# Strong Atiyah conjecture

Who introduced the Strong Atiyah Conjecture?

Recall that the conjecture says the following. Let $G$ be a group, $A$ a $n\times n$-matrix over ${\mathbb Z}G$. We view $A$ as a bounded operator $l^2(G)^n\to l^2(G)^n$. Let $K$ be the kernel of that operator and $p$ be the orthogonal projection onto $K$. Let $\delta$ be the function of $l^2(G)$ which is 1 on $1\in G$, and 0 everywere else. Let $e_i$ be the vector from $l^2(G)^n$ with $i$-th coordinate $\delta$, all other coordinates $0$, and $t_A$ be the sum of dot products $\sum_i \langle p(e_i), e_i\rangle$. The conjecture says that $t_A$ always belongs to $\frac1g\mathbb{Z}$ where $g$ is the least common multiple of the orders of finite subgroups of $G$ (so if $G$ is torsion-free, $t_A$ must be an integer).

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Certainly not Atiyah. –  Benjamin Steinberg May 8 '12 at 23:14
@Ben: Of course not. –  Mark Sapir May 8 '12 at 23:14
My conjecture is that it was Linnell or Schick. –  Mark Sapir May 8 '12 at 23:17
I am pretty sure I heard a talk by Grigorchuk 10 years ago where he said who made the strong conjecture but I can't remember anymore the attribution. –  Benjamin Steinberg May 9 '12 at 0:18
There is also the "super-strong Atiyah conjecture": the one which states that $t_A\in \frac1{g'}{\mathbb Z}$ where $g'$ is the least common multiple of orders of elements of $G$. That conjecture has been solved in negative by many people including Grigorchuk. I think that the "strong Atiyah" is still open. –  Mark Sapir May 9 '12 at 1:15

I received a message from Thomas Schick, which answers my question. The strong Atiyah conjecture was introduced jointly by Lueck and Schick (this was also one of the alternatives in Wolfgang Lueck's email to me), and Thomas Schick is responsible for the super-strong one. He was a postdoc working with Wolfgang Lueck at that time. Peter Linnell came to the $l_2$-Betti theory via Kaplansky's zero divisors conjecture (these are related) and learned about various forms of Atiyah conjectures from Wolfgang Lueck. In any case, the various forms of Atiyah's conjecture have generated a lot of interesting mathematics already and probably will generate more.