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I've seen one proof of $\mathrm{psl}_2(\mathbf Z) \cong C_2 * C_3$ based on the Ping-pong lemma

But what I'm looking for is how this result was historically obtained first.

I've checked Coxeter & Moser's "Generators and relations for discrete groups" and they (§7.2) do things that I find weird, they let the matrices act on circles in the half-plane $y>0$ which in turn they regard as lines of a hyperbolic plane, it then follows this group can be interpreted as triangle group acting on a hyperbolic triangle.

Is this indeed the 'classic' way to arrive at this result? And if so, I'd be happy with a bit of an explanation of what's going on, or a reference to a text where this is explained a bit more in detail, because I'm really not that comfortable with hyperbolic triangles.

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    Well, I've been gazing through it and it's interesting to read even though the style is a bit weird.2011-03-09

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I don't know about the history, but if you want to understand 'what's going on' the keyword is 'orbifold'. I recommend Peter Scott's article 'The geometries of 3-manifolds', available as a pdf on his website, for an introduction to 2-dimensional orbifolds.