Ending laminations for hyperbolic group extensions.

*(English)*Zbl 0880.57001Let \(H\) be a hyperbolic normal subgroup of infinite index in a hyperbolic group \(G\) (hyperbolic in the sense of Gromov). Choosing a finite generating set of \(G\) which contains a finite generating set of \(H\) one has an embedding of the Cayley graph of \(H\) into that of \(G\). By previous work of the author, this embedding extends to a continuous map between the compactified Cayley graphs (every hyperbolic group admits a compactification of its Cayley graph by adjoining the Gromov boundary consisting of classes of asymptotic geodesics). The motivating situation is that of a hyperbolic surface bundle over the circle. It has been shown by Cannon and Thurston that the inclusion of the universal covering of the fiber (which is quasi-isometric to the hyperbolic plane, and also to the Cayley graph of the fundamental group of the fiber) into the universal covering of the 3-manifold (hyperbolic 3-space) extends to a continuous map between compactifications, that is from the closed 2-disk to the 3-disk (the image of the boundary of the 2-disk is a Peano curve covering the whole 2-sphere). Cannon and Thurston gave an explicit description of this extension in terms of ending laminations coming from Thurston’s theory of stable and unstable laminations of pseudo-Anosov diffeomorphisms of surfaces (the holonomy of the fibration).

“In the case of normal hyperbolic subgroups of hyperbolic groups, though existence of a continuous extension between compactifications of Cayley graphs was proven by the author, an explicit description was missing. To fill this gap in the theory, an analog of Thurston’s theory of ending laminations is necessary. In this paper we generalize some part of Thurston’s theory of ending laminations to the context of normal hyperbolic subgroups of hyperbolic groups. Using this we give an explicit description of the continuous extension between the compactified Cayley graphs.” The ending laminations are naturally parametrized by points in the Gromov boundary of the Cayley graph of the quotient group \(G/H\) which also parametrize limiting actions of \(H\) on \(\mathbb{R}\)-trees (in the motivating example of a hyperbolic 3-manifold fibering over the circle there are two ending laminations corresponding to the stable and unstable lamination of the pseudo-Anosov diffeomorphism). In fact the laminations of the paper and small actions on \(\mathbb{R}\)-trees can be regarded as dual objects. As explained in the introduction, by work of Rips and Sela the normal subgroup \(H\), if it is torsion free, is a free product of free groups and surface groups: there exists a limiting small action of \(H\) on an \(\mathbb{R}\)-tree which is free.

“In the case of normal hyperbolic subgroups of hyperbolic groups, though existence of a continuous extension between compactifications of Cayley graphs was proven by the author, an explicit description was missing. To fill this gap in the theory, an analog of Thurston’s theory of ending laminations is necessary. In this paper we generalize some part of Thurston’s theory of ending laminations to the context of normal hyperbolic subgroups of hyperbolic groups. Using this we give an explicit description of the continuous extension between the compactified Cayley graphs.” The ending laminations are naturally parametrized by points in the Gromov boundary of the Cayley graph of the quotient group \(G/H\) which also parametrize limiting actions of \(H\) on \(\mathbb{R}\)-trees (in the motivating example of a hyperbolic 3-manifold fibering over the circle there are two ending laminations corresponding to the stable and unstable lamination of the pseudo-Anosov diffeomorphism). In fact the laminations of the paper and small actions on \(\mathbb{R}\)-trees can be regarded as dual objects. As explained in the introduction, by work of Rips and Sela the normal subgroup \(H\), if it is torsion free, is a free product of free groups and surface groups: there exists a limiting small action of \(H\) on an \(\mathbb{R}\)-tree which is free.

Reviewer: B.Zimmermann (Trieste)