Near full groups of bounded type, II

Abstract

We study when a finitely generated group of bounded type contains the AF alternating group of the tail groupoid of its defining Bratteli diagram. This containment is an input to the full group completion theorem of Part I. We give sufficient conditions for this containment, in which selectors separate inherited diagonal actions, and an exact criterion for finite regular coverings. In the covering case, the intersection with the AF alternating group is the centralizer of the deck transformations commuting with the adjoined AF permutations. The formula classifies all subgroups between the canonical lift of the base AF alternating group and the AF alternating group of the covering. In particular, the lift is maximal exactly when the deck group has prime order, and cyclic coverings yield finitely generated groups whose intersections with the AF alternating group of the covering realize these subgroups. We identify fragmentations that produce the required selectors. For periodic fragmentations of the directed generator in the modified Fabrykowski--Gupta group, surjectivity of the activity subspace onto each pair of consecutive coordinates gives containment with absorption delay at most two. A five-periodic example also satisfies this condition, although its activity subspace has dual distance two. The original four-periodic fragmentation group has index four in its topological full group.

arXiv commentPart II of a two-part series; Part I: arXiv:2607.25729. Revised and Expanded: added subgroup classification for finite regular coverings and new periodic fragmentation results. Refined proofs and attribution