Free limits of Thompson's group $F$

TL;DR

This paper constructs sequences of markings of Thompson's group F that converge to free groups on n generators, providing new insights into the group's limits within the space of marked groups.

Contribution

It introduces explicit sequences of markings of F that converge to free groups, and provides presentations for these limit groups.

Findings

Sequences of markings of F converge to free groups for n ≥ 3

Presentations for limit groups of certain natural sequences

New understanding of the boundary of Thompson's group F in the space of marked groups

Abstract

We produce a sequence of markings $S_k$ of Thompson's group $F$ within the space ${\mathcal G}_n$ of all marked $n$-generator groups so that the sequence $(F,S_k)$ converges to the free group on $n$ generators, for $n \geq 3$. In addition, we give presentations for the limits of some other natural (convergent) sequences of markings to consider on $F$ within ${\mathcal G}_3$, including $(F,\{x_0,x_1,x_n\})$ and $(F,\{x_0,x_1,x_0^n\})$.