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Letter-braiding: bridging combinatorial group theory and topology

Nir Gadish

TL;DR

This work introduces letter-braiding invariants of words in arbitrary groups, constructed via the Bar cohomology of simplicial cochains and yielding a universal, finite-type invariant that extends Magnus expansions to all groups over any PID. The core idea is a dual pairing between the Bar coalgebra $\mathrm{H}^0_{Bar}(\Gamma;A)$ and the group ring $A[\Gamma]$, which encodes information about the lower central series through a complete, functorial framework. The authors develop both a topological (via Bar constructions on cochains) and a combinatorial model (via discrete Chen-like sums) to compute these invariants, prove their completeness, and relate them to Massey products, Massey-length relative products, and Johnson-type filtrations. They further demonstrate substantial applications to combinatorial group theory, geometric topology, and positive characteristic phenomena, including a Johnson homomorphism in general settings and new constraints on automorphisms of finite $p$-groups. Overall, the letter-braiding program provides a concrete, computable bridge between group theory and topology with a unifying perspective on classical invariants and their generalizations.

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.

Letter-braiding: bridging combinatorial group theory and topology

TL;DR

This work introduces letter-braiding invariants of words in arbitrary groups, constructed via the Bar cohomology of simplicial cochains and yielding a universal, finite-type invariant that extends Magnus expansions to all groups over any PID. The core idea is a dual pairing between the Bar coalgebra and the group ring , which encodes information about the lower central series through a complete, functorial framework. The authors develop both a topological (via Bar constructions on cochains) and a combinatorial model (via discrete Chen-like sums) to compute these invariants, prove their completeness, and relate them to Massey products, Massey-length relative products, and Johnson-type filtrations. They further demonstrate substantial applications to combinatorial group theory, geometric topology, and positive characteristic phenomena, including a Johnson homomorphism in general settings and new constraints on automorphisms of finite -groups. Overall, the letter-braiding program provides a concrete, computable bridge between group theory and topology with a unifying perspective on classical invariants and their generalizations.

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.
Paper Structure (26 sections, 25 theorems, 99 equations)

This paper contains 26 sections, 25 theorems, 99 equations.

Table of Contents

  1. Introduction
  2. Main Results
  3. First examples, motivation and background
  4. Current and planned applications
  5. Combinatorial group theory
  6. Geometric topology
  7. Positive characteristic
  8. Relative Massey products vs. relations
  9. Algebraic models for spaces
  10. Johnson homomorphisms
  11. Related work
  12. (Co)algebraic preliminaries
  13. Coassociative Bar construction
  14. Group rings
  15. Tensors and infinite cartesian powers
  16. ...and 11 more sections

Key Result

Theorem 1.1

For any group $\Gamma$, its Bar cohomology $\mathrm{H}^0_{Bar}(\Gamma;A)$ is a filtered coalgebra, functorially in $\Gamma$. Under the letter-braiding pairing, this structure is dual to the filtered algebra structure on $A[\Gamma]$, as follows.

Theorems & Definitions (70)

  • Theorem 1.1
  • Example 1.2
  • Example 1.3: Torsion commutators
  • Proposition 1.4
  • Theorem 1.5
  • Example 1.6
  • Corollary 1.7
  • Corollary 1.8: Compare Mandell mandell, and Dwyer dwyer1975exotic
  • Theorem 1.9
  • Definition 2.1
  • ...and 60 more