Strong convergence of Kleinian groups (Q2571729)

In this paper the author presents a useful characterization of strong convergence of Kleinian groups. Let \(\Gamma\) be a convex cocompact Kleinian group. Let \(\Gamma_A\) be the algebraic limit of a sequence of quasi-conformal deformations of \(\Gamma\). It is shown that for a parabolic element \(\delta\) of \(\Gamma_A\), the conjugacy class of the corresponding element in \(\Gamma\) is represented by a closed curve on the conformal boundary at infinity of \(N=\mathbb H^3/\Gamma\). Let \(l_{X}(\delta)\) be the minimum of the hyperbolic lengths of these curves in the conformal boundary \(X\in \mathcal T(\partial_{\infty}N)\). It is said that \(\delta\) is conformally peripheral if its conjugacy class is represented by a peripheral simple closed curve on the conformal boundary of \(\mathbb H^3/\Gamma_A\). Then the author gives a sufficient condition for the convergence to be strong in terms of the conformally peripheral pinching, i.e., for every conformally peripheral \(\delta \in \Gamma_A\), the hyperbolic lengths \(l_{X_i}(\delta)\) tend to zero, where \(X_i\) denote the conformal boundaries associated to the sequence. The author also shows that this condition is necessary when \(\Gamma\) does not split as a non-trivial free product.