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Hyperbolic groups with 1-dimensional boundary

Michael Kapovich·Bruce Kleiner
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Abstract

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.

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Michael Kapovich, Bruce Kleiner. 1998-06-11. Hyperbolic groups with 1-dimensional boundary. https://arxiv.org/abs/math/9806059

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