## 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

Zbl 0634.20015
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