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Fixed point property for universal lattice on Schatten classes

Author: Masato Mimura
Journal: Proc. Amer. Math. Soc. 141 (2013), 65-81
MSC (2010): Primary 20F65, 20J06; Secondary 20H25, 22D12
Published electronically: May 7, 2012
MathSciNet review: 2988711
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Abstract: The special linear group $ G=\mathrm {SL}_n (\mathbb{Z}[x_1, \ldots , x_k])$ ($ n$ at least $ 3$ and $ k$ finite) is called the universal lattice. Let $ n$ be at least $ 4$, and $ p$ be any real number in $ (1, \infty )$. The main result is the following: any finite index subgroup of $ G$ has the fixed point property with respect to every affine isometric action on the space of $ p$-Schatten class operators. It is in addition shown that higher rank lattices have the same property. These results are a generalization of previous theorems respectively of the author and of Bader-Furman-Gelander-Monod, which treated a commutative $ L^p$-setting.

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Additional Information

Masato Mimura
Affiliation: Graduate School of Mathematical Sciences, University of Tokyo, Komaba, Tokyo, 153-8914, Japan – and – École Polytechnique Fédérale de Lausanne, SB–IMB–EGG, Station 8, Bâtiment MA, Lausanne, Vaud, CH-1015, Switzerland
Address at time of publication: Graduate School of Mathematical Sciences, University of Tokyo, Komaba, Tokyo, 153-8914, Japan – and – Institut de Mathématiques, Faculté des Sciences, Université de Neuchâtel, Rue Emile-Argand 11, CH-2000 Neuchâtel, Switzerland

Keywords: Fixed point property, Kazhdan’s property (T), Schatten class operators, noncommutative $L^{p}$-spaces, bounded cohomology
Received by editor(s): October 22, 2010
Received by editor(s) in revised form: June 7, 2011
Published electronically: May 7, 2012
Additional Notes: The author is supported by JSPS Research Fellowships for Young Scientists No. 20-8313.
Communicated by: Marius Junge
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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