Detection of incompressible surfaces in hyperbolic punctured torus bundles

Abstract

Culler and Shalen, and later Yoshida, give ways to construct incompressible surfaces in 3-manifolds from ideal points of the character and deformation varieties, respectively. We work in the case of hyperbolic punctured torus bundles, for which the incompressible surfaces were classified by Floyd and Hatcher, and independently by Culler, Jaco and Rubinstein. We convert non fiber incompressible surfaces from their form to the form output by Yoshida’s construction, and run his construction backwards to give (for non semi-fibers, which we identify) the data needed to construct ideal points of the deformation variety corresponding to those surfaces via Yoshida’s construction. We use a result of Tillmann to show that the same incompressible surfaces can be obtained from an ideal point of the character variety via the Culler-Shalen construction. In particular this shows that all boundary slopes of non fiber and non semi-fiber incompressible surfaces in hyperbolic punctured torus bundles are strongly detected.