Hyperbolic groups with low-dimensional boundary

If a torsion-free hyperbolic group G has 1-dimensional boundary partial der ivative (infinity)G, then partial derivative (infinity)G is a Menger curve or a Sierpinski carpet provided G does not split over a cyclic group. When partial derivative (infinity)G is a Sierpinski carpet we show that G is a q uasi-convex subgroup of a 3-dimensional hyperbolic Poincare! duality group. We also construct a "topologically rigid" hyperbolic group G: any homeomor phism of partial derivative (infinity)G is induced by an clement of G. (C) 2000 Editions scientifiques et medicales Elsevier SAS.