Property (TT)/T and homomorphism superrigidity to mapping
class groups

Masato MIMURA

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=SL_{m}(Z[x_{1},...,x_{k}])
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 Sp_{2m}(Z[x_{1},..., x_{k}])
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 L^{2}-condition. Namely, for any γ in Γ, |α (γ,
-)|_{Λ} is in L^{2}(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(Σ_{g,l}) (g,l
nonnegative); or into Out(F_{n}) (n>1) has finite image. In
particular, the following holds true: let Γ be a finite index subgroup
of E_{m}(A), m at least 3; or of Ep_{2m}(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(Σ_{g}_{,l}); or into Out(F_{n})
has finite image.
Here E_{m}(A) denotes the elementary group, namely, the
multiplicative group generated by elementary matrices in M_{m}(A),
and Ep_{2m}(A) denotes the elementary symplectic group.