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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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