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Cannon-Thurston maps for hyperbolic free group extensions. (English) Zbl 1361.20030
Let $$H$$ and $$G$$ be two hyperbolic groups such that $$H$$ is a subgroup of $$G$$. If the inclusion of $$H$$ into $$G$$ induces a map from the Gromov boundary of $$H$$ to that of $$G$$, the map goes under the name of Cannon-Thurston map. J. W. Cannon and W. P. Thurston [Geom. Topol. 11, 1315–1355 (2007; Zbl 1136.57009)] studied the case where $$G$$ is the fundamental group of a closed hyperbolic $$3$$-manifold fibring over the circle, $$H$$ a surface group being the fundamental group of the fibre. It was shown by M. Mitra [Topology 37, No. 3, 527–538 (1998; Zbl 0907.20038)] that if $$H$$ is normal in $$G$$ the Cannon-Thurston map exists and is, moreover, surjective. In this setting, it follows that, if $$H$$ is torsion-free, then it is a free product of surface groups and free groups.
In this paper, the authors analyse the Cannon-Thurston map in the situation where $$H$$ is a free group of rank at least $$3$$ and $$G$$ is a hyperbolic extension of $$H$$ by a convex cocompact subgroup of the outer automorphism group of $$H$$, generalising work of I. Kapovich and M. Lustig on free-by-cyclic hyperbolic extensions [J. Lond. Math. Soc., II. Ser. 91, No. 1, 203–224 (2015; Zbl 1325.20035)]. Note that the condition that $$G/H$$ is convex cocompact implies in particular that the quotient is a hyperbolic group.
A first main result established in the paper shows that, under the above hypotheses, the fibres of the Cannon-Thurston map have size bounded by twice the rank of $$H$$. The authors also prove that a point in the Gromov boundary of $$G$$, satisfying the technical condition of being essential, is rational (irrational, resp.) if so is its image in the boundary of $$G/H$$, provided the fibre of the Cannon-Thurston map over the point is “sufficiently large”, i.e., it contains at least $$3$$ ($$2$$, resp.) points. Recall that a point in the boundary of $$G$$ is rational if it is the limit point of an infinite order element of $$G$$, and irrational otherwise.
In the last part of the paper, the authors consider the map that associates to each point in the boundary of $$G/H$$ an ending lamination of $$H$$. Recall that a lamination can be seen as a closed subset (with some extra properties) of the set of pairs of distinct points of the boundary of $$H$$. The collection of lamination is thus endowed with the Chabauty topology. The authors exhibit an exemple of an extension $$G$$ for which the above map is not continuous, answering a question of Mitra. On the other hand, they show that for an extension of a free group by a purely atoroidal, convex cocompact group $$G/H$$ the following property holds: Let $$z$$ and $$z_i$$, $$i\in\mathbb N$$, be points in the boundary of $$G/H$$ such that the sequence $$(z_i)$$ converges to $$z$$. Let $$\Lambda_z$$ and $$\Lambda_i$$, $$i\in\mathbb N$$, be the corresponding ending laminations. Then if $$L$$ is the limit of a convergent subsequence of $$(\Lambda_i)$$, $$L$$ is contained in $$\Lambda_z$$ and contains the accumulation points of $$\Lambda_z$$.

##### MSC:
 20F67 Hyperbolic groups and nonpositively curved groups 20E22 Extensions, wreath products, and other compositions of groups 20E05 Free nonabelian groups 57M07 Topological methods in group theory 20E36 Automorphisms of infinite groups 20F65 Geometric group theory
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