I study geometric group theory, the study of groups through their actions on metric spaces. I am particularly interested in groups that act by isometries on hyperbolic metric spaces.  My research focuses on the following main areas:
  • acylindrically hyperbolic groups and their various acylindrical actions on hyperbolic spaces 
  • hierarchically hyperbolic groups and spaces, which include mapping class groups, fundamental groups of 3-manifolds, and CAT(0) cubical groups 
  • (not necessarily acylindrical) actions of groups on hyperbolic spaces, including solvable Baumslag-Solitar groups, fundamental groups of mapping tori and flip graph manifolds.
  • big mapping class groups (these are mapping class groups of surfaces with infinitely generated fundamental group)

Here is my current research statement.

Papers:
  1. Not all finitely generated groups have universal acylindrical actions. Proc. Amer. Math. Soc., 144(10):4151–4155, 2016, pdf.
  2. (with F. Dahmani) Acylindrically hyperbolic groups have property P_naive. arXiv:1610.04143.  Math. Z., 291(1-2):555-568, 2019.
  3. (with S. Balasubramanya and D. Osin) Hyperbolic structures on groups.  arXiv:1710.05197. Algebr. Geom. Topol., 19-4, 1747-1835, 2019.
  4. (with D. Hume and D. Osin) Extending group actions on metric spaces.   arXiv:1703.03010.  J. Topol. Anal., 12(3): 625--665, 2020, pdf.
  5. (with D. Hume) The geometry of generalized loxodromic elements.  arXiv:1802.03089.  Ann. I. Fourier (Grenoble), 70(4): 1689--1713, 2020.
  6. (with J. Behrstock and M. Durham) Largest acylindrical actions and stability in hierarchically hyperbolic groups, with an appendix by Daniel Berlyne and Jacob Russell. arXiv:1705.06219. Trans. Amer. Math. Soc. Ser. B, 8: 66--104, 2021 
  7. (with D. Hume) Actions of small cancellation groups on hyperbolic spaces.  arXiv:1807.10524. ​Geom. Dedicata, 212(1), 325--363, 2021.
  8. (with M. Hull) Random walks and quasi-convexity in acylindrically hyperbolic groups.  arXiv:1909.10876.  To appear in J. Topol.
  9. (with J. Behrstock) Conjugator lengths in hierarchically hyperbolic groups.  arXiv:1808.09604.
  10. (with A. Rasmussen) Actions of solvable Baumslag-Solitar groups on hyperbolic metric spaces.  arXiv:1906.04227​. To appear in Algebra. Geom. Topol.
  11. (with T. Ng and D. Spriano) Uniform exponential growth in hierarchically hyperbolic groups. arXiv:1909.00439
  12. (with A. Rasmussen) Largest hyperbolic actions and quasi-parabolic actions of groups.  arXiv:1910.14157. To appear in J. Topol. Anal.
  13. (with J. Manning) Acylindrically hyperbolic groups and their quasi-isometrically embedded subgroups, arXiv:2105.02333.
  14. (with S. Balasubramanya and A. Rasmussen) Higher rank confining subsets and hyperbolic actions of solvable groups.  arxiv:2108.08175. To appear in Adv. Math.
  15. (with N. Miller and P. Patel) Infinite-type loxodromic isometries of the relative arc graph.  arXiv:2109.06106
  16. (with H. Hoganson, M. Loving, P. Patel, and R. Skipper) Finding and combining indicable subgroups big mapping class groups.  arXiv: 2109.05976
  17. (with S. Balasubramanya and A. Rasmussen) Valuations, completions, and hyperbolic actions of metabelian groups. arxiv.org/abs/2207.12945
  18. (with J. Behrstock and J. Russell) Structure invariant properties of the hierarchically hyperbolic boundary. arxiv.org/abs/2208.07930
  19. ​(with N. Miller and P. Patel) Shift maps are not type-preserving.  arXiv:2212.09156
  20. (with M. Incerti-Medici) Hyperbolic projections and topological invariance of sublinear Morse boundaries.  arXiv:2212.09539
  21. (with H. T. Nguyen and A. Rasmussen) Largest hyperbolic actions of 3--manifold groups. arXiv:2305.07425
  22. (with M. Hagen, H. Petyt, and A. Zalloum) Uniform undistortion from barycentres, and applications to hierarchically hyperbolic groups. arXiv:2305.097