  Every homomorphism from finite index subgroups of a universal lattices to
mapping class groups of orientable surfaces (possibly with punctures), or to
outer automorphism groups of finitely generated nonabelian free groups must
have finite image. Here the universal lattice denotes the special linear group
G=SL_m(Z[x1,...,xk]) with m at least 3 and k finite. Moreover, the same results
hold ture if universal lattices are replaced with symplectic universal lattices
Sp_{2m}(Z[x1,...,xk]) with m at least 2. These results can be regarded as a
non-arithmetization of the theorems of Farb--Kaimanovich--Masur and
Bridson--Wade. A certain measure equivalence analogue is also established. To
show the statements above, we introduce a notion of property (TT)/T ("/T"
stands for "modulo trivial part"), which is a weakening of property (TT) of N.
Monod. Furthermore, symplectic universal lattices Sp_{2m}(Z[x1,...,xk]) with m
at least 3 has the fixed point property for L^p-spaces for any p in
(1,infinity).
