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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.

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.

Reviewer: Athanase Papadopoulos (Strasbourg)