On free subgroups of $SL(2,\mathbb C)$ with two parabolic generators

M. Yu. Lyubich ; V. V. Suvorov

Zapiski Nauchnykh Seminarov POMI, Differential geometry, Lie groups and mechanics. Part VIII, Tome 155 (1986), pp. 150-155

Résumé

Let $G_\lambda$ be a group generated by a pair of parabolic matrices $A=\left(\begin{smallmatrix}1 0 \\ 1 1 \end{smallmatrix}\right)$ and $B_\lambda=\left(\begin{smallmatrix}1 \lambda \\ 0 1 \end{smallmatrix}\right)$, $\Gamma_0$ be a set of $\lambda\in\mathbb C$ for which $G_\lambda$ is non-free, $\Gamma=\bar\Gamma_0$. Using the theory of Kleinian groups we prove that both sets $\Gamma$ and $\mathbb C\setminus\Gamma$ are connected. Besides we observe that $\mathbb C\setminus\Gamma_0$ is invariant under a large semigroup of polynomial mappings. Using this observation we show that $\Gamma$ coinsides with the closure of non-discrete groups and with the closure of torsion groups. Finally we describe $\Gamma$ in terms of the dynamics of those polynomial mappings.