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Hyperbolic groups with low-dimensional boundary. (English) Zbl 0989.20031
Authors’ abstract: If a torsion-free hyperbolic group \(G\) has 1-dimensional boundary \(\partial_\infty G\), then \(\partial_\infty G\) is a Menger curve or a Sierpiński carpet provided \(G\) does not split over a cyclic group. When \(\partial_\infty G\) is a Sierpiński carpet we show that \(G\) is a quasi-convex subgroup of a 3-dimensional hyperbolic Poincaré duality group. We also construct a “topologically rigid” hyperbolic group \(G\): any homeomorphism of \(\partial_\infty G\) is induced by an element of \(G\).
Reviewer: J.W.Cannon (Provo)

MSC:
20F67 Hyperbolic groups and nonpositively curved groups
57M07 Topological methods in group theory
57M50 General geometric structures on low-dimensional manifolds
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