Locally solvable subgroups of PLo(I) are countable

TL;DR

This paper proves that all locally solvable subgroups of the group PLo(I) are countable, and as a consequence, certain uncountable wreath products cannot embed into PLo(I).

Contribution

It establishes the countability of locally solvable subgroups of PLo(I) and rules out embeddings of specific uncountable wreath products.

Findings

Locally solvable subgroups of PLo(I) are countable.

Uncountable wreath products of Z do not embed into PLo(I).

Abstract

We show every locally solvable subgroup of PLo(I) is countable. A corollary is that an uncountable wreath product of copies of $\mathbb{Z}$ with itself does not embed into PLo(I).