Unsolvable problems about higher-dimensional knot groups

An n-knot is an embedding of the n-sphere in the (n+2)-sphere, and the corresponding n-knot group is the fundamental group of its complement. In contrast to the classical case n = 1, we show that many decision problems about the class of n-knot groups, n>2, (as well as certain classes of groups of other kinds of codimension 2 embeddings) are unsolvable. We also pose some open questions about the class of 2-knot groups, which is still not well understood. (This is joint work with F. Gonzalez-Acuna and J. Simon.)