Uncountable autohomeomorphism groups of thin-tall locally compact scattered spaces (Q1205162)

A space \(X\) is scattered if every subset has a dense set of isolated points. Locally compact scattered spaces will have a base of clopen sets. A Boolean algebra is superatomic if every subalgebra is atomic. The class of locally compact scattered spaces corresponds exactly to the class of open subsets of Stone spaces of superatomic Boolean algebras. Many authors in both disciplines have been interested in these structures. A locally compact scattered space (or corresponding Boolean algebra) is said to be thin-tall if it is uncountable and every co-countable subset is separable. Rather than investigating the group of autohomeomorphisms of such a space it is more informative to consider the quotient by the subgroup of those which fix are the identity on a co-countable set. The author is working towards characterizing those groups which are isomorphic to such quotient groups. It is known that each countable group is such a group. The results of this paper demonstrate that it is consistent (and follows from CH) that a large collection of uncountable groups (including the free group and free abelian group with \(\omega_ 1\) generators) are also representable in this way. Koppelberg developed a technique for representing countable groups which involved suitably representing countable groups in the permutation group of the integers. The technique in the present paper involves showing that the above mentioned free groups satisfy a property which allows the author to significantly modify Koppelberg's technique by inductively lifting suitable embeddings of countable subgroups of the desired group to larger and larger countable subgroups.