  As a strengthening of Kazhdan's property (T) for locally compact groups,
property (TT) was introduced by Burger and Monod. In this paper, we add more
rigidity and introduce property (TTT). This property is suited for the study of
rigidity phenomena for quasi-homomorphisms with noncommutative targets.
Partially upgrading a result of Burger and Monod, we will prove that SL(n,R)
with n at least 3 and their lattices have property (TTT). As a corollary, we
generalize the well-known fact that every homomorphism from such a lattice into
an amenable group or a hyperbolic group has finite image to the extent that it
includes a quasi-homomorphism.
