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Dynamics of holomorphic transformations; Mandelbrot and Julia sets.
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When are those subgroups of $\mathrm{SL}(2, \mathbb{C})$ discrete?
Let $A = \pmatrix{1 & 0 \\ \alpha & 1} $ and $ B = \pmatrix{1 & 1 \\ 0 & 1}$, where $\alpha \in \mathbb{C}$ is a complex parameter.
Now consider the family of representations $r_{\alpha}$ of the fre …