Hall's universal group is a subgroup of the abstract commensurator of a free group (Q6588724)

Let \( G \) be a group. Consider the set \( \Omega(G) \) of all isomorphisms between subgroups of finite index of \( G \). Two such isomorphisms \( \varphi_1 : H_1 \to H_1' \) and \( \varphi_2 : H_2 \to H_2' \) are called \textit{equivalent}, written \( \varphi_1 \sim \varphi_2 \), if there exists a subgroup \( H \) of finite index in \( G \) such that both \( \varphi_1 \) and \( \varphi_2 \) are defined on \( H \) and \( \varphi_1|_H = \varphi_2|_H \). For any two isomorphisms \( \alpha : G_1 \to G_1' \) and \( \beta : G_2 \to G_2' \) in \( \Omega(G) \), we define their product \( \alpha \beta : \alpha^{-1}(G_1' \cap G_2) \to \beta(G_1' \cap G_2) \) in \( \Omega(G) \). The factor-set \( \Omega(G) /\! \sim \) inherits the multiplication \( [\alpha][\beta] = [\alpha \beta] \) and is a group, called the \textit{abstract commensurator} of \( G \) and denoted \( \mathrm{Comm}(G) \).\N\NLet \(\mathbb{F}_k\) be the free group of rank \(k\geq2\). It is easy to see that \(\mathrm{Comm}(\mathbb{F}_k)\simeq\mathrm{Comm}(\mathbb{F}_2)\) for every \(k\geq2\). Therefore, one usually refers to \(\mathrm{Comm}(\mathbb{F}_k)\), say, as \(\mathrm{Comm}(\mathbb{F})\).\N\NRecall that a countable locally finite group, which contains an isomorphic copy of every finite group and such that any two isomorphic finite subgroups of it are conjugate, is called an \textit{Hall universal group}. \textit{P. Hall} [J. Lond. Math. Soc. 34, 305--319 (1959; Zbl 0088.02301)] proved that such a group exists and that it is unique up to isomorphisms.\N\NMaking use of the realization of the abstract commensurator \(\mathrm{Comm}(\mathbb{F})\) as the homotopy equivalence group of a full solenoid \(\hat{\Gamma}\), namely the inverse limit of all finite-sheeted covers of a finite connected graph \(\Gamma\) with non-abelian fundamental group, the authors prove the following\N\NMain Theorem. Hall's universal group can be embedded into \(\mathrm{Comm}(\mathbb{F})\).\N\NFrom this result and from the properties of Hall's universal group in particular, one immediately gets the following\N\NCorollary. Every countable locally finite group can be embedded into \(\mathrm{Comm}(\mathbb{F})\).