What I find most fascinating about the unsolvability of the word problem and related unsolvability problems in group theory is how readily they tell us how hard a problem is in other branch of math. For example, it seems intractable to develop methods of determining whether two -manifolds are homeomorphic since given any free group one can construct a -manfiold such that . Thus, to be able to solve this problem would imply that we would be able to tell from a presentation whether two free groups are isomorphic.
Are you aware of any more examples of this? Do you do work in this area? I didn't think people were doing this kind of group theory as much anymore.