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Stacks of hyperbolic spaces and ends of 3-manifolds. (English) Zbl 1297.57044
Hodgson, Craig D. (ed.) et al., Geometry and topology down under. A conference in honour of Hyam Rubinstein, Melbourne, Australia, July 11–22, 2011. Proceedings. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-8480-5/pbk). Contemporary Mathematics 597, 65-138 (2013).
The paper under review discusses an alternative approach to the construction of Cannon-Thurston maps and to the proof of the ending lamination theorem of Brock-Canary-Minsky
The general framework of this paper is given by the notion of a “hyperbolic stack”. Here a stack is a geodesic space $$\Xi$$ together with a sequence of pairwise uniformly quasi-isometric spaces $$X_i$$ and embeddings $$f_i: X_i\to\Xi$$ that are uniformly straight (distances in $$\Xi$$ are bounded above and below in terms of distances in $$X_i$$), whose union is quasi-dense in $$\Xi$$, and such that the Hausdorff-distances between the images of consecutive $$X_i$$ are bounded above while the distance between $$X_i$$ and $$X_j$$ is bounded away from zero by a linear function of $$\mid i-j\mid$$. A stack can be constructed from a sequence of uniformly quasi-isometric spaces and it is unique up to quasi-isometry.
A hyperbolic stack is a stack, where $$\Xi$$ and $$X_i$$ are uniformly hyperbolic spaces. (One may think of surfaces $$\widetilde{\Sigma}\times \left\{i\right\}\subset\widetilde{\Sigma}\times{\mathbb R}$$ in the universal covering of a hyperbolic $$3$$-manifold $$\Sigma\times{\mathbb R}$$.) Since there is a well-defined (i.e., up to bounded distance) quasi-isometry $$X_i\to X_0$$, the Gromov-boundaries of all $$X_i$$ have a canonical homeomorphism to $$\partial^0\Xi:=\partial X_0$$. Section 2 of the paper under review constructs a “Cannon-Thurston map” $$\partial^0\Xi\to\partial\Xi$$ such that $$X_0\cup\partial X_0\to\Xi\cup\partial\Xi$$ is continuous. It is proved that $$\partial\Xi$$ is a metrisable Peano continuum.
When $$\Xi$$ is decomposed into semi-infinite stacks $$\Xi=\Xi^+\cup\Xi^-$$ with $$\Xi^+\cap\Xi^-=X_0$$, then $$\Xi^\pm$$ are hyperbolic, $$\partial^0\Xi\to\partial\Xi$$ factors over $$\partial\Xi^\pm$$ and $$\partial\Xi^\pm$$ is a dendrite, i.e. every pair of points is separated by a cut point. If all $$X_i$$ are hyperbolic planes, then the Cannon-Thurston map $$\partial^0\Xi\to\partial\Xi$$ is proven to be surjective and the maps $$\partial^0\Xi\to\partial\Xi^\pm$$ are the quotient maps for the equivalence relations given by unique laminations $$\Lambda^\pm$$ of $$\partial H^2$$.
These laminations are called the ending laminations of the stack and Section 3 of the paper under review is devoted to their study in the case that the $$X_i$$ are universal covers of hyperbolic surfaces. It is shown that any ending lamination of a hyperbolic surface stack is regular, i.e. there is a linear function, $$f$$, such that for any interval $$E$$ contained in a leaf and any essential curve $$\gamma$$ in the complement of $$E$$ one has $$length(E)\leq f(length(\gamma))$$. It is shown that regular laminations have a transverse measure, unique up to scaling.
Section 4 gives an independent proof of a result of L. Mosher [Geom. Topol. 7, 33–90 (2003; Zbl 1021.57009)]: a path $$\beta$$ in the thick part of Teichmüller space is bounded distance from a Teichmüller geodesic if and only if the universal covering of $$P(\beta)$$ is Gromov hyperbolic. Here $$P(\beta)$$ means the (up to bilipschitz equivalence canonical) Riemannian manifold $$\Sigma\times {\mathbb R}$$ such that $$\Sigma\times\left\{t\right\}$$ is uniformly bilipschitz equivalent to the hyperbolic metric $$\beta(t)$$. The universal covering of $$P(\beta)$$ is equivariantly quasi-isometric to the stack $$\Xi$$ build from the $$X_i=\widetilde{\Sigma}\times\left\{i\right\}$$, so that this result fits in the above-mentioned more general setting. In general, a stack turns out to be hyperbolic if the functions $$i\to d_i(x_i,y_i)$$ are uniformly quasiconvex for all chains $$x_i,y_i\in X_i$$.
In the final Section 4.9 the author discusses how this can be applied to prove the Ending Lamination Theorem in the case of positive injectivity radius.
For the entire collection see [Zbl 1272.57002].

##### MSC:
 57M60 Group actions on manifolds and cell complexes in low dimensions 32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables) 57M50 General geometric structures on low-dimensional manifolds