Injective realizations of bending transverse cocycles for geodesic laminations.
Abstract
Given a Fuchsian group $G$ uniformizing a closed hyperbolic surface $S$
together with the corresponding real moduli, a complex deformation of the real
moduli gives a conjugation between $G<PSL_2(\mathbf{R})$ and a subgroup of
$PSL_2(\mathbf{C})$. If the conjugating map is injective then the image is a
quasiFuchsian group and the quotient hyperbolic three-manifold is called a
quasiFuchsian manifold. We consider shear cocycles for a maximal geodesic
lamination on $S$ defined by Thurston and Bonahon, and their complexification
called the shear-bend cocycles for $S$. Our main result gives a genus
independent geometric conditions on the bending cocycles in terms of
underlining shear cocycles such that the image group is quasifuchsian. The
shear-bend cocycles translate into complex Fenchel-Nielsen coordinates and our
result gives a version of a theorem of Kahn-Markovic \cite{KahnMark} (which is
used in their proof of the surface subgroup conjecture).
Вход в систему
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