Fixed point properties and second bounded cohomology of universal lattices on Banach space

TL;DR

This paper proves that universal lattices in high dimensions have fixed point properties on Banach spaces, strengthening known properties and implications for bounded cohomology, with new generalizations introduced.

Contribution

It establishes fixed point properties (F_B) and a new boundedness property (FF_B) for universal lattices on Banach spaces, extending previous results and introducing novel concepts.

Findings

Universal lattices have property (F_B) for Banach spaces isomorphic to Hilbert spaces.

Universal lattices possess property (FF_B) modulo trivial parts.

Injectivity of the comparison map in bounded cohomology is established under certain conditions.

Abstract

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 Gamma =SL_n (Z[x1,...,xk]) has property (F_B) in the sense of Bader--Furman--Gelander--Monod. Namely, any affine isometric action of Gamma 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)$ for a group: the boundedness property of all affine quasi-actions on B. We name it property (FF_B) and prove that the group Gamma above also has this property modulo trivial part. The conclusion above in particular implies that the comparison map in degree two H^2_b (Gamma; B) \to H^2(Gamma; B) from bounded to ordinary cohomology is injective, provided that the associated linear representation does not contain the trivial representation.