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Quasiregular self-mappings of manifolds and word hyperbolic groups

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Martin R. Bridson, Aimo Hinkkanen and Gaven Martin

#### to appear in Compositio

An extension of a result of Sela shows that
if $\G$ is a torsion-free word hyperbolic group, then the only
homomorphisms $\G\to\G$ 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 ($n\neq 4$),
then every quasiregular mapping $f:M \to M$ 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 automorphism.
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 $\pi_1$-injective proper quasiregular mappings $f:M\to N$ between hyperbolic $3$-manifolds $M$ and $N$ with non-elementary fundamental group.