On the Andreadakis equality for a subgroup of the McCool group

TL;DR

This paper proves the generalized Andreadakis equality for a subgroup of the McCool group, specifically the partial inner automorphism group, and establishes an isomorphism with the inner automorphism group of the upper triangular McCool group.

Contribution

It extends the validity of the Andreadakis conjecture to the partial inner automorphism subgroup and links it to the inner automorphism group of the upper triangular McCool group.

Findings

Proves the Andreadakis equality for the partial inner automorphism group.

Establishes an isomorphism with the inner automorphism group of the upper triangular McCool group.

Extends known results to new subgroups of the McCool group.

Abstract

The McCool group has families of subgroups such as the ordinary pure braid group, the upper triangular McCool group and the partial inner automorphism group. The generalized Andreadakis conjecture holds for the ordinary pure braid group and the upper triangular McCool group. In this paper, we prove a similar result for the partial inner automorphism group and also establish an isomorphism between this group and the inner automorphism group of the upper triangular McCool group.