Conjugacy equivalence relation on subgroups (Q2717647)

This paper is a contribution to the ongoing study of definable equivalence relations on Polish spaces. If \(G\) is a countable group, the space \(\text{Sgr} (G)\) of all of its subgroups is Polish, and a natural equivalence relation on \(\text{Sgr} (G)\) is conjugacy. This equivalence relation is obviously induced by a continuous action of \(G\) on the space \(\text{Sgr} (G)\) and is denoted by \(E_{\text{c}} (G, \text{Sgr} (G))\). \(E_{\text{c}} (G, \text{Sgr} (G))\) is Borel (as a subset of \(\text{Sgr} (G) \times \text{Sgr} (G)\)) and countable (because all of its equivalence classes are countable). The authors prove, improving preceding results of \textit{S. Thomas} and \textit{B. Veličković} [J. Algebra 217, 352-373 (1999; Zbl 0938.03060)] and of \textit{S. Gao} [Bull. Lond. Math. Soc. 32, 653-657 (2000; Zbl 1025.03044)], that if \(G\) contains the free group on two generators as a subgroup then \(E_{\text{c}} (G, \text{Sgr} (G))\) is universal for countable Borel equivalence relations (i.e.\ any countable Borel equivalence relation is Borel reducible to \(E_{\text{c}} (G, \text{Sgr} (G))\)). NEWLINENEWLINENEWLINEThe proof of the main theorem uses a result which is interesting in its own right: if \(G\) is a countable group of permutations of \(\mathbb N\) which includes all recursive permutations (actually a slightly weaker hypothesis suffices), then the equivalence relation induced by the action of right composition of \(G\) on \(5^{\mathbb N}\) is universal. This theorem improves the result by \textit{R. Dougherty} and \textit{A. S. Kechris} [Contemp. Math. 257, 83-94 (2000; Zbl 0962.03033)] where \(5^{\mathbb N}\) is replaced by \(\mathbb N^{\mathbb{N}}\).