Weakly type-preserving sequences and strong convergence
Suppose that \(\rho_i: G\to\text{PSL}(2,\mathbb{C})\) is a sequence of discrete, faithful representations, where \(G\) is finitely generated and each of the groups \(\rho_i(G)\) is a nonelementary Kleinian group. Suppose further that the sequence \(\rho\) converges both algebraically and geometrically to nonelementary Kleinian groups \(\rho_{\text{alg}}(G)\) \((\approx G)\) and \(\rho_{\text{geo}}(G)\), respectively. Assume finally that the limit sets \(\Lambda(\rho_i(G))\) converge to a set \(\overline\Lambda\) in the Riemann sphere. Then we have obvious inclusions: \[ \begin{aligned} \rho_{\text{alg}}(G) &\subset \rho_{\text{geo}}(G),\\ \Lambda(\rho_{\text{alg}}(G)) &\subset \Lambda(\rho_{\text{geo}}(G))\subset\overline\Lambda.\end{aligned} \] In keeping with a conjecture of Jørgensen, the author's main results show that all of these set inclusions are in fact set equalities provided that the following two additional conditions are satisfied: (1) The algebraic limit \(\rho_{\text{alg}}(G)\) has non empty domain of discontinuity. (2) There are no new parabolics in the algebraic limit. [Note: The author uses the Chabauty topology to define set convergence. In the settings considered, this topology is a generalization of the following classical notion of set convergence: Sets \(X_1,X_2,\dots\) converge to \(X\) (denoted \(\lim X_i\)) in \(Y\) iff \(X= \limsup_i\{X_i\}= \liminf_i\{X_i\}\), where \(\limsup_i\{X_i\}= \{x\in Y\mid\) every neighborhood of \(x\) intersects infinitely many \(X_i\}\) and \(\liminf_i\{X_i\}= \{x\in Y\mid\) every neighborhood of \(x\) intersects all but finitely many \(X_i\}\). A fairly easy classical theorem states that every sequence of sets in a separable metric space has a convergent subsequence. Then \(\rho_{\text{alg}}(G)= \{\lim_{i\to\infty}(\rho_i(g))\mid g\in G\}\) (elementwise limit); \(\rho_{\text{geo}}(G)= \lim_{i\to\infty}\rho_i(G)\) (setwise limit), and \(\overline\Lambda= \lim_{i\to\infty} \Lambda(\rho_i(g))\) (setwise limit)].
scientific article; zbMATH DE number 2145809
scientific article