Fixed point property for universal lattice on schatten classes

The special linear group G = SL _n(ℤ[x ₁,..., x _k]) (n at least 3 and k finite) is called the universal lattice. Let n be at least 4, and p be any real number in (1,∞). 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 a generalization of previous theorems respectively of the author and of Bader-Furman-Gelander- Monod, which treated a commutative L ^p-setting.