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Algebraic limits of Kleinian groups which rearrange the pages of a book. (English) Zbl 0874.57012
The authors provide a series of new unexpected examples of algebraic limits of geometrically finite groups in $$\text{PSL}_2 C$$. They show that there exists a sequence $$\rho_n:\pi_1(M)\to \text{PSL}_2 C$$ of discrete faithful representations of the fundamental group of a compact oriented 3-manifold $$M$$ converging to $$\rho$$ such that the limit manifold $$N_\rho=H^3/\rho(\pi_1(M))$$ is not homeomorphic to $$N_{\rho_{n}}=H^3/\rho_{n}(\pi_1(M))$$ for any $$n$$. The manifold $$M$$ is obtained by gluing a collection of $$I$$-bundles to a solid torus $$V$$ along a family of parallel annuli. The above sequence $$\rho_n$$ is constructed in several steps: firstly, a sequence $$\rho'_n$$ is given by doing hyperbolic Dehn $$(1,n)$$-filling on the manifold obtained from $$M$$ by removing the core curve of $$V$$; and secondly one obtains $$\rho_n$$ by precomposing $$\rho'_n$$ with an automorphism of $$\pi_1(M)$$ which “permutes” the $$I$$-bundles. Finally the authors prove that the sequence $$\rho_n$$ converges to $$\rho$$ (all groups $$\rho_n(\pi_1(M))$$ and $$\rho(\pi_1(M))$$ are geometrically finite) such that $$N_\rho$$ is obtained from $$M$$ by rearranging the $$I$$-bundles by the same permutation. It then follows from Johansson-Jaco-Shalen characteristic submanifold theory that the limit manifold $$N_\rho$$ is homotopy equivalent but not homeomorphic to $$N_{\rho_n}$$ unless the permutation above is cyclic.
The authors further show that the closure of the space $$CC(M)$$ of convex cocompact representations of $$\pi_1(M)$$ is connected whereas $$CC(M)$$ can have arbitrary many components. They also announce that there is a manifold $$M$$ with incompressible boundary whose deformation space $$D(\pi_1(M))$$ (which is the set of discrete, faithful representations of $$\pi_1(M)$$ into $$\text{PSL}_2 C$$) has infinitely many components.

##### MSC:
 57N10 Topology of general $$3$$-manifolds (MSC2010) 30F40 Kleinian groups (aspects of compact Riemann surfaces and uniformization)
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