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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.

MSC:
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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