Embeddable box spaces of free groups (Q464136)

The main goal of the paper is to generalize the following result of \textit{G. Arzhantseva} et al. [Geom. Funct. Anal. 22, No. 1, 22--36 (2012; Zbl 1275.46013)]: Let \(\mathbb F_n\) be a free group with \(n\) generators; \(G^{(2)}\) denotes the subgroup of a group \(G\) generated by the squares of elements of \(G\); and define iteratively \(\Gamma_0=\mathbb F_n\) and \(\Gamma_{i+1}=\Gamma_i^{(2)}\). Then the quotients \(\{\mathbb F_n/\Gamma_i\}_{i=1}^\infty\) endowed with the metrics induced by the images of the generators of \(\mathbb F_n\) are uniformly coarsely embeddable into an (infinite-dimensional) Hilbert space. The author proves that the statement remains true in the case where, instead of \(\Gamma_{i+1}=\Gamma_i^{(2)}\), we let \(\Gamma_{i+1}=[\Gamma_i,\Gamma_i]\Gamma_i^{(m)}\), where \(G^{(m)}\) denotes the subgroup of a group \(G\) generated by the \(m\)th powers of elements of \(G\), and \(m\in \mathbb N\), \(m\geq 2\). The result is proved in a somewhat more general context.