Topological Full Groups of Minimal Subshifts and Just-infinite Groups

In [24], confirming a conjecture of Hjorth-Kechris [16], Thomas-Velickovic proved that the isomorphism relation on the space Gfg of finitely generated groups is a universal countable Borel equivalence relation. (Here Gfg denotes the Polish space of finitely generated groups introduced by Grigorchuk [12]; i.e. the elements of Gfg are the isomorphism types of marked groups (G, c ), where G is a finitely generated group and c is a finite sequence of generators.) This result suggests the project of analyzing the Borel complexity of the isomorphism relation for various restricted classes of finitely generated groups; and the main result in this paper can be regarded as the first step in this analysis for both the class of infinite finitely generated simple groups and the class of infinite finitely generated amenable groups.