Research Statement (in progress)

Curriculum Vitae

The main fields of my research interest are Geometric Group Theory and Low-Dimesional Topology. Particularly, I am interested in finding (or disproving the existence of) closed, hyperbolic surface subgroups in finitely generated groups. The main fuel in this direction is an intriguing question raised by M. Gromov:

Does every one-ended word-hyperbolic group contain a closed, hyperbolic surface group?

The answer to this question is known for Coxeter groups (Gordon-Long-Reid ’04), graphs of free groups with non-trivial rational homologies (Calegari ’08), 3-manifold groups with positive virtual betti numbers or more generally, hyperbolic 3-mainfold groups (Kahn-Markovic ’09). However, the general approach is still widely open. What Henry Wilton and I recently discovered is there is a simple combinatorial property for a word in a free group that provides a sufficient condition for a double of a free group amalgamated along that word contains a surface group. This serves a model case for general one-ended word-hyperbolic groups according to the JSJ-theory of Sela and Bowditch.