Distortion of Surface Groups in CAT(0) free-by-cyclic Groups

Given a non-positively curved 2-complex with a circle-valued Morse function satisfying some extra combinatorial conditions, we describe how to locally isometrically embed this in a larger non-positively curved 2-complex with free-by-cyclic fundamental group. This embedding procedure is used to produce examples of CAT(0) free-by-cyclic groups that contain closed hyperbolic surface subgroups with polynomial distortion of arbitrary degree. We also produce examples of CAT(0) hyperbolic free-by-cyclic groups that contain closed hyperbolic surface subgroups that are exponentially distorted.