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Abstract

We first motivate the study of proper isometric actions on Banach spaces,
and we survey some recent results. Then we introduce the class (BP_0) of groups such
that every isometric action on a Banach space, with linear part a
$C_0$-representation, is either proper or bounded. We prove that this class contains
all solvable groups, and all simple algebraic groups over local fields (this is
joint work with Y. de Cornulier and R. Tessera).