If a torsion-free hyperbolic group G has 1-dimensional boundary ∂∞G, then ∂∞G is a Menger curve or a Sierpinski carpet provided G does not split over a cyclic group. When ∂∞G is a Sierpinski 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 ∂∞G is induced by an element of G.
|Number of pages
|Annales Scientifiques de l'Ecole Normale Superieure
|Published - Sep 2000
ASJC Scopus subject areas
- General Mathematics