Uncountable Groups with Property (fh)

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 give two constructions. The first is a (non-explicit) transfinite induction on countable ordinals, embedding every countable group in a group of cardinality א1 with Property (FH). The second construction requires the following definition: we say that a group G has the CIE Property if, somewhat informally, every embedding of a countable subgroup of G into another countable group can be realized inside G. We embed every group in a group with the CIE Property. Using Delzant’s Theorem, such a group has Property (FH). Finally, we prove that, if G is a finite perfect group, and I is a set, then G has Property (FH). We prove, in fact, something stronger. A group is strongly bounded if every isometric action on a metric space has bounded orbits; we show that G even satisfies this property. This strengthens a result of Koppelberg and Tits.