Groups not acting on compact metric spaces by homeomorphisms (Q2407271)

The author principally studies embedability properties of a particular class of groups given as direct products of uncountably many non-abelian groups. It is shown that such groups cannot appear as subgroups of the homeomorphism group of a compact metric space. It is noted that the proof of the main result extends equally well from homeomorphism groups of compact metric spaces to any second countable Hausdorff topological group. Two interesting examples are discussed: the group of invertible bounded operators on an infinite dimensional Banach space and the group of germs at infinity of homeomorphisms of the real line. In both cases, it is shown that the considered groups admit a subgroup isomorphic to an uncountable direct sum of non-abelian groups. The ability for a group to admit a subgroup isomorphic to an uncountable direct sum of non-abelian groups is a special instance of a more general property (introduced by the author) that a group either does or does not satisfy; one property for every (reduced) word in the free group generated by two elements. It is shown that if a group satisfies any of these more general properties for some non-trivial reduced word, then it cannot embed as a subgroup of a second countable Hausdorff topological group.