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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.

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