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Divergent sequences of function groups. (English) Zbl 1166.58008

To describe the main result of the paper under review, let us denote by \(N\) a hyperbolic 3-manifold with a compressing boundary and consider the space \({\mathcal P}{\mathcal M}{\mathcal L}(\partial N)\) of all projective measured laminations on \(\partial N\). Denote by \({\mathcal E}\) the set of all \(\lambda\in{\mathcal P}{\mathcal M}{\mathcal L}(\partial N)\) such that there exists an essential disk or annulus \(A\) satisfying \(i(\lambda,\partial A)=0\), where \(i(\cdot,\cdot)\) stands for the intersection form. Then for every \(\lambda\) in the closure of \({\mathcal E}\) in \({\mathcal P}{\mathcal M}{\mathcal L}(\partial N)\) there exists a sequence in a convex compact representation space that converges to \(\lambda\) and diverges in the space of conjugation classes of discrete faithful representations from \(\pi_1(N)\) into \(\text{PSL}(2,{\mathbb C})\).

MSC:

58D19 Group actions and symmetry properties
53C20 Global Riemannian geometry, including pinching
57S25 Groups acting on specific manifolds
53C23 Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces
30F40 Kleinian groups (aspects of compact Riemann surfaces and uniformization)
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