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The universal Cannon-Thurston map and the boundary of the curve complex. (English) Zbl 1248.57003
The original Cannon-Thurston map is a quotient map from the boundary $$\partial \mathbb{H}$$ of the hyperbolic plane onto the limit set of a Kleinian group $$\Gamma$$. It was constructed by J. W. Cannon and W. P. Thurston in their paper [“Group invariant Peano curves”, Geom. Topol. 11, 1315–1355 (2007; Zbl 1136.57009)] for the fiber subgroup of the fundamental group of a closed hyperbolic 3-manifold fibering over the circle. It was later on extended by Y. N. Minsky in his paper [“Teichmüller geodesics and ends of hyperbolic 3-manifolds”, Topology 32, No. 3, 625–647 (1993; Zbl 0793.58010)] and by M. Mj in the preprint [“Ending laminations and Cannon-Thurston maps”, arXiv:math/0701725 (2007)]. In this quotient map, distinct points are identified if and only if they are ideal points of a leaf of an ending lamination for $$\Gamma$$. Other kinds of “Cannon-Thurston maps” were constructed later on, and they are mentioned in the introduction of the paper under review.
In this paper, the authors construct a map they call the Universal Cannon-Thurston map. For this, they consider a closed hyperbolic surface $$S$$ of genus $$\geq 2$$ with a distinguished point $$z\in S$$. The curve complexes of $$S$$ and $$(S,z)$$ are denoted respectively by $$\mathcal{C}(S)$$ and $$\mathcal{C}(S,z)$$. The fundamental group $$\pi_1(S)$$ acts on $$\mathcal{C}(S,z)$$ via the inclusion into the mapping class group of $$(S,z)$$ and this action gives rise to map $\Phi:\mathcal{C}(S)\times \mathbb{H}\to \mathcal{C}(S,z)$ which leads, by restriction and for any given vertex $$v$$ of $$\mathcal{S}$$, to a map $\Phi_v: \mathbb{H}\to \mathcal{C}(S,z).$ Here the authors show that if $$r\subset \mathbb{H}$$ is a geodesic ray that eventually lies in the preimage of some proper essential subsurface of $$S$$ then $$\Phi_v(r) \subset \mathcal{C}(S,z)$$ has finite diamater. The remaining rays define a subset $$\mathbb{A}\subset \partial \mathbb{H}$$ which is of full measure. The authors then show that this map $$\Phi_v$$ has a unique continuous $$\pi_1(S)$$-equivariant extension $\overline{\Phi}_v:\mathbb{H}\cup \mathbb{A}\to \overline{\mathcal{C}}(S,z)$ and that the map $$\partial \Phi= \overline{\Phi}_v|_{\mathbb{A}}$$ does not depend on $$v$$ and that it is a quotient map onto the Gromov boundary $$\partial \mathcal{C}(S,z)$$ of the curve complex. Furthermore, they show that for given distinct points $$x$$ and $$y$$ in $$\mathbb{A}$$, one has $$\partial \Phi(x)=\partial \Phi(y)$$ if and only if $$x$$ and $$y$$ are ideal endpoints of a leaf (or ideal vertices of a complementary polygon) of the lift of any ending lamination on $$S$$. The last property is the one that makes the map $$\partial \Phi$$ universal. The authors also prove that the quotient map $\partial \Phi: \mathbb{A}\to \partial \mathcal{C}(S,z)$ is equivariant with respect to the action of the mapping class group of $$(S,z)$$. Finally, they prove that the Gromov boundary $$\partial \mathcal{C}(S,z)$$ is path-connected and locally path-connected. This strengthens a result in [C. J. Leininger and S. Schleimer, “Connectivity of the space of ending laminations”, Duke Math. J. 150, No. 3, 533–575 (2009; Zbl 1190.57013)] in a special case.

MSC:
 57M07 Topological methods in group theory 20F67 Hyperbolic groups and nonpositively curved groups 57M50 General geometric structures on low-dimensional manifolds
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