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

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Published/Copyright: January 12, 2011
Journal für die reine und angewandte Mathematik
From the journal Volume 2011 Issue 653

Abstract

Let B be any Lp 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 (FB) 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 (FB) for all B above is known to be strictly stronger than Kazhdan's property (T). We also define the following generalization of property (FB) for a group: the boundedness property of all affine quasi-actions on B. We name it property (FFB) 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 from bounded to ordinary cohomology is injective, provided that the associated linear representation does not contain the trivial representation.

Received: 2009-05-14
Revised: 2010-10-28
Published Online: 2011-01-12
Published in Print: 2011-April

© Walter de Gruyter Berlin · New York 2011

Articles in the same Issue

  1. Modular symbols for reductive groups and p-adic Rankin–Selberg convolutions over number fields
  2. Wandering vectors and the reflexivity of free semigroup algebras
  3. On the equivariant main conjecture for imaginary quadratic fields
  4. Fixed point properties and second bounded cohomology of universal lattices on Banach spaces
  5. On the existence of certain affine buildings of type E6 and E7
  6. Endotrivial modules for finite group schemes
  7. Curvature bound for curve shortening flow via distance comparison and a direct proof of Grayson's theorem
  8. On a class of fully nonlinear flows in Kähler geometry
  9. Colocalizing subcategories of D(R)
https://doi.org/10.1515/crelle.2011.021

Articles in the same Issue

  1. Modular symbols for reductive groups and p-adic Rankin–Selberg convolutions over number fields
  2. Wandering vectors and the reflexivity of free semigroup algebras
  3. On the equivariant main conjecture for imaginary quadratic fields
  4. Fixed point properties and second bounded cohomology of universal lattices on Banach spaces
  5. On the existence of certain affine buildings of type E6 and E7
  6. Endotrivial modules for finite group schemes
  7. Curvature bound for curve shortening flow via distance comparison and a direct proof of Grayson's theorem
  8. On a class of fully nonlinear flows in Kähler geometry
  9. Colocalizing subcategories of D(R)
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