Balanced finite presentations of the trivial group

TL;DR

This paper constructs a sequence of balanced finite presentations of the trivial group where the minimal number of relations needed to prove triviality grows faster than any fixed-height exponential tower relative to the presentation length.

Contribution

It introduces a novel sequence of balanced presentations demonstrating extremely complex triviality proofs, surpassing previous complexity bounds.

Findings

Minimal relations grow faster than any fixed-height exponential tower

Demonstrates extreme complexity in trivial group presentations

Provides new insights into the complexity of group triviality proofs

Abstract

We construct a sequence of balanced finite presentations of the trivial group with two generators and two relators with the following property: The minimal number of relations required to demonstrate that a generator represents the trivial element grows faster than the tower of exponentials of any fixed height of the length of the finite presentation.