To cheer you up in difficult times 21: Giles Gardam lecture and new result on Kaplansky’s conjectures

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 $(1-g)(1+g+g^2+\cdots +g^{n-1})=0$.

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 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!