Fixed point property for universal lattice on Schatten classes

Mimura, Masato

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

Fixed point property for universal lattice on Schatten classes
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by Masato Mimura

Proc. Amer. Math. Soc. 141 (2013), 65-81

DOI: https://doi.org/10.1090/S0002-9939-2012-11711-3

Published electronically: May 7, 2012

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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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