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

Abstract

Let B be any Lp space for p in (1,infty) or any Banach space isomorphic to a Hilbert space, and k be a nonnegative integer. We show that if n is at least 4, then the universal lattice Gamma =SL_n (Z[x1,...,xk]) has property (F_B) in the sense of Bader--Furman--Gelander--Monod. Namely, any affine isometric action of Gamma 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 Gamma above also has this property modulo trivial part. The conclusion above in particular implies that the comparison map in degree two H^2_b (Gamma; B) \to H^2(Gamma; B) from bounded to ordinary cohomology is injective, provided that the associated linear representation does not contain the trivial representation.