Research interests

My research interests are low-dimensional topology, geometric topology, geometric group theory, and Teichmüller theory.

You can find short descriptions of my recent research projects below.

See also my research description here.

On the left, it is a short clip explaining my work with Sang-hyun Kim (KIAS) and Thomas Koberda (U of Virginia) (I apologize that the video is in Korean).

Asymptotic translation lengths and normal closures of pseudo-Anosov mapping classes

For the mapping class group actions on either Teichmüller space or curve complex, one can study the asymptotic translation length of the mapping classes. There are many evidence that the minimal translation length among the elements in a proper normal subgroup of the mapping class group, it is bounded from below and the bound can be given by the topological complexity of the surface. In a joint paper with H.Shin, we verify this conjecture for Torelli groups and pure braid groups. Asymptotic translation lengths for the monodromies of the surface-bundles over the circle have been also studied in recent joint works with H.Shin-C.Wu, E.Kin-H.Shin-C.Wu. The case for Torelli groups of the surfaces with boundary has been considered in H.Shin-P.Tranchida and a generalization in the context of Out(F_n) is obtained in D.M.Kim-C.Wu.

Laminar Groups: Recognition of Fuchsian/Kleinian subgroups of Homeo(S^1)

It is a question with a long history to ask which group of circle homeomorphisms is conjugate to a Fuchsian group. A famous theorem of Tukia, Gabai, Casson-Jungreis, Hinkkanen shows that such a group is a Fuchsian group if and only if it is a convergence group. Main theorem of my PhD thesis gives another way of recognizing Fuchsian groups among groups of circle homeomorphisms. Namely, a torsion-free discrete subgroup of Homeo(S^1) is a Fuchsian group (which is not the thrice-punctured sphere group) if and only if it admits three very-full laminations such that their shared endpoints are precisely fixed points of parabolic elements of the group. As a quick corollary, a torsion-free discrete subgroup of Homeo(S^1) is a Fuchsian group which purely consists of hyperbolic elements if and only if it admits three very-full laminations with no shared endpoints. As a result, one can recover the geometric data of a group from topological invariant laminations. Ongoing project is to ask if there is a similar characterization of fibered hyperbolic 3-manifold groups. There are various partial results, and some of them are due to my collaboration with J. Alonso and E. Samperton.

Subgroups in Diffeo^k(S^1)

Every right-angled Artin group (RAAG) admits a faithful action on S^1 by C^1 diffeomorphisms, by a theorem of Farb–Franks. Kim–Koberda showed that every RAAG embeds into MCG(S_g) for some g. My collaboration with Kim, Koberda showed that every RAAG embeds into Diffeo^\infty(R). It is not a priori clear if one can always compactify R to S^1 while keeping the embedding smooth at the point at infinity. In our more recent paper, we show that this is not possible, even virtually. More precisely, we show that no finite-index subgroup of MCG(S_{g,n,b}), where S_{g,n,b} is an orientable surface of genus g with n marked points and b boundary components and 3g−3+n+b ≥ 2, can act faithfully on the closed interval I or S^1 by orientation-preserving C^{1+bv} diffeomorphisms (C^1 diffeomorphisms whose first derivatives have bounded variation). For comparison, Farb–Franks proved that the full mapping class group MCG(S_{g,n,0}), with g ≥ 3 and n = 0 or 1, has no nontrivial C^2 action on the circle or the closed interval; Ghys independently established the circle case. Parwani proved that every C^1 action on the circle of the mapping class group of a closed surface of genus at least six is trivial. These statements about all actions of full groups are distinct from our obstruction to faithful actions of finite-index subgroups. Our proof uses the RAAG whose commutation graph is the four-vertex path P_4 (also denoted A_4): it does not virtually embed into the orientation-preserving C^{1+bv} diffeomorphism group of a compact one-manifold. (See also the above youtube video)

Fried's Conjecture on pseudo-Anosov dilatations

It is a classical fact that the dilatation constant of a pseudo-Anosov surface diffeomorphism is a bi-Perron number, i.e., it is a positive real algebraic integer such that all its Galois conjugates have modulus between the number and its reciprocal. Fried conjectured that the converse of this statement is also true, but this conjecture is widely open. W. Thurston constructed some examples of pseudo-Anosov diffeomorphisms from 1-dimensional dynamical systems which is analogous to the construction due to T. Hall and A. de Carvalho. In the joint work with A. Rafiqi and C. Wu, we gave a similar but different construction of a family of surface diffeomorphisms from certain type of non-negative integer matrices.

Spaces of Circular Orders of Groups

A group is circularly orderable if it admits a circular order as a set which is invariant under left-multiplication. For countable groups, this is equivalent to admitting a faithful action on the circle by orientation-preserving homeomorphisms. In the joint work with E. Samperton, we study various aspects of the space of all circular orders of given groups, and seek for possible generalizations of precedent works for the space of linear orders of groups. In particular, a classification of circular orders of finitely generated abelian groups is given.

Random 3-manifolds

In a reading group seminar with David Bauer, Ilya Gekhtman, Ursula Hamenstaedt, Sebastian Hensel, Thorben Kastenholz, Mark Pedron, Bram Petri, Daniel Valenzuela, we studied abelian covers of random 3-manifolds. Here, a random 3-manifold means taking a random Heegaard splitting, i.e., gluing two handlebodies along the boundary with a random mapping class of the surface. The seminar was fruitful, and as a result, we could show that a random 3-manifold with genus g and with first betti number g admits a tower of cyclic covers where the torsion of the first homology grows exponentially fast relative to the degree of the covers. The paper can be found here.