In my last post I wrote about Nielsen equivalence classes of generating sets for groups as invariants of Heegaard splittings. It turns out that similar ideas also come up in a purely group theoretic context. Alex Lubotsky gave a six week course on homomorphisms from free-like groups into finite simple groups. By free-like, I mean free groups, pro-finite free groups and surface fundamental groups. The main motivation for the course was the Weigold conjecture, which states that the number homomorphisms from a rank n free group to a finite simple group, modulo automorphisms of the free group and automorphisms of the finite simple group, should be one for n greater than or equal to 3.
As mentioned last time, two generating sets are Nielsen equivalent if the homomorphisms they induce from a free group are related by an automorphism of the free group. The Weigold conjecture viewpoint is different from the Heegaard splitter’s view on Nielsen equivalence in two ways: First, very few 3-manifold fundamental groups are finite and simple. Second, we don’t normally consider things modulo automorphisms of the 3-manifold’s fundamental group. (I guess this would reduce things to studying homeomorphism classes of Heegaard splittings instead of isotopy classes.) But there are some interesting (if vague) parallels between the two areas.
I mentioned in the last post the idea of redundant generating sets, which are sets in which a proper subset of the generating set is itself a generating set. There’s a second conjecture (I can’t remember who it’s named after) that every order n generating set for a non-abelian finite simple group with n at least 3 is Nielsen equivalent to a redundant generating set. This would follow from the Weigold conjecture since every non-abelian finite simple group has rank two, so for n greater than two there is a redundant n element generating set with two non-trivial elements and the rest trivial. The handlebodies of a stabilized Heegaard splitting induce generating sets that are redundant (up to Nielsen equivalence). The converse of this is not true, but there is still a rough analogy between Nielsen classes of redundant generating sets and stabilized Heegaard splittings.
The point of all of this is that these two conjectures in group theory are (roughly) parallel to things that have been proved (or one might try to prove) for Heegaard splittings of a different classes of 3-manifolds. The Weigold conjecture is parallel to proving that some 3-manifold has (up to isotopy) exactly one Heegaard splitting for each genus greater than two. The second conjecture is parallel to proving that a certain 3-manifold has no irreducible Heegaard splittings of genus greater than two. I don’t know if there’s a more concrete connection along these lines, but even as a vague analogy, I think it’s interesting.