By definition a finitely generated group G is coarsely embedded into Hilbert space if there is a function $F: G\to \ell_2$, such that $\|F(g_n)-F(h_n)\|\to\infty$ iff $d(g_n,h_n)\to\infty$, where $d$ is word metric with respect to some generating set. Is it true that residually finite groups are coarsely embeddable into Hilbert space?
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2$\begingroup$ I think it's an open question $\endgroup$– YCorCommented Jun 23, 2016 at 13:45
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