On certain homeomorphism groups (Q1321042)

Let \(G\) be group of homeomorphisms of the reals, the rationals, the irrationals or the Cantor set. The authors' main result shows that the free product of \(G\) with a free group of cardinality equal to that of the reals embeds in \(G\). The proof relies on a procedure known as Mather's Trick. [\textit{J.N. Mather}, Topology 10, 297-298 (1971; Zbl 0207.21903)]. This work has its origin in the theory of independent sets. (Kuratowski, Micielski and others), and the history of this problem is recounted in the introduction of the paper.