Hyperbolic Manifolds and Discrete Groups pp 383-395 | Cite as

# The Bounded Image Theorem

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## 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
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 (

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

*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
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