Home List of Papers

List of Papers

[1] Fixed point properties and second bounded cohomology of universal lattices on Banach spaces,

J. reine angew. Math. (Crelle's journal), Vol. 2011, No. 653, 115--134, 2011; arXiv:0904.4650

Let B be any Lp space for p in (1, infty) or any Banach space isomorphic to a Hilbert space, and k be a nonnegative integer. We show that if n is at least 4, then the universal lattice Γ=SL_n(Z[x1,…, xk]) has property (F_B) in the sense of Bader--Furman--Gelander--Monod. Namely, any affine isometric action of Γ on B has a global fixed point. The property of having (F_B) for all B above is known to be strictly stronger than Kazhdan’s property (T). We also define the following generalization of property (F_B): the boundedness property of all affine isometric quasi-actions on B. We name it property (FF_B) and prove that the group Γ above also has this property for non-trivial linear part. The conclusion above implies that the comparison map H^2_b (Γ;B) -> H^2 (Γ;B) from bounded to ordinary cohomology is injective, provided that the associated linear representation does not contain the trivial representation.

[2] On quasi-homomorphisms and commutators in the special linear group over a euclidean ring,

Int. Math. Res. Not. IMRN, Vol. 2010, No. 18, 3519--3529, 2010; arXiv:0911.1341

We prove that for any euclidean ring R and n at least 6, Γ=SL_n(R) has no unbounded quasi-homomorphisms. By Bavard's duality theorem, this means that the stable commutator length vanishes on Γ. The result is particularly interesting for R = F[x] for a certain field F (such as the field C of complex numbers), because in this case the commutator length on Γ is known to be unbounded. This answers a question of M. Abert and N. Monod (ICM, 2006) for n at least 6.

[3] Fixed point property for universal lattice on Schatten classes,

to appear in Proc. Amer. Math. Soc.; arXiv:1010.4532

The special linear group G=SL_n (Z[x1,...,xk]) (n at least 3 and k finite) is called the universal lattice. Let n be at least 4, p be any real number in the open interval (1, infty), and C_p denote the space of p-Schatten class operators on a separable Hilbert space. The main result is the following: G has the fixed point property with respect to every affine isometric action on C_p. Moreover, under the additional assumption below, the comparison map in degree 2 from bounded to ordinary cohomology of G with C_p isometric coefficient is injective: here the associated isometric linear representation is assumed not to have non-zero G-invariant vectors.

[4] Property $(TT)$ modulo $T$ and homomorphism superrigidity into mapping class groups,

Preprint, 2011; arXiv:1106.3769

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).