One dimension higher of the word problem

Abstract

Just as a group presentation 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.