Recognizing geometric 3-manifold groups using the word problem

Adyan and Rabin showed that most properties of groups cannot be algorithmically recognized from a finite presentation alone. We prove that, if one is also given a solution to the word problem, then the class of fundamental groups of closed, geometric 3-manifolds is algorithmically recognizable. In our terminology, the class of geometric 3-manifold groups is `recursive modulo the word problem'.