Conformal interbreeding, Teichmüller spaces and applications

Abstract: We present a new effect in the theory of deformations of hyperbolic manifolds/orbifolds or their uniform hyperbolic lattices (i.e. in the Teichmüller spaces of conformally flat structures on closed hyperbolic 3-manifolds). We show that such varieties may have connected components whose dimensions differ by arbitrary large numbers. This is based on our "Siamese twins construction" of non-faithful discrete representations of hyperbolic lattices related to non-trivial "symmetric hyperbolic 4-cobordisms" and the Gromov–Piatetski-Shapiro interbreeding construction. There are several applications of this result, from new non-trivial hyperbolic homology 4-cobordisms and wild 2-knots in the 4-sphere, to bounded quasiregular locally homeomorphic mappings, especially to their asymptotics in the unit 3-ball solving well known conjectures in geometric function theory.

differential geometrygeometric topologymetric geometry

Audience: researchers in the topic

Topology and geometry: extremal and typical

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