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Cannon-Thurston maps for hyperbolic group extensions. (English) Zbl 0907.20038
For a hyperbolic group \(K\) in the sense of M. Gromov [in Essays in group theory, Publ., Math. Sci. Res. Inst. 8, 75-263 (1985; Zbl 0634.20015)], the Cayley graph \(\Gamma_K\) with respect to a finite set of generators admits a compactification \(\widehat\Gamma_K\); this compactification is obtained by adding the Gromov boundary, which consists of classes of asymptotes of geodesics, to \(\Gamma_K\). Given a hyperbolic group \(G\) and a normal subgroup \(H\) thereof which is itself a hyperbolic group, together with a finite set of generators for \(G\) and one for \(H\), there is a continuous proper embedding \(i\) of the Cayley graph \(\Gamma_H\) into the Cayley graph \(\Gamma_G\) of \(G\); the main result of the paper says that the inclusion \(i\) extends to a continuous map from \(\widehat\Gamma_H\) to \(\widehat\Gamma_G\) (which is necessarily unique). When \(G\) is the fundamental group of a closed hyperbolic 3-manifold fibering over the circle and when \(H\) is the fundamental group of the fibre, this result reduces to one of Cannon and Thurston.

20F65 Geometric group theory
57M07 Topological methods in group theory
20F34 Fundamental groups and their automorphisms (group-theoretic aspects)
20E22 Extensions, wreath products, and other compositions of groups
57M50 General geometric structures on low-dimensional manifolds
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