One dimension higher of the word problem

Abstract

Just as a group presentation ${\mathcal P}$ can be regarded as a $2$-complex with a single $0$-cell, so we can consider a $3$-complex with a single $0$-cell, known as a $3$-presentaton. In this paper, by using a geometric way, called spherical pictures, we show that there exist a finite $3$-presentation which has unsolvable generalised identity problem which can be thought as one-dimension higher of the word problem.