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The Bounded Image Theorem

  • Michael Kapovich
Chapter
Part of the Modern Birkhäuser Classics book series (MBC)

Abstract

We assume that the manifold N is connected. The other case is completely similar (with the exception of a few notational differences).

Proof of Theorem 15.14. Recall that all of the representations \([\rho_n] \epsilon D(G, PSL(2, \mathbb{C}))\) that we constructed in Chapter 16 are induced by quasiconformal conjugations of a single Kleinian group G (since we assume that N is connected). Each nontrivial element in \(\pi_1(Q_j)\) is parabolic (for every component \(Q_j \subset Q\)).

Thus
$$[\rho_n] \epsilon D_{{\rm par}}(G, PSL(2, \mathbb{C})),\quad\rho_n\,:\,G \rightarrow \Gamma_n,$$
Where \(G = \pi_1(N) \subset PSL(2, \mathbb{C})\). Every component of the boundary \(\partial_0 N\,:=\partial N - Q\) is incompressible. If the pared manifold (N, Q) were acylindrical as well, then the compactness theorem (Theorem 12.91) would imply that the sequence \([\rho_n]\) is subconvergent in \(D(G, PSL(2, \mathbb{C}))\) since that space \(D_{{\rm par}}(G, PSL(2, \mathbb{C}))\) is compact.

Keywords

Conjugacy Class Fundamental Group Klein Bottle Quasi Conformal Mapping Parabolic Element 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Birkhäuser Boston 2009

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of California, DavisDavisU.S.A.

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