Letter-braiding: bridging combinatorial group theory and topology

TL;DR

This paper introduces letter-braiding invariants that connect combinatorial group theory and topology, providing a universal, functorial tool to analyze group structures and automorphisms across various contexts.

Contribution

It defines new invariants that detect the dimension series of groups, generalizing Magnus coefficients to all groups and PIDs, and links them to topological constructions.

Findings

Invariants coincide with Magnus coefficients on free groups.

Invariants are complete for the dimension series.

Applications include constraining automorphisms of finite p-groups.

Abstract

We define invariants of words in arbitrary groups, measuring how letters in a word are interleaving, perfectly detecting the dimension series of a group. These are the letter-braiding invariants. On free groups, braiding invariants coincide with coefficients in the Magnus expansion. In contrast with Magnus' coefficients, our invariants are defined on all groups and over any PID. They respect products in the group and are a complete invariant of the dimension series, so they are the coefficients of a universal multiplicative finite-type invariant, depending functorially on the group. Letter-braiding invariants arise from the bar construction on a cochain model of a space with a prescribed fundamental group. This approach specializes to simplicial presentations of a group as well as to more geometric contexts, which we illustrate in examples. As an application, we define variants of the Johnson filtration and the Johnson homomorphism on the automorphisms of arbitrary groups, and use them to constrain automorphisms of finite p-groups.