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Rigidity and Quasisymmetric Uniformization of Higher-Dimensional Thurston-Type Maps

For expanding, postcritically-finite branched covers (Thurston-type maps) on closed oriented Riemannian n-manifolds with n at least 3, this work proves the No Invariant Line Fields conjecture — such a map is uniformly quasiregular if and only if it is a Lattes map — and a companion quasisymmetric uniformization theorem: the identity from the manifold with its visual metric to the manifold with its Riemannian metric is quasisymmetric exactly when the map is uniformly quasiregular.