Complex Fenchel-Nielsen coordinates with small imaginary parts

Šarić, Dragomir

doi:10.1090/S0002-9947-2014-06101-1

Complex Fenchel-Nielsen coordinates with small imaginary parts
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Trans. Amer. Math. Soc. 366 (2014), 6541-6565 Request permission

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.

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