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On free subgroups of \(\text{SL}(2,\mathbb{C})\) with two parabolic generators. (Russian. English summary) Zbl 0624.20030
The paper studies the so-called “eye-problem” [see R. Lyndon and J. Ullman, Can. J. Math. 21, 1388-1403 (1969; Zbl 0191.019)] i.e. the following problem. Let \(G_{\lambda}\subset SL(2,{\mathbb{C}})\) be the group generated by the parabolic matrices \[ A=\left( \begin{matrix} 1\\ 1\end{matrix} \begin{matrix} 0\\ 1\end{matrix} \right)\quad\text{ and }\quad B_{\lambda}=\left( \begin{matrix} 1\\ 0\end{matrix} \begin{matrix} \lambda \\ 1\end{matrix} \right), \] \(\Gamma_ 0=\{\lambda \in {\mathbb{C}}:G_{\lambda}\) is non-free} is then \(\Gamma ={\bar \Gamma}_ 0=?\)
Using Kleinian groups theory, the authors prove that \({\mathbb{C}}\setminus \Gamma\) is the Schottky space \(QC(G_{\lambda_ 0})\) of quasi-conformal deformations of a marked Schottky-type group \(G_{\lambda_ 0}\), \[ QC(G_{\lambda_ 0})=T({\mathbb{C}}\setminus \{0,\pm 1\}/Mod(G_{\lambda_ 0})=\{z: Im z>0\}/Mod(G_{\lambda_ 0}), \] and therefore \({\mathbb{C}}\setminus \Gamma\) and \(\Gamma\) are connected (\({\mathbb{C}}\setminus \Gamma\) is doubly connected). Besides the authors observe that \({\mathbb{C}}\setminus \Gamma_ 0\) is invariant under a large semigroup of polynomial mappings, and show that \(\Gamma\) coincides with the closure of non-discrete groups and with the closure of torsion groups. Finally, they describe \(\Gamma\) in terms of dynamics of those polynomial mappings.
Reviewer: B.N.Apanasov

MSC:
20G20 Linear algebraic groups over the reals, the complexes, the quaternions
30F40 Kleinian groups (aspects of compact Riemann surfaces and uniformization)
30D05 Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable
20E07 Subgroup theorems; subgroup growth
20E05 Free nonabelian groups
20H10 Fuchsian groups and their generalizations (group-theoretic aspects)
22E40 Discrete subgroups of Lie groups
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