We show that
every non-precompact topological group admits a fixed point-free continuous action by affine isometries on a suitable Banach space. Thus, precompact groups are defined by the fixed point property for affine isometric actions on Banach spaces. For separable topological groups, in the above statements it is enough to consider affine actions on one particular Banach space: the unique Banach space envelope $\langle\mathbb U\rangle$ of the universal Urysohn metric space $\mathbb U$, known as the Holmes space. At the same time, we show that Polish groups need not admit topologically proper (in particular, free) affine isometric actions on Banach spaces (nor even on complete metric spaces): this is the case for the unitary group $U(\ell^2)$ with strong operator topology, the infinite symmetric group $S_\infty$, etc.
@article{1267798497,
author = {Nguyen Van Th\'e, Lionel and Pestov, Vladimir G.},
title = {Fixed point-free isometric actions of topological groups on Banach spaces},
journal = {Bull. Belg. Math. Soc. Simon Stevin},
volume = {17},
number = {1},
year = {2010},
pages = { 29-51},
language = {en},
url = {http://dml.mathdoc.fr/item/1267798497}
}
Nguyen Van Thé, Lionel; Pestov, Vladimir G. Fixed point-free isometric actions of topological groups on Banach spaces. Bull. Belg. Math. Soc. Simon Stevin, Tome 17 (2010) no. 1, pp. 29-51. http://gdmltest.u-ga.fr/item/1267798497/