Research

My main fields of study are 3-dimensional topology, hyperbolic geometry, Riemann surface theory (Teichmüller theory) and mapping class groups. I've also dabbled in holomorphic dynamics and combinatorial group theory.

For a (somewhat out of date) collection of pictures of the kind of objects I am interested in, see these lecture slides.

• My PhD students

Papers, roughly in reverse chronological order: 

1. Transverse surfaces and pseudo-Anosov flows. with M. Landry and S. Taylor. Arxiv

2. Endperiodic maps via pseudo-Anosov flows. with M. Landry and S. Taylor. Arxiv

3. Minimizing laminations in regular covers, horospherical orbit closures, and circle-valued Lipschitz maps. with J. Farre and O. Landesberg. Arxiv

4. Subsurface distances for hyperbolic 3-manifolds fibering over the circle. with S. Taylor. Groups, Geometry and Dynamics 18 (2024) 963-1006.  Arxiv

5. Hausdorff dimension of directional limit sets for self-joinings of hyperbolic manifolds. with D. Kim and H. Oh. J. Modern Dynamics 19 (2023) 433-453. Arxiv.

6. Tent property of the growth indicator functions and applications. with D. Kim and H. Oh. Geom. Dedicata 218 (2024)   Arxiv

7. Flows, growth rates, and the veering polynomial. with M. Landry and S. Taylor. Ergodic Th. and Dyn. Sys. 43 (2023) 3026-3107.  Arxiv

8. Stable cubulations, bicombings and barycenters. with M. Durham and A. Sisto. Geom.  Topol. 27 (2023) 2383-2478.  Arxiv.

9. A polynomial invariant for veering triangulations. with M. Landry and S. Taylor. J. European Math. Soc. 26 (2024) 731-788. Arxiv.

10. Bottlenecks for Weil-Petersson geodesics. with B. Modami. Adv. Math. 381 (2021) 49 pp. Arxiv

11. Weil-Petersson translation length and manifolds with many fibered fillings. with C. Leininger, J. Souto and S. Taylor. Adv. Math. 376 (2021), 54 pp.  Arxiv

12. Surface groups are flexibly stable. with N. Lazarovich and A. Levit. To appear, J. of the European Math. Soc.  Arxiv

13. Skinning bounds along thick rays. with K. Bromberg and A. Kent. J. Top. Anal. 13 (2021) 591-605.  Arxiv

14. Local topology in deformation spaces of hyperbolic 3-manifolds II. with J. Brock, K. Bromberg, R. Canary and C. Lecuire. Groups, Geom. and Dyn. 13 (2019) no. 3, 767--793. Arxiv

15. Fibered faces, veering triangulations and the arc complex. with S. Taylor. (Arxiv)  Geom. Funct. Anal. 27 (2017), 1450–1496.

16. Windows, cores and skinning maps.  with J. Brock, K. Bromberg and R. Canary. Ann. Sci. Éc. Norm. Sup. 53 (2020) 173--216. Arxiv. 

17. Bounded combinatorics and uniform models for hyperbolic 3-manifolds.  with J. Brock, H. Namazi and J. Souto. J. Topol. (2016). Arxiv 

18. A brief introduction to mapping class groups. Moduli spaces of Riemann Surfaces, 5-44, IAS/Park City Math. Ser., 20. Amer. Math. Soc. (2013). Preprint 

19. Thick-skinned 3-manifolds.  with A. Kent. Geom. Funct. Anal. 24 (2014) 1981-2001. Arxiv 

20. Convergence properties of end invariants.  with J. Brock, K. Bromberg and R. Canary. ArXiv. Geom. Topol. 17 (2013) 2877–2922.  

21. Picture-hanging puzzles.  with E. Demaine, M. Demaine, J. Mitchell, R. Rivest and M. Patrascu. ArXiv. Theory Comput. Syst. 54 (2014), no. 4, 531-550. 

22. Cohomology classes represented by measured foliations, and Mahler's question for interval exchanges. with B. Weiss. ArXiv. Ann. Sci. Ec. Norm. Super. (4) 47 (2014), no. 2, 245-284. 

23. The dynamics of Aut(F_n) on redundant representations.  with T. Gelander. ArXiv. Groups, Geometry and Dynamics 7 (2013), 557-576. 

24. Extending pseudo-Anosov maps to compression bodies.  with I. Biringer and J. Johnson. ArXiv. J. Topol. 6 (2013), no. 4, 1019-1042. 

25. Discrete primitive-stable representations with large rank surplus.  with Y. Moriah. ArXiv. Geom. Topol. 17 (2013) 2223-2261. 

26. Asymptotics of Weil-Petersson geodesics II: bounded geometry and unbounded entropy.  with J. Brock and H. Masur. ArXiv. Geom. Funct. Anal. 21 (2011), 820-850. 

27. Heegaard splittings with large subsurface distances.  with J. Johnson and Y. Moriah. ArXiv. Algebr. Geom. Topol. 10 (2010), 2251-2275. 

28. Local topology in deformation spaces of hyperbolic 3-manifolds.  with J. Brock, K. Bromberg and R. Canary. ArXiv Geom. Topol. 15 (2011), 1169-1224. 

