## Abstract

Let B be any L^{p} space for p ∈ (1, ∞) or any Banach space isomorphic to a Hilbert space, and k ≧ 0 be integer. We show that if n ≧ 4, then the universal lattice Γ = SLn(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}) for a group: the boundedness property of all affine quasi-actions on B. We name it property (FF_{B}) and prove that the group Γ above also has this property modulo trivial part. The conclusion above implies that the comparison map in degree two H _{b}^{2} (Γ;B) → H^{2}(ΓB) from bounded to ordinary cohomology is injective, provided that the associated linear representation does not contain the trivial representation.

Original language | English |
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Pages (from-to) | 115-134 |

Number of pages | 20 |

Journal | Journal fur die Reine und Angewandte Mathematik |

Issue number | 653 |

DOIs | |

Publication status | Published - 2011 Apr |

Externally published | Yes |

## ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics