# The word problem of the Brin-Thompson group is coNP-complete

## TL;DR

This paper establishes that the word problem for the Brin-Thompson groups nV and the Thompson group V is coNP-complete, highlighting the computational complexity of these algebraic problems.

## Contribution

It proves coNP-completeness of the word problem for nV groups for all n ≥ 2 and for Thompson group V over a specific generator set, advancing understanding of their computational complexity.

## Key findings

- Word problem of nV is coNP-complete for all n ≥ 2
- Word problem of Thompson group V over certain generators is coNP-complete
- Highlights computational complexity of these algebraic problems

## Abstract

We prove that the word problem of the Brin-Thompson group nV over a finite generating set is coNP-complete for every n \ge 2. It is known that the groups nV are an infinite family of infinite, finitely presented, simple groups. We also prove that the word problem of the Thompson group V over a certain infinite set of generators, related to boolean circuits, is coNP-complete.