The word problem of the Brin-Thompson groups is coNP-complete

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.