29. On dynamics of Out(Fn) on PSL(2,C) characters.  Arxiv. Israel J. Math. 193 (2013), 47-70. 

30. Centroids and the rapid decay property in mapping class groups.  with J. Behrstock. Arxiv. J. Lond. Math. Soc. (2) 84 (2011), 765--784. 

31. Asymptotics of Weil-Petersson geodesics I: ending laminations, recurrence and flows.  with J. Brock and H. Masur. Geom. Funct. Anal. 19 (2010), 1229-1257. Arxiv Preprint 

32. Geometry and rigidity of mapping class groups.  with J. Behrstock, B. Kleiner and L. Mosher. Arxiv. Geom. Topol. 16 (2012), 781-888. 

33. High distance knots.  with Y. Moriah and S. Schleimer. ArXiv. Algebraic & Geometric Topology 7 (2007), 101–999.

34. Dimension and rank for mapping class groups.  with J. Behrstock. Ann. of Math. 167 (2008), 1055-1077. Arxiv.

35. Curve complexes, surfaces and 3-manifolds. Proceedings of the ICM 2006. PDF

36. The classification of Kleinian surface groups II: The Ending Lamination Conjecture.  with J. Brock and R. Canary. PDF. Ann. of Math. (2) 176 (2012), 1-149. 

37. End invariants and the classification of hyperbolic 3-manifolds.  Lecture notes from the Current Developments in Mathematics conference, Harvard, Nov. 2002. PDF.

38. Quasiconvexity in the curve complex.  with H. Masur. In the Tradition of Ahlfors and Bers, III (W. Abikoff and A. Haas, eds.), Contemporary Mathematics 355, Amer. Math. Soc. (2004), 309-320. PDF. 

39. The classification of Kleinian surface groups I: models and bounds. Ann. of Math. (2) 171 (2010), 1-107. Arxiv.

40. Combinatorial and Geometrical Aspects of Hyperbolic 3-Manifolds. PDF. in Kleinian Groups and Hyperbolic 3-Manifolds (Y. Komori, V. Markovic and C. Series, Eds.) London Math. Soc. Lec. Notes, 299 (2003), 3-40. 

41. Nondivergence of horocyclic flows on Moduli space,   with B. Weiss. IMS Preprint 2000-08. J. Reine Angew. Math. 552 (2002), 131-177. PDF.

42. Bounded geometry for Kleinian groups.   Invent. Math. 146 (2001), 143-192. Arxiv.

43. Short geodesics and end invariants.   Expository article, Comprehensive Research on Complex Dynamical Systems and Related Fields, RIMS Kokyuroku #1153 (2000) 1-20, M. Kisaka and S. Morosawa, eds. Preprint.

44. Rank one phenomena in mapping class groups,  with B. Farb and A. Lubotzky. Duke Math. J. 106 (2001), 581-597.  PDF.

45. Unstable quasi-geodesics in Teichmuller space, with H. Masur. In the tradition of Ahlfors and Bers: Proceedings of the first Ahlfors-Bers colloquium, I. Kra and B. Maskit, eds. AMS Contemp. Math. 256 (2000), 239-241.  Preprint.

46. Kleinian groups and the complex of curves, Geometry and Topology, 4 (2000), 117-148. Arxiv.

47. Geometry of the complex of curves II: Hierarchical structure, with H. Masur. Geom. Funct. Anal. 10 (2000), 902-974.  Arxiv.

48. The classification of punctured-torus groups, Annals of Math. 149 (1999), 559-626. Arxiv.

49. Geometry of the complex of curves I: Hyperbolicity, with H. Masur. Invent. Math. 138 (1999), 103-149. Arxiv.

50. Laminations in holomorphic dynamics, with M. Lyubich. J. Diff. Geometry. 47 (1997), 17-94. PDF.

51. A geometric approach to the complex of curves on a surface,Proceedings of the Taniguchi Symposium, Finland 1995.  Preprint.

52. Spectral theory, Hausdorff dimension and the topology of hyperbolic 3-manifolds, with R. Canary and E. Taylor. J. Geom. Analysis 9 (1999), 18-40. PDF

53. On Thurston's ending lamination conjecture, Proceedings of the Knoxville Conference on Low-Dimensional Topology, 1992. International Press 1994. Preprint.

54. Extremal length estimates and product regions in Teichmuller space, Duke Math. J. 83 (1996), 249-286. Preprint.

55. On limits of tame hyperbolic 3-manifolds, with R. Canary, J. Differential Geom. 43 (1996), 1-41. PDF

56. Quasi-projections in Teichmuller space,  J. Reine Angew. Math. 473 (1996), 121-136. Preprint. 

57. On rigidity, limit sets and end invariants of hyperbolic 3-manifolds, J. Amer. Math. Soc. 7 (1994), 539-588. PDF

58. Teichmuller geodesics and ends of hyperbolic 3-manifolds, Topology 32 (1993), 625-647. PDF

59. Harmonic maps into hyperbolic 3-manifolds, Trans. Amer. Math. Soc. 332 (1992), 607-632. PDF

60. Harmonic maps, length and energy in Teichmuller space,  J. Differential Geom. 35 (1992) 151-217. PDF

Some of the work shown here was supported by grants from the National Science Foundation and a fellowship from the Alfred P. Sloan foundation. 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.