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The matrices $\begin{pmatrix}1&2\\0&1\end{pmatrix}$ and $\begin{pmatrix}1&0\\2&1\end{pmatrix}$ are well-known to (freely) generate a free group. Some years ago, I read a paper that, if memory serves, sort of summarised what was known about generalisations to pairs of the form $\begin{pmatrix}1&\lambda\\0&1\end{pmatrix}$ and $\begin{pmatrix}1&0\\\lambda&1\end{pmatrix}$. (Or perhaps even more general such pairs.) I recall that there were results there describing regions in $\mathbb{C}$ from which $\lambda$ chould be chosen so that such a pair freely generated a free group, and there were plots of these regions (at least one, anyway). Unfortunately, I cannot recall the title or author(s) of the paper, and Googling didn't turn up anything that looked familiar.

Does anyone recall such a paper and be able to provide a reference.

(I did find a 1967 paper by Lyndon and Ullman, but that wasn't it, and I think the one I'm after was later, and probably described further progress on the question, probably with its own new results.)

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There's a 1969 paper by Lyndon and Ullman that appears to be it, plots and all.

R. C. Lyndon and J. L. Ullman, Groups generated by two parabolic linear fractional transformations. Canad. J. Math. 21 (1969) 1388-1403

The article itself is available online, but I'm not sure whether it's free, or I'm seeing it because of my University's subscription. (Most likely the former applies.)

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  • $\begingroup$ The PDF seems to be freely available. $\endgroup$ – lhf Jul 12 '14 at 11:46
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For a slightly more recent reference see

M. Yu. Lyubich, V. V. Suvorov, Free subgroups of $SL_2(C)$ with two parabolic generators Journal of Soviet Mathematics, 1988, Volume 41, Issue 2, p. 976-979.

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FREE AND NONFREE SUBGROUPS OF $PSL_2(\mathbf{C})$ GENERATED BY TWO PARABOLIC ELEMENTS by Ju A Ignatov

1979 Math. USSR Sb. 35 49. doi:10.1070/SM1979v035n01ABEH001449

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