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

TL;DR

This paper proves that homomorphisms from certain universal lattices to mapping class groups or outer automorphism groups have finite images, introducing a new property (TT)/T and establishing fixed point properties for symplectic universal lattices.

Contribution

It introduces property (TT)/T, a new weakening of property (TT), and applies it to prove superrigidity results for universal lattices into mapping class groups.

Findings

Homomorphisms from universal lattices to mapping class groups have finite images.

Universal lattices satisfy a new property (TT)/T, leading to superrigidity.

Symplectic universal lattices have fixed point properties for L^p-spaces.

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).