Free subgroups of the homeomorphism group of the reals (Q1083074)

The authors give a short self-contained proof of the known result that the homeomorphism group \({\mathcal H}({\mathbb{R}})\) of the reals contains a free subgroup of rank equal to the cardinality of the continuum. They apply similar techniques to show that, in many cases, a subgroup G of \({\mathcal H}({\mathbb{R}})\) can be enlarged by the choice of an independent generator h to get a subgroup of \({\mathcal H}({\mathbb{R}})\) isomorphic to the free product G* \({\mathbb{Z}}\), and to give criteria for the existence of many (in the sense of a comeager set in a natural complete metric topology) homeomorphisms h independent of G.