TY - JOUR

T1 - Fixed point properties and second bounded cohomology of universal lattices on Banach spaces

AU - Mimura, Masato

PY - 2011/4

Y1 - 2011/4

N2 - 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 H b2 (Γ;B) → H2(ΓB) from bounded to ordinary cohomology is injective, provided that the associated linear representation does not contain the trivial representation.

AB - 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 H b2 (Γ;B) → H2(ΓB) from bounded to ordinary cohomology is injective, provided that the associated linear representation does not contain the trivial representation.

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U2 - 10.1515/CRELLE.2011.021

DO - 10.1515/CRELLE.2011.021

M3 - Article

AN - SCOPUS:79956352160

SN - 0075-4102

SP - 115

EP - 134

JO - Journal fur die Reine und Angewandte Mathematik

JF - Journal fur die Reine und Angewandte Mathematik

IS - 653

ER -