The Open Conjecture

Here is the problem I am attempting to understand:

Conjecture: All freely indecomposable one-relator groups with torsion are co-hopfian.

Until I come up to speed (or rather, once I get anywhere near the ballpark), I’ll start looking at possible ways to attack the problem. For now, I’ll be investigating the different parts of this problem individually.

The Silly Case of Abelian Groups

A one-relator groups we’re treating here are really in a special class of groups. For instance, any abelian group which satisfies the hypotheses must be . Since we only have room for one relator, we can’t include commutators of any additional generators, restricting us to just two.

However, as per the naive attempt, in any one-relator group with torsion we must have the relator as a proper power. Since is not a proper power, we conclude that no abelian groups which satisfy the other hypotheses can have torsion. Hence, no abelian group works.

This could work someway into a method of contradiction in the original proof: supposing we have a group which satisfies the hypotheses but is not cohopfian, perhaps one could prove it is commutative. But then again that seems trivial because we already know there is only one relator, so any route by which we could deduce commutativity would likely provide a contradiction sooner anyway. So that is disappointingly superfluous. But it rules out one more case.

Naive Attempt to Reduce the Problem

As we noted in the previous post, one relator groups which satisfy the hypotheses of the open conjecture must satisfy certain conditions. We naively wish to utilize these conditions to reduce the problem to a simpler class of groups. To recall, we present the following proposition:

Proposition: If is a freely-indecomposable one relator group with torsion, then embeds in the two-generator group , where is a proper power which involves both and .

That embeds in a two-generator group follows from the result of the Freiheitssatz, the proper power condition is a result of torsion (a theorem of Grünberg as mentioned last time), and the trivial note that if has torsion then any group in which embeds also has torsion. Finally, the requirement that involve both generators follows from the construction of the embedding. So the proposition holds.

However, even if is the embedding above, it is unlikely that a statement about the cohopficity of will yield a statement about cohopficity of . In other words, even if is cohopfian, then may not be; and if is not cohopfian, still could be. But of course, if one produced an example of the latter case, the conjecture would be solved in the negative.

So the approach of embedding into a two-generator group seems inherently flawed. Unless we find some relevant theorems on cohopficity, or we are able to prove the larger group is itself, our approach does not seem to bear fruit.

Alas, the search continues.