Abstract: We discuss the unique product property, a natural group property that implies the Kaplansky zerodivisor conjecture. This conjecture states that the group algebra of a torsion-free group has no zero-divisors. Another statement that implies the conjecture of Kaplansky is Atiyah’s conjecture on the l2-betti numbers.
We give new constructions of torsion-free hyperbolic groups without the unique product property that combine further additional algebraic, algorithmic and geometric properties. We stress a new viewpoint in our construction, the graphical small cancellation theory over the free product. This allows us to combine two famous group theoretic constructions, our versions of the Rips construction and the Rips-Segev construction of groups without the unique product property.
We contrast the groups resulting from our construction with our recent positive result on the CAT(0)–cubulation of classical small cancellation groups over the free product. This result implies the Kaplansky zero-divisor conjecture for a new large class of hyperbolic groups given by the classical small cancellation presentations over the free product.
This talk will be on joint works with Goulnara Arzhantseva and with Alexandre Martin. I am recipient of the DOC fellowship of the Austrian Academy of Sciences, and partially supported by the ERC-grant ANALYTIC no. 269527 of Goulnara Arzhantseva.