The Finitary Andrews-Curtis Conjecture

TL;DR

This paper introduces a finite version of the Andrews-Curtis Conjecture by characterizing the connected components of its associated graphs in finite groups, impacting computational group theory and addressing open questions.

Contribution

It establishes the finite version of the conjecture, describing the connected components of Andrews-Curtis graphs for finite groups, and shows finite computations cannot disprove the classical conjecture.

Findings

Finite Andrews-Curtis graphs are characterized by their connected components.

Finite group computations cannot produce counterexamples to the classical conjecture.

The work resolves an open question from previous research.

Abstract

The well known Andrews-Curtis Conjecture [2] is still open. In this paper, we establish its finite version by describing precisely the connected components of the Andrews-Curtis graphs of finite groups. This finite version has independent importance for computational group theory. It also resolves a question asked in [5] and shows that a computation in finite groups cannot lead to a counterexample to the classical conjecture, as suggested in [5].