Property (TT)/T and homomorphism superrigidity to mapping class groups
We show the following: 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=SLm(Z[x1,...,xk]) with m at least 3 and k any finite. We show that the results above remain true when universal lattices are replaced with symplectic universal lattices Sp2m(Z[x1,..., xk]) with m at least 2. These results can be regarded as a non-arithmetization of homomorphism superrigidity of Farb-Kaimanovich-Masur and Bridson-Wade for higher rank lattices. Moreover, we obtain a similar assertion in a certain measure equivalent setting. To show the statements above, we introduce a notion of property (TT)/T (``/T" stands for ``modulo trivial part"), which is a strengthing of Kazhdan's property (T) and a weakening of property (TT) of N. Monod. The following is the precise statement of our results: see arXiv:1106.3769
Theorem: Let Γ be a quotient group of a universal lattice or of a symplectic universal lattice. Suppose a countable group Λ satisfies the following two conditions:
(i) Λ is measure equivalent to Γ
(ii) For an ergodic ME-coupling Ω of (Γ,Λ), there exists a Λ -fundamental domain X(=Ω/Λ) in Ω such that the associated ME-cocycle 
α :Γ *X → Λ satisfies the L2-condition. Namely, for any γ in Γ, |α (γ, -)|Λ is in L2(X), where | - |Λ denotes a word metric on Λ with respect to a finite generating set for Λ. Then every homomorphism from Λ into MCG(Σg,l) (g,l nonnegative); or into Out(Fn) (n>1) has finite image. In particular, the following holds true: let Γ be a finite index subgroup of Em(A), m at least 3; or of Ep2m(A), m at least 2, where A is a finitely generated commutative (unital and associative) ring. Suppose G be a locally compact and second countable group and G contains a group isomorphic to Γ as a lattice. Then for any cocompact lattice Λ in G, every homomorphism from Λ into MCG(Σg,l); or into Out(Fn) has finite image. Here Em(A) denotes the elementary group, namely, the multiplicative group generated by elementary matrices in Mm(A), and Ep2m(A) denotes the elementary symplectic group.