Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

Request Permissions   Purchase Content 
 
 

 

Complex Fenchel-Nielsen coordinates with small imaginary parts


Author: Dragomir Šarić
Journal: Trans. Amer. Math. Soc. 366 (2014), 6541-6565
MSC (2010): Primary 30F40, 32G15
DOI: https://doi.org/10.1090/S0002-9947-2014-06101-1
Published electronically: September 4, 2014
MathSciNet review: 3267018
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Kahn and Markovic (2012) proved that the fundamental group of each closed hyperbolic three manifold contains a closed surface subgroup. One of the main ingredients in their proof is a theorem which states that an assignment of nearly real, complex Fenchel-Nielsen coordinates to the cuffs of a pants decomposition of a closed surface $ S$ induces a quasi-Fuchsian representation of the fundamental group of $ S$. We give a new proof of this theorem with slightly stronger conditions on the Fenchel-Nielsen coordinates and explain how to use the exponential mixing of the geodesic flow on a closed hyperbolic three manifold to prove that our theorem is sufficient for the applications in the work of Kahn and Markovic.


References [Enhancements On Off] (What's this?)

  • [1] Alan F. Beardon, The geometry of discrete groups, Graduate Texts in Mathematics, vol. 91, Springer-Verlag, New York, 1983. MR 698777 (85d:22026)
  • [2] Francis Bonahon, Transverse Hölder distributions for geodesic laminations, Topology 36 (1997), no. 1, 103-122. MR 1410466 (97j:57015), https://doi.org/10.1016/0040-9383(96)00001-8
  • [3] Francis Bonahon, Shearing hyperbolic surfaces, bending pleated surfaces and Thurston's symplectic form, Ann. Fac. Sci. Toulouse Math. (6) 5 (1996), no. 2, 233-297 (English, with English and French summaries). MR 1413855 (97i:57011)
  • [4] L. Bowen, Weak Forms of the Ehrenpreis Conjecture and the Surface Subgroup Conjecture, arXiv:math/0411662.
  • [5] C. J. Earle, I. Kra, and S. L. Krushkal, Holomorphic motions and Teichmüller spaces, Trans. Amer. Math. Soc. 343 (1994), no. 2, 927-948. MR 1214783 (94h:32035), https://doi.org/10.2307/2154750
  • [6] D. B. A. Epstein and A. Marden, Convex hulls in hyperbolic space, a theorem of Sullivan, and measured pleated surfaces, Analytical and geometric aspects of hyperbolic space (Coventry/Durham, 1984), London Math. Soc. Lecture Note Ser., vol. 111, Cambridge Univ. Press, Cambridge, 1987, pp. 113-253. MR 903852 (89c:52014)
  • [7] D. B. A. Epstein, A. Marden, and V. Markovic, Quasiconformal homeomorphisms and the convex hull boundary, Ann. of Math. (2) 159 (2004), no. 1, 305-336. MR 2052356 (2005d:30067), https://doi.org/10.4007/annals.2004.159.305
  • [8] Linda Keen and Caroline Series, How to bend pairs of punctured tori, Lipa's legacy (New York, 1995) Contemp. Math., vol. 211, Amer. Math. Soc., Providence, RI, 1997, pp. 359-387. MR 1476997 (98m:30063), https://doi.org/10.1090/conm/211/02830
  • [9] J. Kahn and V. Markovic, Immersing almost geodesic surfaces in a closed hyperbolic 3-manifold, Ann. of Math. (2) 175 (2012), no. 3, 1127-1190. MR 2912704
  • [10] Christos Kourouniotis, Complex length coordinates for quasi-Fuchsian groups, Mathematika 41 (1994), no. 1, 173-188. MR 1288062 (96g:30079), https://doi.org/10.1112/S0025579300007270
  • [11] R. Mañé, P. Sad, and D. Sullivan, On the dynamics of rational maps, Ann. Sci. École Norm. Sup. (4) 16 (1983), no. 2, 193-217. MR 732343 (85j:58089)
  • [12] Calvin C. Moore, Exponential decay of correlation coefficients for geodesic flows, Group representations, ergodic theory, operator algebras and mathematical physics (Berkeley, Calif., 1984),, Math. Sci. Res. Inst. Publ., vol. 6, Springer, New York, 1987, pp. 163-181. MR 880376 (89d:58102), https://doi.org/10.1007/978-1-4612-4722-7_6
  • [13] Dragomir Šarić, Real and complex earthquakes, Trans. Amer. Math. Soc. 358 (2006), no. 1, 233-249 (electronic). MR 2171231 (2006i:30063), https://doi.org/10.1090/S0002-9947-05-03651-2
  • [14] D. Šarić, Bendings by finitely additive transverse cocycles, available on arXiv.
  • [15] Zbigniew Slodkowski, Holomorphic motions and polynomial hulls, Proc. Amer. Math. Soc. 111 (1991), no. 2, 347-355. MR 1037218 (91f:58078), https://doi.org/10.2307/2048323
  • [16] Ser Peow Tan, Complex Fenchel-Nielsen coordinates for quasi-Fuchsian structures, Internat. J. Math. 5 (1994), no. 2, 239-251. MR 1266284 (94m:32030), https://doi.org/10.1142/S0129167X94000140

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 30F40, 32G15

Retrieve articles in all journals with MSC (2010): 30F40, 32G15


Additional Information

Dragomir Šarić
Affiliation: Department of Mathematics, Queens College of CUNY, 65-30 Kissena Boulevard, Flushing, New York 11367
Email: Dragomir.Saric@qc.cuny.edu

DOI: https://doi.org/10.1090/S0002-9947-2014-06101-1
Received by editor(s): April 25, 2012
Received by editor(s) in revised form: February 2, 2013
Published electronically: September 4, 2014
Additional Notes: This research was partially supported by National Science Foundation grant DMS 1102440.
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society