Some Applications of Tree-Limits to Groups. Part 1

Abstract

Sharper applications to group theory are given of an elegant construction -the tree-limit-which S. Shelah circulated as a preprint in 1977 and used to obtain oo-co-enlargements to power 2@ of certain countable homogeneous groups and skew fields. In this paper we enlarge the class of groups to which this construction can be interestingly applied and we obtain permutation representations of countable degree of the groups; we obtain uncountable subgroup-incomparability for enlargements of countable existentially closed groups and even in nonhomogeneous cases we obtain the very strong direct limit (which implies oo-X-equivalence (see (1.0)) of the permutation representations). We are able to control the permutation representations which get stretched by the by varying the point-stabilizer subgroups (see (5.5)). In particular we can archetypally stretch in 2@ subgroup-incomparable ways any homogeneous permutation representation of a countable locally finite group in which every finite subgroup has infinitely many regular orbits (Theorem 4). We discuss cases where tree-limits are subgroups of inverse limits. 1. Orientation and results. Here we give a number of new group-theoretical applications of an elegant construction of S. Shelah [11] which we call the tree-limit construction. This construction was circulated in preprint form in 1977 by Shelah (in fact, the present author had to decline an offer of coauthorship at that time). The version [3] is formally quite different from Shelah's original presentation which we have chosen to follow because of its naturalness. The progress we have made comes mainly from the special types of amalgamations we use which provide an easy way to verify Shelah's crucial lemma giving the tree limit group power 2@ and which also provide a way of obtaining an embedding of the group in Sym(X). We are also able to relax the requirement of homogeneity on the groups which get enlarged by the by making use of special properties of wreath products. We also formulate and study the countable direct limit property which the group often satisfies. This, we feel, is of such interest that it merits some new terminology. (1.0) DEFINITION. Suppose G is a countable group and H is a group. We say that H is an archetypal limit of G iff H = U{Fs Is E P(I)} where P(I) is the set of finite subsets of I, FS is finitely generated for all s E P(I), s c t E P(I) implies F5 c Ft, and there exists S C I with ISI = X such that, for all S C T C I with