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For a (somewhat out of date) collection of pictures of the kind of objects I am interested in, see these lecture slides.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

**Surface groups are flexibly stable.**with N. Lazarovich and A. Levit.**Arxiv****Skinning bounds along thick rays.**with K. Bromberg and A. Kent. Arxiv**Local topology in deformation spaces of hyperbolic 3-manifolds II.**with J. Brock, K. Bromberg, R. Canary and C. Lecuire. Arxiv**Fibered faces, veering triangulations and the arc complex.**with S. Taylor. (Arxiv)*Geom. Funct. Anal.***Windows, cores and skinning maps.**with J. Brock, K. Bromberg and R. Canary. Arxiv. To appear,*Ann. Sci. Ec. Norm. Super.***Bounded combinatorics and uniform models for hyperbolic 3-manifolds.**with J. Brock, H. Namazi and J. Souto.*J. Topol.*(2016). Arxiv**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**Thick-skinned 3-manifolds.**with A. Kent.*Geom. Funct. Anal.***24**(2014) 1981-2001. Arxiv**Convergence properties of end invariants.**with J. Brock, K. Bromberg and R. Canary. ArXiv.*Geom. Topol.***17**(2013) 2877–2922.**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.**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.**The dynamics of Aut(F_n) on redundant representations.**with T. Gelander. ArXiv.*Groups, Geometry and Dynamics***7**(2013), 557-576.**Extending pseudo-Anosov maps to compression bodies.**with I. Biringer and J. Johnson. ArXiv.*J. Topol.***6**(2013), no. 4, 1019-1042.**Discrete primitive-stable representations with large rank surplus.**with Y. Moriah. ArXiv.*Geom. Topol.***17**(2013) 2223-2261.**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.**Heegaard splittings with large subsurface distances.**with J. Johnson and Y. Moriah. ArXiv.*Algebr. Geom. Topol.***10**(2010), 2251-2275.**Local topology in deformation spaces of hyperbolic 3-manifolds.**with J. Brock, K. Bromberg and R. Canary. ArXiv*Geom. Topol.***15**(2011), 1169-1224.**On dynamics of**Arxiv. Israel J. Math.*Out(F*on_{n})*PSL(2,C)*characters.**193**(2013), 47-70.**Centroids and the rapid decay property in mapping class groups.**with J. Behrstock. Arxiv.*J. Lond. Math. Soc.*(2)**84**(2011), 765--784.**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**Geometry and rigidity of mapping class groups.**with J. Behrstock, B. Kleiner and L. Mosher. Arxiv.*Geom. Topol.***16**(2012), 781-888.**High distance knots.**with Y. Moriah and S. Schleimer. math.GT/0607265.*Algebraic & Geometric Topology***7**(2007), 101–999.**Dimension and rank for mapping class groups.**with J. Behrstock. math.GT/0512352.*Ann. of Math.***167**(2008), 1055-1077.**The classification of Kleinian surface groups II: The Ending Lamination Conjecture.**with J. Brock and R. Canary. dvi, ps. Ann. of Math. (2)**176**(2012), 1-149.**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. Ann. of Math. (2)**171**(2010), 1-107.**Combinatorial and Geometrical Aspects of Hyperbolic 3-Manifolds.**Postscript. in Kleinian Groups and Hyperbolic 3-Manifolds (Y. Komori, V. Markovic and C. Series, Eds.) London Math. Soc. Lec. Notes,**299**(2003), 3-40.**Nondivergence of horocyclic flows on Moduli space,**with B. Weiss. IMS Preprint 2000-08. J. Reine Angew. Math.**552**(2002), 131-177.**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. Preprint.**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.

*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.*