Locally Solvable Subgroups of PLo(I)

TL;DR

This paper characterizes locally solvable subgroups of PLo(I), constructs uncountably many non-isomorphic such subgroups of Thompson's Group F, and develops tools for understanding their embeddings and invariants.

Contribution

It proves that locally solvable subgroups of PLo(I) are countable and constructs uncountably many non-isomorphic locally solvable subgroups of Thompson's Group F.

Findings

Locally solvable subgroups of PLo(I) are countable.

Constructed uncountably many non-isomorphic locally solvable subgroups of Thompson's Group F.

Developed machinery for understanding embeddings and invariants of these groups.

Abstract

We show that locally solvable subgroups of PLo(I) are countable. Then for each countable ordered set, we construct a locally solvable subgroup of Thompson's Group F. We develop machinery for understanding embeddings from solvable subgroups into solvable subgroups. Finally, we apply this machinery to show the ordered sets used in our construction are invariant under isomorphisms between the groups constructed. Therefore, we effectively distinguish the groups and provide uncountably many non-isomorphic locally solvable, hence elementary amenable, subgroups of Thompson's Group F.