# Research

I am interested in Geometric Topology, Geometric Group Theory, Ergodic Theory and Dynamics. In particular;

• Mapping class groups, Teichmüller spaces, Moduli spaces, Geodesic currents on surfaces
• Veech Surfaces, Quadratic and Abelian Differentials, Dynamics on translation surfaces
• Out(FN), Culler-Vogtmann’s Outer Space, Geometry of free factor and free splitting complexes, Geodesic currents on free groups
• Substitutions and symbolic dynamics
• Random walks on groups and spaces, hyperbolic groups

## Papers:

1. Length functions on currents and applications to dynamics and counting
(with V. Erlandsson)
submitted, 2018. [pdf] [arxiv]
2. Exceptional directions for Teichmüller geodesic flow and Hausdorff dimension
(with H. Al-Saqban, P. Apisa, A. Erchenko, O. Khalil, S. Mirzadeh)
submitted, 2018. [pdf] [arxiv]
3. Hyperbolic extensions of free groups from atoroidal ping-pong
submitted, 2018. [pdf] [arxiv]
4. Simultaneous construction of hyperbolic isometries (with M. Clay)
Pacific J. Math. 294 (2018), no. 1, 71-88.  [pdf] [arxiv] [journal]
5. North-South dynamics of hyperbolic free group automorphisms on the space of currents (with M. Lustig)
Journal of Topology and Analysis, to appear, 2017. [pdf] [arxiv] [journal]
6. Perron-Frobenius theory and frequency convergence for reducible substitutions (with M. Lustig)
Discrete Contin. Dyn. Syst. 37 (2017), no. 1, 355-385. [pdf] [arxiv] [journal]
7. The distribution of gaps for saddle connections on the octagon (with G. Work)
Int. Math. Res. Notices, 18 (2016), 5569-5602. [pdf] [arxiv] [journal]
8. Generalized north-south dynamics on the space of geodesic currents
Geom. Dedicata, 177 (2015), 129-148. [pdf] [arxiv] [journal]
9. Dynamics of hyperbolic iwips
Conform. Geom. Dyn. 18 (2014), 192-216. [pdf] [arxiv] [journal]

## Ph.D. Thesis:

• Dynamics of free group automorphisms and a subgroup alternative for Out(FN) [pdf]

## Notes:

Notes on geodesic currents on free groups from a mini-course given by Ilya Kapovich.