Quasiregular self-mappings of manifolds and word hyperbolic groups

Martin Bridson, Aimo Hinkkanen, Gaven Martin

Research output: Contribution to journalArticlepeer-review


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 languageEnglish (US)
Pages (from-to)1613-1622
Number of pages10
JournalCompositio Mathematica
Issue number6
StatePublished - Nov 2007


  • Hopfian
  • Hyperbolic group
  • Hyperbolic manifold
  • Open mapping
  • Quasiregular

ASJC Scopus subject areas

  • Algebra and Number Theory


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