We investigate fixed point properties for isometric actions of topological groups on a wide class of metric spaces, with a particular emphasis on Hilbert spaces. Instead of requiring the action to be continuous, we assume that it is ``controlled", i.e. compatible with respect to some natural left-invariant coarse structure. For locally compact groups, we prove that these coarse fixed point properties are equivalent to the usual ones, defined for continuous actions. We deduce generalisations of two results of Gromov originally stated for discrete groups. For Polish groups with bounded geometry (in the sense of Rosendal), we prove a version of Serre's theorem on the stability of coarse property FH under central extensions, and apply it to the group of homeomorphisms of the line commuting with integral translations. Finally, we characterise geometric property (T) for sequences of finite Cayley graphs in terms of coarse property FH of a certain ``large'' group. This is a joint work with Jeroen Winkel.
