Abstract
Let B be any Lp 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 (FB) 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 (FB) for all B above is known to be strictly stronger than Kazhdan's property (T). We also define the following generalization of property (FB) for a group: the boundedness property of all affine quasi-actions on B. We name it property (FFB) 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 
from bounded to ordinary cohomology is injective, provided that the associated linear representation does not contain the trivial representation.
© Walter de Gruyter Berlin · New York 2011
