On the acylindrical hyperbolicity of the tame automorphism group of SL2(C)

We introduce the notion of über-contracting element, a strengthening of the notion of strongly contracting element which yields a particularly tractable criterion to show the acylindrical hyperbolicity, and thus a strong form of non-simplicity, of groups acting on non-locally compact spaces of arbitrary dimension. We also give a simple local criterion to construct über-contracting elements for groups acting on complexes with unbounded links of vertices.As an application, we show the acylindrical hyperbolicity of the tame automorphism group of SL2(C)SL2(C), a subgroup of the 3-dimensional Cremona group Bir(P3(C))Bir(P3(C)), through its action on a CAT(0) square complex recently introduced by Bisi–Furter–Lamy.