Topological full groups of minimal subshifts and the classification problem for finitely generated complete groups (Q1733172)

The aim of this paper is to analyze the Borel complexity of the isomorphism relation on $\mathcal{FG}_{\mathrm{cmp}}$, where $\mathcal{FG}$ denotes the Polish space of marked finitely generated groups and $\mathcal{FG}_{\mathrm{cmp}}=\{G\in\mathcal{FG}: G$ is complete$\}$. The main result is as follows: Theorem. The isomorphism relation on the space $\mathcal{FG}_{\mathrm{cmp}}$ of finitely generated complete groups is not smooth. Recall that a Borel equivalence relation $E$ on a Polish space $X$ is smooth if $E$ is Borel reducible to the identity relation Id$_Y$ on some (equivalently every) uncountable standard Borel space $Y$. It should be noticed that in this paper the author is only concerned with countable Borel equivalence relations, i.e., Borel equivalence relations $E$ such that every $E$-equivalence class is countable. The author proves the above theorem by making use of the following results on the structure of the topological full groups of minimal subshifts. If $(X,T)$ is a minimal subshift and $[[T]]$ is the corresponding topological full group, then the commutator subset $[[T]]'$ is an infinite finitely generated simple group and hence $\mathrm{Aut}([[T]]')$ is a (not necessarily finitely generated) complete group. Furthermore, if $(Y,S)$ is another minimal subshift, then $\mathrm{Aut}([[T]]')\simeq\mathrm{Aut}([[S]]'])$ if and only if $(X,T)$ and $(Y,S)$ are flip conjugate. Therefore, in order to prove the above theorem, it is enough to find a standard Borel space $\mathcal{M}$ of minimal subshifts such that $\mathrm{Aut}([[T]]')$ is finitely generated for each $(X,T)\in\mathcal{M}$ and the flip conjugacy relation on $M$ is nonsmooth.