Property $(TT)$ modulo $T$ and homomorphism superrigidity into mapping class groups

Abstract

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).