My research interests include low-dimensional topology, geometric group theory, and Teichmüller theory. You can find short descriptions of my recent research projects below. For more details, please see my research statement

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

RAAGs in Diffeo^k(S^1) 
It is well known that mapping class groups MCG(S_g) embed into Diffeo^1(S^1). Kim-Koberda showed every RAAG embeds into MCG(S_g) for some g, hence embeds into Diffeo^1(S^1). 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 3g-3+n+b is at least 2 can act on I or S^1 by Diff^{1+bv} (for the full mapping class groups with a slightly more restrictive assumption on the topology of S, it was already known that there is non-trivial action at all by Farb-Franks, Ghys, and also Parwani). This is proved by showing that RAAG with the commutativity graph A_4 cannot be embedded into Diffeo^{1+bv}. 

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