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The nonabelian product modulo sum

Samuel M. Corson

TL;DR

The paper analyzes the nonabelian analogue of product modulo sum by constructing the topologist's product $\circledast_{n\in\omega} H_n$ and the quotient $\mathcal{A}(\{H_n\})$ by the normal closure of the free product, proving that when each $H_n$ has no involutions and satisfies $1<|H_n|\le 2^{\aleph_0}$, the isomorphism type of $\mathcal{A}(\{H_n\})$ is independent of the sequence (up to deleting finitely many factors). The core method is a transfinite back-and-forth achieved via coherent coi (close-order isomorphism) collections that glue local partial isomorphisms into a global one, extended to a $\mathbb{Q}$-type concatenation to handle dense index sets. The results connect to the fundamental group of the harmonic archipelago/Griffiths space and yield local-freeness properties, contrasting with the abelian setting where sequence choice can affect the quotient’s structure. Overall, the work provides a robust, nonconstructive framework for showing sequence-independence in the nonabelian setting and clarifies the role of involutions and dense-index concatenations in the construction.

Abstract

It is shown that if $\{H_n\}_{n \in ω}$ is a sequence of groups without involutions, with $1 < |H_n| \leq 2^{\aleph_0}$, then the topologist's product modulo the finite words is (up to isomorphism) independent of the choice of sequence. This contrasts with the abelian setting: if $\{A_n\}_{n \in ω}$ is a sequence of countably infinite torsion-free abelian groups, then the isomorphism class of the product modulo sum $\prod_{n \in ω} A_n/\bigoplus_{n \in ω} A_n$ is dependent on the sequence.

The nonabelian product modulo sum

TL;DR

The paper analyzes the nonabelian analogue of product modulo sum by constructing the topologist's product and the quotient by the normal closure of the free product, proving that when each has no involutions and satisfies , the isomorphism type of is independent of the sequence (up to deleting finitely many factors). The core method is a transfinite back-and-forth achieved via coherent coi (close-order isomorphism) collections that glue local partial isomorphisms into a global one, extended to a -type concatenation to handle dense index sets. The results connect to the fundamental group of the harmonic archipelago/Griffiths space and yield local-freeness properties, contrasting with the abelian setting where sequence choice can affect the quotient’s structure. Overall, the work provides a robust, nonconstructive framework for showing sequence-independence in the nonabelian setting and clarifies the role of involutions and dense-index concatenations in the construction.

Abstract

It is shown that if is a sequence of groups without involutions, with , then the topologist's product modulo the finite words is (up to isomorphism) independent of the choice of sequence. This contrasts with the abelian setting: if is a sequence of countably infinite torsion-free abelian groups, then the isomorphism class of the product modulo sum is dependent on the sequence.
Paper Structure (6 sections, 24 theorems, 56 equations)

This paper contains 6 sections, 24 theorems, 56 equations.

Table of Contents

  1. Introduction
  2. The topologist's product
  3. Coi collections
  4. Some straightforward coi extensions
  5. Extension to a $\mathbb{Q}$-type concatenation
  6. Conclusion of the proof

Key Result

Lemma 2.1

Each $\sim$ class includes a reduced word, and this word is unique up to $\equiv$. If $W_0$ and $W_1$ are reduced and $W_0W_1 \sim E$ then $W_1 \equiv W_0^{-1}$. If $W$ and $W'$ are reduced then there exist reduced words $W_0$, $W_1$, $W_0'$, and $W_1'$ such that Given a word $W$ we will let $\operatorname{Red}(W)$ denote the reduced word such that $W \sim \operatorname{Red}(W)$.

Theorems & Definitions (50)

  • Lemma 2.1
  • Definition 2.3
  • Definition 2.5
  • Proposition 2.6
  • proof
  • Definition 2.7
  • Lemma 2.8
  • proof
  • Definition 2.9
  • Lemma 2.11
  • ...and 40 more