The boundary of the Gieseking tree in hyperbolic three-space

We give an elementary proof of the Cannon–Thurston Theorem in the case of the Gieseking manifold. We do not use Thurstonʹs structure theory for Kleinian groups but simply calculate with two-by-two complex matrices. We work entirely on the boundary, using ends of trees, and obtain pictures of the regions which are successively filled in by the Peano curve of Cannon and Thurston.