Some applications of tree-limits to groups. I

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 $\infty$-$\omega$-enlargements to power ${2^\omega }$ 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 tree-limit groups; we obtain uncountable subgroup-incomparability for enlargements of countable existentially closed groups and even in nonhomogeneous cases we obtain the very strong "archetypal direct limit property" (which implies $\infty$-$\omega$-equivalence (see (1.0)) of the permutation representations). We are able to control the permutation representations which get stretched by the tree-limit by varying the point-stabilizer subgroups (see (5.5)). In particular we can archetypally stretch in ${2^\omega }$ 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.