Research
My main fields of study are 3-dimensional topology, hyperbolic geometry,
Riemann surface theory (Teichmuller theory). 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For a (somewhat out of date) collection of pictures of the kind of objects I am interested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.
A current list of papers, roughly in reverse chronological order:
- Fibered faces, veering triangulations and the arc complex.
with S. Taylor.
Arxiv
- Windows, cores and skinning maps.
with J. Brock, K. Bromberg and R. Canary.
Arxiv
- 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 R. 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 Out(F_{n}) on PSL(2,C) characters
Arxiv.
Israel J. Math. 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
- 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.
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.
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.
"Good enough for Government work."
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