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. Published

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. Published

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. Embedability between right-angled Artin groups, Geometry & Topology 17 (2013) 493--530. Published

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. Published

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). Published

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. Published

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. Published

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), June 21-24, 2009 (Perugia, Italy) (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), Special Issue on Managing Complexity in Multiuser MIMO Systems, 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. Published

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.

## accepted/submitted

2. Nhat Minh Doan, **Sang-hyun Kim**, Mong Lung Lang, Ser Peow Tan. Optimal independent generating system for the congruence subgroups Γ₀(p). [Link]

We prove that if n is a prime or its square then the congruence subgroup Gamma_0(n) admits a freely independent set of generators whose (2,1) components are exactly 0 or n.

1. Jaewon Chang, **Sang-hyun Kim** and Thomas Koberda. Algebraic Structure of Diffeomorphism Groups of One--Manifolds (submitted)

Mather proved that the group of *C*^{1,}^{k}** --diffeomorphisms of an n--manifold is simple, provided that a mild isotopy condition is satisfied, with the possible exception of k=n+1$. We give a detailed account of Mather's proof in the case when n=1 and extend this result to a slightly larger class of diffeomorphism groups of certain ``tame* regularities.*

## For general public

3. 튜링과 정지문제, HORIZON, KIAS Webzine (2021). [Link] [Naver]

2. 수학은 상상 (대중교양서적), EBS 출판부, 2021년 9월 30일. [Link]

(Imagine Mathematics), Korean Educational Broadcasting System Publication, September 30, 2021.

1. A group seen from its boundary, Horizon (a KIAS magazine in Korean), April 2019.

## unpublished notes

1. (With Thomas Koberda and Juyoung Lee), Finite subgraphs of extension graphs.

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.

## Reviews

Mathematical Reviews by American Mathematical Society

MR3033518 ; MR3125410 ; MR3158758 ; MR3158775 ; MR3668054 ; MR3728497 ; MR3738334 ; MR3797073 ; MR3858768 ; MR3864538 ; MR3909234 ; MR3954281 ; MR4009420 ; MR4020679 ; MR3993762

## 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)