  If a torsion-free hyperbolic group G has 1-dimensional boundary, then the
boundary is a Menger curve or a Sierpinski carpet provided G does not split
over a cyclic group. When the boundary of G is a Sierpinski carpet we show that
G is a quasi-convex subgroup of a 3-dimensional hyperbolic Poincare duality
group. We also construct a ``topologically rigid'' hyperbolic group G: any
homeomorphism of the boundary of G is induced by an element of G.
