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.