Research

My main fields of study are 3-dimensional topology, hyperbolic geometry, Riemann surface theory (Teichmuller theory). I've also been dabbling in holomorphic dynamics and combinatorial group theory.

For a collection of pictures of the kind of objects I am interested in, see these lecture slides.

My collaborators past and present:

• Jason Behrstock
• Jeff Brock
• Richard Canary
• Benson Farb
• Alex Lubotzky
• Howard Masur
• Yoav Moriah
• Saul Schleimer
• Ed Taylor
• Barak Weiss

A current list of papers, roughly in reverse chronological order:

• Local topology in deformation spaces of hyperbolic 3-manifolds
  with J. Brock, K. Bromberg and R. Canary. Preprint or via ArXiv
  
  
• On dynamics of Out(F_n) on PSL(2,C) characters
  Arxiv Preprint
  
  
• Centroids and the rapid decay property in mapping class groups
  with J. Behrstock. Arxiv Preprint
  
  
• Asymptotics of Weil-Petersson geodesics I: ending laminations, recurrence and flows
  with J. Brock and H. Masur. To appear in GAFA. Arxiv Preprint
  
  
• Geometry and rigidity of mapping class groups
  with J. Behrstock, B. Kleiner and L. Mosher. Arxiv Preprint
  
  
• High distance knots
  with Y. Moriah and S. Schleimer. math.GT/0607265
  
  
• Dimension and rank for mapping class groups
  with J. Behrstock. math.GT/0512352
  
  
• The classification of Kleinian surface groups II: The Ending Lamination Conjecture
  with J. Brock and R. Canary. dvi, ps.
  
  
• End invariants and the classification of hyperbolic 3-manifolds
  Lecture notes from the Current Developments in Mathematics conference, Harvard, Nov. 2002. Postscript.
  
  
• 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. Postscript.
  
  
• The classification of Kleinian surface groups I: models and bounds
  math.GT/0302208
  
  
• Combinatorial and Geometrical Aspects of Hyperbolic 3-Manifolds
  Lecture notes from the Workshop on Kleinian Groups and Hyperbolic 3-Manifolds, Warwick, England Sept. 2001. Postscript.
  
  
• Nondivergence of horocyclic flows on Moduli space
  with B. Weiss. IMS Preprint 2000-08, to appear in J. Reine Angew. Math. Latest revision: PS or DVI.
  
  
• Bounded geometry for Kleinian groups
  Invent. Math. 146 (2001), 143-192. (May 10 2001 Preprint in Postscript)
  
  
• 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.
  DVI or Postscript.
  
  
• Rank one phenomena in mapping class groups
  with B. Farb and A. Lubotzky. Duke Math. J. 106 (2001), 581-597. Preprint: DVI or Postscript.
  
  
• 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: DVI or Postscript.
  
  
• Kleinian groups and the complex of curves,
  Geometry and Topology, 4 (2000), 117-148.
  
  
• Geometry of the complex of curves II: Hierarchical structure,
  with H. Masur. Geom. Funct. Anal. 10 (2000), 902-974. (Abstract and preprint, revised July 2000).
  
  
• The classification of punctured-torus groups,
  Annals of Math. 149 (1999), 559-626. May 1998 Version.
  
  
• Geometry of the complex of curves I: Hyperbolicity,
  with H. Masur. Invent. Math. 138 (1999), 103-149. (January 1998 preprint here.)
  
  
• Laminations in holomorphic dynamics,
  with M. Lyubich. J. Diff. Geometry. 47 (1997), 17-94. Preprint (DVI)
  
  
• A geometric approach to the complex of curves on a surface,
  Proceedings of the Taniguchi Symposium, Finland 1995. Preprint.
  
  
• Spectral theory, Hausdorff dimension and the topology of hyperbolic 3-manifolds,
  with R. Canary and E. Taylor. J. Geom. Analysis 9 (1999), 18-40. Preprint.
  
  
• On Thurston's ending lamination conjecture,
  Proceedings of the Knoxville Conference on Low-Dimensional Topology, 1992. International Press 1994. DVI or Postscript.
  
  
• Extremal length estimates and product regions in Teichmuller space,
  Duke Math. J. 83 (1996), 249-286. Preprint.
  
  
• On limits of tame hyperbolic 3-manifolds,
  with R. Canary, J. Differential Geom. 43 (1996), 1-41.
  
  
• Quasi-projections in Teichmuller space,
  J. Reine Angew. Math. 473 (1996), 121-136. Preprint.
  
  
• On rigidity, limit sets and end invariants of hyperbolic 3-manifolds,
  J. Amer. Math. Soc. 7 (1994), 539-588.
  
  
• Teichmuller geodesics and ends of hyperbolic 3-manifolds,
  Topology 32 (1993), 625-647.
  
  
• Harmonic maps into hyperbolic 3-manifolds,
  Trans. Amer. Math. Soc. 332 (1992), 607-632.
  
  
• Harmonic maps, length and energy in Teichmuller space,
  J. Differential Geom. 35 (1992) 151-217.

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