Uncountable groups with Property (FH)

Abstract. A group has Property (FH) if every isometric action on a Hilbert space has a fixed point. We exhibit some uncountable groups with Property (FH). In particular, these groups do not have Kazhdan’s Property (T), which is known to be equivalent to Property (FH) for countable groups. Our first examples rely on a theorem of Delzant, which states that every countable group embeds in a group with Property (T). We deduce that every ω1-existentially closed group has Property (FH), so that every group embeds in a group with Property (FH). Next we prove that, if G is a finite perfect group, and I is a set, then G I has Property (FH). We actually prove something stronger. We say that a group is strongly bounded if every isometric action on a metric space has bounded orbits. This latter property is equivalent, for infinite groups, to the so-called uncountable strong cofinality. We show that G I is strongly bounded. This strengthens a result of Koppelberg and Tits. In this paper, all groups are discrete. 1.