Box spaces of the free group that neither contain expanders nor embed into a Hilbert space (Q1790251)

The first examples of metric spaces with bounded geometry which do not embed coarsely into a Hilbert space were suggested by \textit{M. Gromov} [in: GAFA 2000. Visions in mathematics -- Towards 2000. Proceedings of a meeting, Tel Aviv, Israel, August 25--September 3, 1999. Part I. Basel: Birkhäuser. 118--161 (2000; Zbl 1006.53035)], namely, he observed that a metric space $M$ of bounded geometry containing a sequence $\{G_n\}_{n=1}^\infty$ of expanders weakly (this means that there exist maps $f_n:G_n\to M$ with uniformly bounded Lipschitz constants and cardinalities of pre-images $f_n^{-1}(m)$ of points $m\in M$ being $o(|G_n|)$) does not admit a coarse embedding into a Hilbert space. This observation plays a very important role in topology and group theory (see [\textit{M. Gromov}, Geom. Funct. Anal. 13, No. 1, 73--146 (2003; Zbl 1122.20021)] and [\textit{N. Higson} et al., Geom. Funct. Anal. 12, No. 2, 330--354 (2002; Zbl 1014.46043)]). \par For a long time, weakly embeddable expanders were the only known obstacle for coarse embeddability of metric spaces with bounded geometry into a Hilbert space. (Without the requirement of bounded geometry other obstacles are known, for example, metric cotype, see [\textit{M. Mendel} and \textit{A. Naor} [Ann. Math. (2) 168, No. 1, 247--298 (2008; Zbl 1187.46014)].) In this connection, it was asked, see Problem 11.9 in [the reviewer, Metric embeddings. Bilipschitz and coarse embeddings into Banach spaces. Berlin: de Gruyter (2013; Zbl 1279.46001)]: are there other obstacles for embeddings of bounded geometry metric spaces into $\ell_2$? \par This question was answered in the negative in a very important paper by \textit{G. Arzhantseva} and \textit{R. Tessera} [Geom. Funct. Anal. 25, No. 2, 317--341 (2015; Zbl 1325.46022)]. \par The present paper is devoted to another construction of spaces giving a negative answer to this question. The authors do this by constructing two different box spaces of the free group with three generators -- one of which is an expander and the other of which is coarsely embeddable into a Hilbert space, and then by ``mixing'' these constructions. The main part of the proof is to show that both box spaces and their mixture can be constructed in such a way that the mixture will inherit from the sequence of expanders coarse nonembeddability into a Hilbert space, and from coarsely embedded sequences lack of weakly embeddable expanders. The construction sheds a new light on the structure of spaces with bounded geometry which do not admit coarse embeddings into a Hilbert space.