HLS groupoids are named after three mathematicians: Nigel Higson, Vincent Lafforgue, and Georges Skandalis, which first appear as counterexamples to the Baum- Connes conjecture for groupoids.

Let (Γ , N = {Nk}) be a residually finite group. We show that the associated HLS groupoid GN (Γ) is a-T-manable if and only if the associated box space □N (Γ) coarsely embeds into Hilbert space. Moreover, we prove that a-T-menability and metric amenability are independent for HLS groupoids: all four possibilities occur among HLS groupoids associated with approximating sequences of the free group F2 . The two separating examples are constructed using relative expansion and twisted spectral-gap techniques, respectively. It is joint work with Hao SONG, Yan WANG and Jiawen ZHANG.