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Negatively curved groups have the convergence property. I. (English) Zbl 0847.20031
It is known that the Cayley graph \(\Gamma\) of a negatively curved (Gromov-hyperbolic) group \(G\) has a well-defined boundary at infinity \(\partial\Gamma\). Furthermore, \(\partial\Gamma\) is compact and metrizable. In this paper it is shown that \(G\) acts on \(\partial\Gamma\) as a convergence group. This implies that if \(\partial\Gamma\simeq\partial\Gamma{\mathbf S}^1\), then \(G\) is topologically conjugate to a cocompact Fuchsian group.
Reviewer: E.M.Freden (Provo)

MSC:
20F65 Geometric group theory
57S05 Topological properties of groups of homeomorphisms or diffeomorphisms
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