Confined subgroups of Thompson's group $F$ and its embeddings into wobbling groups

TL;DR

This paper characterizes confined subgroups of Thompson's group F and explores its embeddings into wobbling groups, revealing conditions for subexponential growth and non-embeddability.

Contribution

It provides a new characterization of confined subgroups of F and establishes criteria for embedding F into wobbling groups based on growth properties.

Findings

Orbital graphs of points fixed by the commutator subgroup have subexponential growth.

F cannot be embedded into wobbling groups of graphs with uniformly subexponential growth.

Characterization of confined subgroups of Thompson's group F.

Abstract

We obtain a characterisation of confined subgroups of Thompson's group $F$. As a result, we deduce that orbital graph of a point under action of $F$ has uniformly subexponential growth if and only if this point is fixed by the commutator subgroup. This allows us to prove non-embeddability of $F$ into wobbling groups of graphs with uniformly subexponential growth.