Fixed point property for universal lattice on Schatten classes

Mimura, Masato

doi:10.1090/S0002-9939-2012-11711-3

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ISSN 1088-6826(online) ISSN 0002-9939(print)

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
Posted: May 7, 2012
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Abstract: The special linear group $G=\mathrm {SL}_n (\mathbb{Z}[x_1, \ldots , x_k])$ ( at least and finite) is called the universal lattice. Let be at least , and be any real number in $(1, \infty )$ . The main result is the following: any finite index subgroup of has the fixed point property with respect to every affine isometric action on the space of -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 -setting.

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