Two-chains and square roots of Thompson's group $F$

TL;DR

This paper explores the structure and diversity of certain subgroups of homeomorphisms related to Thompson's group F, revealing uncountably many types with complex properties and behaviors of their square roots.

Contribution

It introduces a new class of subgroups with rich isomorphism types, demonstrating their complex actions and properties, and addresses open questions about square roots of Thompson's group F.

Findings

Uncountably many isomorphism types of subgroups with specific properties.

Existence of subgroups with nonabelian free subgroups and complex actions.

Many groups of homeomorphisms can have highly complicated square roots.

Abstract

We study two--generated subgroups $\langle f,g\rangle<\mathrm{Homeo}^+(I)$ such that $\langle f^2,g^2\rangle$ is isomorphic to Thompson's group $F$, and such that the supports of $f$ and $g$ form a chain of two intervals. We show that this class contains uncountably many isomorphism types. These include examples with nonabelian free subgroups, examples which do not admit faithful actions by $C^2$ diffeomorphisms on $1$--manifolds, examples which do not admit faithful actions by $PL$ homeomorphisms on an interval, and examples which are not finitely presented. We thus answer questions due to M. Brin. We also show that many relatively uncomplicated groups of homeomorphisms can have very complicated square roots, thus establishing the behavior of square roots of $F$ as part of a general phenomenon among subgroups of $\mathrm{Homeo}^+(I)$.