The Bounded Image Theorem

Part of the Modern Birkhäuser Classics book series (MBC)


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 \(Q_j \subset Q\)).

$$[\rho_n] \epsilon D_{{\rm par}}(G, PSL(2, \mathbb{C})),\quad\rho_n\,:\,G \rightarrow \Gamma_n,$$
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.


Conjugacy Class Fundamental Group Klein Bottle Quasi Conformal Mapping Parabolic Element 
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© Birkhäuser Boston 2009

Authors and Affiliations

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

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