Fixed point properties and second bounded cohomology of universal lattices on Banach spaces

Let B be any L^p space for p ∈ (1, ∞) or any Banach space isomorphic to a Hilbert space, and k ≧ 0 be integer. We show that if n ≧ 4, then the universal lattice Γ = SLn(ℤ[x1,..., xk]) has property (F_B) in the sense of Bader-Furman-Gelander-Monod. Namely, any affine isometric action of Γ on B has a global fixed point. The property of having (F_B) for all B above is known to be strictly stronger than Kazhdan's property (T). We also define the following generalization of property (F_B) for a group: the boundedness property of all affine quasi-actions on B. We name it property (FF_B) and prove that the group Γ above also has this property modulo trivial part. The conclusion above implies that the comparison map in degree two H _b² (Γ;B) → H²(ΓB) from bounded to ordinary cohomology is injective, provided that the associated linear representation does not contain the trivial representation.