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19.42 (2018)

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Suppose that $H$ is a word-hyperbolic subgroup of a word-hyperbolic group $G$ such that the inclusion of $H$ to $G$ extends to a continuous $H$-equivariant map $j : \partial H \to \partial G$ between their hyperbolic boundaries. If such an extension exists, it is unique and $j$ is called the Cannon–Thurston map.
$\qquad$ a) Is it true that for every point $p \in \partial G$ its full preimage $j^{-1}(p)$ is finite?
$\qquad$ b) Moreover, is it true that there is a number $N = N(G, H) < \infty$ such that for every $p \in \partial G$ the full preimage $j^{-1}(p)$ consists of at most $N$ points?

Contributor: I. Kapovich

Progress

The answer to both questions is “yes” in all the cases where the Cannon–Thurston map is known to exist and where it has been possible to analyze the multiplicity of this map. This includes the original set-up considered by Cannon and Thurston where $H$ is the surface group, and $G$ is the fundamental group of a closed hyperbolic 3-manifold fibering over the circle with that surface as a fiber. In this case, $j : S^1 \to S^2$ is a uniformly finite-to-one continuous surjective ‘Peano curve’.

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