Locally Homogeneous Actions and Universal Extensions of Groups

We study further the class of universal locally finite (u.l.f.) M ‐extensions E of a countable group G in which the centralizer C E ( G ) is countable. These were introduced in [8]: M ⊂ Aut G /Inn G is the fixed l.f. group of outer automorphisms of G realized in E and E / G is locally finite and has the ‘ω‐injective property’ with respect to finite M ‐extensions F of G , that is, F / GC p ( G ) ⊂ M and F / G is finite. Here we shall prove that for most G , if | M | = N 1 andN 1 = 2 N 0then there are N 2 non‐isomorphic u.l.f. M ‐extensions E of G with | C E ,( G )| = N 0 . We also complete the presentation of [8] by showing that every countably infinite l.f. group L has 2 N 0 locally homogeneous actions (on U ( A ), the countable u.l.f. central extension of any countable periodic abelian group A [6, 11]) which extend any given action of L on A and such that no infinite subactions of any two of these locally homogeneous actions are equivalent. We also construct 2 N 0 nonembeddable locally homogeneous actions of many L on U ( A ) which have ∞‐ω‐equivalent actions on U ( A ) of power 2 M 0 . Finally we produce 2 N 1 complete u.l.f. groups each of which has 2 N 0 inequivalent universal actions on the countable u.l.f. group. (Universal actions were introduced in [8, Theorem 8] and are a categorical type of locally homogeneous action; the method of obtaining complete u.l.f. groups is that of [4].)