 
Summary: THE BOUNDARY OF THE GIESEKING
TREE IN HYPERBOLIC THREESPACE
R. C. Alperin*, W. Dicks**, J. Porti**
Abstract. We give an elementary proof of the CannonThurston Theorem in
the case of the Gieseking manifold. 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.
1. Introduction
The Gieseking manifold M is the threemanifold fibered over the circle S1
with fibre F a punctured torus, and with homological monodromy 0 1
1 1
.
This monodromy determines the manifold M, which is unorientable, and hy
perbolic, with an unoriented cusp.
Let Hn
denote ndimensional hyperbolic space. Its boundary, Hn
, is
homeomorphic to the (n1)sphere Sn1
, and its compactification, Hn
Hn
