A Bending Norm Criterion for Coarsely Embedded Pleated Planes

Given a surface, $S$, of negative Euler characteristic and some maximal lamination, $\lambda \subset S$, Bonahon developed shear-bend coordinates, which identify an open subset of the character variety, $\chi(\pi_1(S), \mrm{PSL}_2\C)$, whose images contain the quasi-Fuchsian representations with an open subset of a finite dimensional $\C$-vector space, $\mc{H}(\lambda, \C)$. We show that at every Fuchsian representation, there is some definite radius depending only the choice of a train track (which yields a norm on $\mc{H}(\lambda,\C)$) and on the injectivity radius determined by the Fuchsian representation so that the ball of this radius in the bend coordinates is entirely contained in the quasi-Fuchsian locus in $\chi(\pi_1(S), \mrm{PSL}_2\C)$. This work is foreshadowed by the work of Epstein, Marden, and Markovich in which they prove a similar result in the special case that the bending cocycle is also a transverse measure.