## Abstract

An extension of a result of Sela shows that if is a torsion-free word hyperbolic group, then the only homomorphisms with finite-index image are the automorphisms. It follows from this result and properties of quasiregular mappings, that if M is a closed Riemannian n-manifold with negative sectional curvature (n ≠ q 4), then every quasiregular mapping f:MM is a homeomorphism. In the constant-curvature case the dimension restriction is not necessary and Mostow rigidity implies that f is homotopic to an isometry. This is to be contrasted with the fact that every such manifold admits a non-homeomorphic light open self-mapping. We present similar results for more general quotients of hyperbolic space and quasiregular mappings between them. For instance, we establish that besides covering projections there are no 1-injective proper quasiregular mappings f:MN between hyperbolic 3-manifolds M and N with non-elementary fundamental group.

Original language | English (US) |
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Pages (from-to) | 1613-1622 |

Number of pages | 10 |

Journal | Compositio Mathematica |

Volume | 143 |

Issue number | 6 |

DOIs | |

State | Published - Nov 2007 |

## Keywords

- Hopfian
- Hyperbolic group
- Hyperbolic manifold
- Open mapping
- Quasiregular

## ASJC Scopus subject areas

- Algebra and Number Theory