The Andreadakis Problem for the McCool groups

TL;DR

This paper investigates the Andreadakis problem for McCool groups, demonstrating that the equality fails from degree 7 and providing bounds on the difference between filtrations, showing the problem does not stabilize.

Contribution

It establishes the failure of the Andreadakis equality for McCool groups from degree 7 and offers a lower bound on the filtration difference, advancing understanding of their structure.

Findings

Andreadakis equality fails from degree 7

Provides a lower bound for the filtration difference

Shows the Andreadakis problem does not stabilize

Abstract

In this short paper, we show that the McCool group does not satisfy the Andreadakis equality from degree $7$, and we give a lower bound for the size of the difference between the two relevant filtrations. As a consequence, we see that the Andreadakis problem for the McCool group does not stabilize.