Fixed point property for universal lattice on Schatten classes

The special linear group G=SL_n(Z[x1,…,xk]) (n at least 3 and k finite) is called the universal lattice. Let n be at least 4, p be any real number in (1,\infty). The main result is the following: any finite index subgroup of G has the fixed point property with respect to every affine isometric action on the space of p-Schatten class operators. It is in addition shown that higher rank lattices have the same property. These results are generalization of previous theorems repsectively of the author and of Bader–Furman–Gelander–Monod, which treated commutative Lp-setting.