I work in low-dimensional and geometric topology, and in geometric group theory. Much of my research focuses on mapping class groups, with particular interest in pseudo-Anosov surface homeomorphisms and convex cocompact subgroups. This research also involves the geometry of Teichmüller spaces, which are the parameter spaces of hyperbolic structures on surfaces. Recently, I have become very interested in free group automorphisms. I am involved in an ongoing project to study the various ways a given group can split as a free by cyclic group, and I am also studying hyperbolicity in the setting of free group extensions.

Publications and Preprints

All of my papers are available on the arXiv.
  1. Lattice point counting for finite-order mapping classes, 102 pages.
       (with Howard Masur)       preprint: arXiv:2305.03721

  2. Orientable maps and polynomial invariants of free-by-cyclic groups, 35 pages.
       (with Radhika Gupta, and Samuel J. Taylor)       preprint: arXiv:2203.12134
    To appear in Advances in Mathematics

  3. Extensions of Veech groups II: Hierarchical hyperbolicity and quasi-isometric rigidity, 57 pages.    (with Matthew G. Durham, Christopher J. Leininger, and Alessandro Sisto)      arXiv:2111.00685
    To appear in Commentarii Mathematici Helvetici

  4. Extensions of Veech groups I: A hyperbolic action, 45 pages.    (with Matthew G. Durham,
          Christopher J. Leininger, and Alessandro Sisto)       preprint: arXiv:2006.16425
    Journal of Topology (2023) 16: 757–805.     doi: 10.1112/topo.12296

  5. Discretely shrinking targets in moduli space, 18 pages.
         (with Grace Work)       preprint: arXiv:1909.05817
    Geometriae Dedicata (2022) 216:53, 17pp.     doi: 10.1007/s10711-022-00716-4 / Springer SharedIt

  6. Isomorphisms between big mapping class groups, 11 pages.
         (with Juliette Bavard and Kasra Rafi)       preprint: arXiv:1708.08383
    Int. Math. Res. Not. IMRN (2020), no. 10, 3084–3099.     doi: 10.1093/imrn/rny093

  7. Rank and Nielsen equivalence in hyperbolic extensions, 9 pages.
         (with Samuel J. Taylor)       preprint: arXiv:1706.02368
    Internat. J. Algebra Comput. (IJAC) 29 (2019) no. 4, 615–625.     doi: 10.1142/S0218196719500176

  8. The co-surface graph and the geometry of hyperbolic free group extensions, 33 pages.
         (with Samuel J. Taylor)       preprint: arXiv:1601.00101
    Journal of Topology 10 (2017), 447–482.     doi: 10.1112/topo.12013

  9. Endomorphisms, train track maps, and fully irreducible monodromies, 18 pages.
         (with Ilya Kapovich and Christopher J. Leininger)       preprint: arXiv:1507.03028
    Groups, Geometry, and Dynamics 11 (2017) no. 4, 1179–1200.     doi: 10.4171/GGD/425

  10. Cannon–Thurston maps for hyperbolic free group extensions, 27 pages.
         (with Ilya Kapovich and Samuel J. Taylor)       preprint: arXiv:1506.06974
    Israel Journal of Mathematics 216 (2016), 753–797.     doi: 10.1007/s11856-016-1426-2

  11. Contracting orbits in Outer space, 20 pages.
         (with Samuel J. Taylor)       preprint: arXiv:1502.04053
    Math. Z.. 293 (2019) no.1-2, 767–787.     doi: 10.1007/s00209-019-02227-7 / Springer SharedIt

  12. Hyperbolic extensions of free groups, 50 pages.
         (with Samuel J. Taylor)       preprint: arXiv:1406.2567
    Geometry & Topology 22 (2018), no. 1, 517–570.     doi: 10.2140/gt.2018.22.517

  13. Unbounded asymmetry of stretch factors, 2 pages.
         (with Ilya Kapovich and Christopher J. Leininger)       preprint: arXiv:1405.0771
    C. R. Acad. Sci. Paris, Ser. I Vol. 352 (2014), 885–887.     doi: 10.1016/j.crma.2014.09.007

  14. McMullen polynomials and Lipschitz flows for free-by-cyclic groups, 73 pages.
         (with Ilya Kapovich and Christopher J. Leininger)       preprint: arXiv:1310.7481
    J. Eur. Math. Soc. (JEMS) 19 (2017), no. 11, 3253–3353.     doi: 10.4171/JEMS/739

  15. Dynamics on free-by-cyclic groups, 67 pages.
         (with Ilya Kapovich and Christopher J. Leininger)       preprint: arXiv:1301.7739
    Geometry & Topology 19-5 (2015), 2801–2899.     doi: 10.2140/gt.2015.19.2801

  16. Pseudo-Anosov subgroups of fibered 3-manifold groups, 29 pages.
         (with Autumn Kent and Christopher J. Leininger)       preprint: arXiv:1208.2495
    Groups, Geometry, and Dynamics 8 (2014) no. 4, 1247–1282.     doi: 10.4171/GGD/302

  17. Spheres in the curve complex, 8 pages.
         (with Moon Duchin and Howard Masur)       preprint: arXiv:1109.6338
    Contemporary Mathematics, vol. 590, Amer. Math. Soc., Providence, RI, 2013, pp. 1–8.     
    doi: 10.1090/conm/590/11727

  18. Statistical hyperbolicity in Teichmüller space, 41 pages.
         (with Moon Duchin and Howard Masur)       preprint: arXiv:1108.5416
    Geometric and Functional Analysis Vol. 24 (2014) 748–795.     doi: 10.1007/s00039-014-0265-8

  19. Dilatation versus self-intersection number for point-pushing pseudo-Anosov
    homeomorphisms    , 50 pages.
          preprint: arXiv:1004.3936
    Journal of Topology 4 (2011) no. 4: 942–984.     doi: 10.1112/jtopol/jtr023

Part of this material is based upon work supported by the National Science Foundation under Grants DMS-1204814, DMS-1711089, and DMS-2005368. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation or of Vanderbilt University.