The Free Group of Rank 2 is a Limit of Thompson's Group F

TL;DR

This paper demonstrates that the free group of rank 2 can be approximated as a limit of Thompson's group F by constructing a sequence of generating pairs with relations growing arbitrarily long.

Contribution

It introduces a novel method to approximate the free group of rank 2 using Thompson's group F through specific generating pairs.

Findings

The free group of rank 2 is a limit of 2-marked groups derived from Thompson's group F.

A sequence of generating pairs for F with relations tending to infinity was constructed.

Thompson's group F can approximate free groups in the space of marked groups.

Abstract

We show that the free group of rank 2 is a limit of 2-markings of Thompson's group F in the space of all 2-marked groups. More specifically, we find a sequence of generating pairs for F so that as one goes out the sequence, the length of the shortest relation satisfied by the generating pair goes to infinity.