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The boundary of the Gieseking tree in hyperbolic three-space. (English) Zbl 0926.57008
In 1989, J. W. Cannon and W. P. Thurston proved that: Every lift to the universal cover of the embedding $$F \to M$$ induces an embedding $${\mathbb H}^2 \to {\mathbb H}^3$$ which extends continuously to a $$\pi _1(F)$$-equivariant quotient map $$\Psi : \partial {\mathbb H}^2 \to \partial {\mathbb H}^3$$. In this paper, the authors give an elementary proof of the Cannon-Thurston theorem in the case of the Gieseking manifold, the three-manifold fibred over the circle $$S^1$$ with fibre $$F$$ a punctured torus, and with homological monodromy $$\left(\begin{smallmatrix} 0 & 1 \\ 1 &1 \end{smallmatrix}\right)$$.

MSC:
 57M50 General geometric structures on low-dimensional manifolds 57M60 Group actions on manifolds and cell complexes in low dimensions
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