There is a very famous conjecture of Irving Kaplansky that asserts that the group ring of a torsion free group does not have zero-divisors. Given a group G and a ring R, the group ring R[G] consists of formal (finite) linear combinations of group elements with coefficients in the ring. You can easily define additions in R[G], and can extend the group multiplication to R[G], which makes the group ring a ring. (And if R is a field, R[G] is an algebra, called group-algebra.)

**Kaplansky’s zero divisor conjecture** asserts that if G is torsion-free and K is a field then K[G] has no zero-divisors.

Irving Kaplansky

This conjecture was made in the 1950s or even 1940s. It is known to hold for many classes of groups. If G has torsion, namely an element g of order n, then .

Kaplansky made also in the 50s two other conjectures.

**Kaplansky’s unit conjecture** asserts that if G is torsion-free and K is a field the only units in the face ring of K[G] are of the form kg where k in a non-zero element in the field and g is an element of the group,

**Kaplansky’s idempotents conjecture** asserts that if G is torsion-free and K is a field then the only idempotents in K[G] are 0 and 1.

If G has torsion, namely an element g of order n, then $(1-g)(1+g+g^2+\cdots +g^{n-1})=0$. It is not very hard to see that the unit conjecture implies the zero-divisor conjecture which in turn implies the idempotent conjecture.

A few days ago, Giles Gardam gave a great, very pleasant, lecture about Kaplansky’s conjectures.

After introducing the conjectures themselves, Giles explained that these conjectures are related to several other conjectures like the Baum-Connes conjecture or Farrell-Jones and a conjecture of Atiyah. So this implies, for example that the zero divisors conjecture holds for residually torsion free solvable group. Now, as an aside let me say that it is good to know what does it mean for a property X to say that a groups G is “residually X”. I tried to explain it in this post. But I myself forgot, so together with you, devoted readers, I will go to the old post to be reminded. Let’s get reminded also of the easier concept of “virtually X”.

The assertion of Kaplansky’s unit conjecture holds for torsion-free unique-product groups. The unique product property says the following if A and B are finite subsets of the group there is an element c that can be written in a unique way as c=ab where a belongs to A and b belongs to B.

This concept was defined in 1964 by Rudin and Schneider and for two decades it was not even known that there are groups without the unique-product property. The first example of a group without this property was discovered by Rips and Segev.

Let me make a small diversion here. From time to time I talk about results by people I personally know and usually I don’t mention that in the posts. For example, in the post about Yuansi Chen’s work on Bourgain’s slicing conjecture and the KLS conjecture I personally knew about 70% of the heroes in that story (update: 19:25). In fact, both Bo’az Klartag and me are living in the very same apartment building in Tel Aviv. (Officially, I am in number 9 and Bo’az is in number 7 but topologically it is the same building.)

But here I must mention that Yoav Segev is my class mate in undergraduate years and is now a Professor at Ben Gurion University in Beer-Shava. He is responsible to one of the final steps in the classification program, to knocking down several other conjectures in algebra, and also to works on fixed-point free actions of non solvable groups on simplicial complexes. This last topic is close to my own interests and from time to time we chat about it. And of course, Ilya Rips is extraordinary mathematician and we are emeritus colleagues at HUJI – see this post about Ripsfest.

So let me go back to Giles’s lecture. Giles discussed two classes of examples of groups without the unique product property. And at minute 49 of the (50-minute) lecture Giles said: “I am nearly at the end of the talk and its time for me to tell you what’s new, um, what’s my contribution to this story, um, so I am really happy to be able to announce today for the first time, that, in fact, the unit conjecture is false.”

**Theorem (Giles Gardam, 2021):** Let P be the torsion-free virtually abelian group

And lt K be the field with two elements. Then there is a nontrivial unit in K[G], so that both and have support of size 21.

(This group does satisfy the zero-divisor conjecture.)

Here is the paper: Giles Gardam, A counterexample to the unit conjecture for group rings. Congratulations, Giles!

Today, for the first time in several months, I visited the Givat Ram campus of the Hebrew University, and spent some time at my office. One of the things that I did there was to print out Giles Gardam;s paper, I have it here, but have not studied it yet. Like Gil, I attended Giles’ talk (via zoom), and was impressed by him holding till the last minute telling about his remarkable example.

And speaking about people that we know, I well remember the discovery by Segev and Rips of groups without the unique product property. Segev started out as a Ph.D. student under the guidance of Rips, later he preferred to transfer to me, and naturally I am proud to have had such a student. So I guess I was one of the first people to hear about their breakthrough, they even consulted me about their paper. Not about the mathematics, I had nothing to do with that, just about the form of the paper.

Dear Avinoam, thanks for sharing these memories. There is a story related to you, me and Kaplansky: Namely, Yuval Flicker and I took a reading seminar with you on Kaplansky’s book “commutative rings”. I suppose that Both Kaplansky and Jackobson, visited the Hebrew University in my student years, but I don’t remember them.

Sure, Gil, I remember that seminar. That was the first time that I’ve met Yuval, you were my student earlier in a number theory course. And you are right, both Kaplansky and Jacobson visited Jerusalem, which really meant visiting Amitsur, Jacobson came quite a few times.

Let us try to think where we go now from here. One may think that this example closes the matter. Quite the contrary, Gardam’s result opens a new area of study: what is the structure of the unit group of a group algebra, now that we know that it’s far from trivial. The example is over the field of two elements, so it is also an example over any field of characteristic 2, and the first question is: what about other characteristics? In particular characteristic zero, and in particular what about the integral group ring ZG (though Z is not a field)?

And of course what about the zero-divisors conjecture? The tendency now would be to look for counter examples also for that, but where to look? As Giles Gardam pointed out, his group does satisfy the zero divisors conjecture, so where should we look for a counter example?

We see that there is yet a lot to be done. And that is a good thing.

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A comment and a question 🙂

1. Your link to an explanation of residuality is incomplete, I believe. I referred to this: https://en.wikipedia.org/wiki/Residual_property_(mathematics)

2. Did Gardam show the explicit unit he discovered as the finale of his talk? It would seem that it would be likely to fit on one screen. I haven’t read his paper yet.

Thanks and best wishes for you and yours, sir.

The paper is only seven pages long, and yes, the unit is completely explicit. The support is of cardinality 21, which makes a nice coincidence with the numbering of Gil’s post.