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].

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].

Reviewer: Thilo Kuessner (Seoul)