First uses of the Ping-Pong Lemma

Question

I am interested in knowing the origins of this useful result, but I haven't been able to precisely pinpoint the context of its first use. Most texts seem to indicate the result originally comes from Felix Klein, though the lemma is sometimes referred as Schottky's criterion. I would also like to know what the original formulation looks like, if anyone knows it.

Thank you so much for your help!

Edit: this was crossposted to History of Science and Mathematics Stackexchange.

Answer 1

Tits in the famous "Tits alternative" paper (1972) provides a "free product" version of the ping-pong lemma. He then mentions in an "added in proof" footnote: "R. Lyndon has pointed out to the author that a similar criterion has been given by A. Macbeath (Proc Cambridge Ph. Soc. 1963). Cf also R. Lyndon and J. Ullman (Mich. Math J 1968) (...)."

Here's the abstract of Macbeath's paper (link behind paywall)

In this note a simple principle is explained for constructing a transformation group which is a free product of given transformation groups. The principle does not seem to have been formulated explicitly, though it has been used in a more or less vague form in the theory of discontinuous groups (see, for instance, L. R. Ford, Automorphic functions, vol. I, pp. 56–59). It is perhaps of interest that the formulation given here is purely set-theoretic, without any topology, and that it can apply to any free product, whatever the cardinal number of the set of factors. The principle is used to establish the closure under the formation of countable free product of the family of groups which can be represented as discontinuous subgroups of a certain group of rational projective transformations. (The word ‘discontinuous’ is used here in a weak sense, defined later.) Finally, these results are applied to give a new proof of the theorem of Gruenberg that a free product of residually finite groups is itself residually finite (K. W. Gruenberg: Residual properties of groups, Proc. London Math. Soc. (3), 7 (1957), 29–62. See Corollary (ii) of Theorem 4.1, p. 44). The present proof is completely different from Gruenberg's and seems to be of interest for its own sake [...]

Possibly the principle appeared much earlier. One often refers to Schottky subgroups, so a look at Schottky's works at the beginning of the 20th century, would be worth.