Hyperbolic groups with low-dimensional boundary

Michael Kapovich; Bruce Kleiner

doi:10.1016/S0012-9593(00)01049-1

Hyperbolic groups with low-dimensional boundary

Michael Kapovich, Bruce Kleiner

Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

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.

Original language	English (US)
Pages (from-to)	647-669
Number of pages	23
Journal	Annales Scientifiques de l'Ecole Normale Superieure
Volume	33
Issue number	5
DOIs	https://doi.org/10.1016/S0012-9593(00)01049-1
State	Published - Sep 2000

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General Mathematics

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10.1016/S0012-9593(00)01049-1

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title = "Hyperbolic groups with low-dimensional boundary",

abstract = "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 Poincar{\'e} duality group. We also construct a {"}topologically rigid{"} hyperbolic group G: any homeomorphism of ∂∞G is induced by an element of G.",

author = "Michael Kapovich and Bruce Kleiner",

note = "Funding Information: * Supported by NSF grant DMS-96-26633. 1Supported by a Sloan Foundation Fellowship, and NSF grants DMS-95-05175, DMS-96-26911, DMS-9022140.",

year = "2000",

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doi = "10.1016/S0012-9593(00)01049-1",

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AU - Kleiner, Bruce

N1 - Funding Information: * Supported by NSF grant DMS-96-26633. 1Supported by a Sloan Foundation Fellowship, and NSF grants DMS-95-05175, DMS-96-26911, DMS-9022140.

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AB - 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.

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Hyperbolic groups with low-dimensional boundary

Abstract

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