1. Quotients of the curve complex, with Hidetoshi Masai and Saul Schleimer, arXiv:2010.13360.

  2. Random trees in the boundary of Outer space, with Ilya Kapovich, Catherine Pfaff, and Samuel J. Taylor, Geometry and Topology, 26 (2022) 127—162, arXiv:1904.10026.

  3. Random walks, WPD actions, and the Cremona group, with Giulio Tiozzo, Proceedings of the London Mathematical Society, Volume 123, Issue 2, August 2021, 153—202, arXiv:1807.10230.

  4. Random outer automorphisms of free groups: Attracting trees and their singularity structures, with Ilya Kapovich, Catherine Pfaff, and Samuel J. Taylor, Transactions of the American Mathematical Society, 375 (2022), 525—557, arXiv:1805.12382.

  5. The compression body graph has infinite diameter, with Saul Schleimer, Algebraic and Geometric Topology, 21 (2021) 1817—1856, arXiv:1803.06065.

  6. Morse functions to graphs and topological complexity for hyperbolic 3-manifolds, with Diane Hoffoss, Communications in Analysis and Geometry, Vol. 30 (2022), no. 4, 843—868, arXiv:1708.04140.

  7. Recurrence of quadratic differentials for harmonic measure, with Vaibhav Gadre, Mathematical Proceedings of the Cambridge Philosophical Society, 169 (2020), no. 2, 299—305, arXiv:1706.01926.

  8. Random subgroups of acylindrically hyperbolic groups and hyperbolic embeddings, with Alessandro Sisto, International Mathematics Research Notices, Vol. 2019 (13) 3941—3980, arXiv:1701.00253.

  9. The stratum of random mapping classes, with Vaibhav Gadre, Ergodic Theory and Dynamical Systems, (2018) Vol. 38 (7) 2666—2682, arXiv:1607.01281.

  10. Morse area and Scharlemann-Thompson width for hyperbolic 3-manifolds, with Diane Hoffoss, Pacific Journal of Mathematics, Vol. 281 (2016) No. 1, 83—102,

  11. Random walks on weakly hyperbolic groups, with Giulio Tiozzo, Journal für die reine und angewandte Mathematik (Crelle’s Journal), Vol. 2018, Issue 742, 187—239, arXiv:1410.4173.

  12. Random methods in 3-manifold theory, with Alexander Lubotzky and Conan Wu, Proceedings of the Steklov Institute of Mathematics, 2016, Vol. 292, 118—142, arXiv:1405.6410.

  13. Word length statistics for Teichmüller geodesics and singularity of harmonic measure, with Vaibhav Gadre and Giulio Tiozzo, Commentarii Mathematici Helvetici 92 (2017), no. 1, 1—36., arXiv:1212.1481.

  14. Word length statistics and Lyapunov exponents for Fuchsian groups with cusps, with Vaibhav Gadre and Giulio Tiozzo, New York Journal of Mathematics, Volume 21 (2015) 511—531.

  15. Exponential decay in the mapping class group, Journal of the London Mathematical Society, (2012) 86(2), 366—386, arXiv:1104.5543, correction.

  16. Statistics and compression of scl, with Danny Calegari, Ergodic Theory and Dynamical Systems, (2015) volume 35(1), issue 01, pp. 64—110, arXiv:1008.4952.

  17. Asymptotics for pseudo-Anosov elements in Teichmuller lattices, Geometric and Functional Analysis, Vol. 20 (2010) 527—544, arXiv:0901.2679.

  18. Random Heegaard splittings, Journal of Topology, (2010) 3 (4), 997—1025, arXiv:0809.4881.

  19. Linear progress in the complex of curves, Transactions of the American Mathematical Society, 362 (2010), 2963—2991, arXiv:0802.0467.

  20. Random walks on the mapping class group, Duke Mathematical Journal, 156, Number 3 (2011), 429—468, arXiv:math/0604433.

  21. Heegaard gradient and virtual fibers, Geometry and Topology, Vol. 9 (2005), 2227—2259, arXiv:math/0411219.

  22. Period three actions on lens spaces, Algebraic and Geometric Topology, Vol 7. (2007), 2021—2102, arXiv:math/0311009.

  23. Period three actions on the three sphere, with Hyam Rubinstein, Geometry and Topology, Vol. 7 (2003), 329—397, arXiv:math/0204077.

  24. Virtually embedded boundary slopes, Topology And Its Applications, (95)1 (1999), 63—74, arXiv:math/9901041.


  • Research supported by NSF grant DMS-070674, Simons Foundation CGM 234477, CGM 579762, PSC-CUNY grants PSCOOC-41-80, TRADB-43-20, TRADB-44-178, TRADB-45-17, TRADB-46-3, TRADB-47-2, TRADB-50-260 and TRADB-51-242, TRADB-53-2.

Graduate Students.


Normally I try and write on blackboards, but here are the slides I’ve made for various talks, and some video recordings.


Here is a short overview of my thesis, intended for a general audience.

Here is the preprint version of the period three actions paper, with the original pictures, and an index.

My ORCID ID is 0000-0003-3391-1671.

My Erdös number is 3: Maher - Rubinstein - Wormald - Erdös.