Igor Mineyev: The Atiyah Conjecture and submultiplicativity

Abstract: The Strengthened Hanna Neumann Conjecture (SHNC) is a question about free groups and graphs, and it can be stated in analytic terms using l2 Betti numbers. This gives a generalization of the statement of SHNC from graphs to complexes: submultiplicativity. This also relates SHNC to the integral Atiyah Conjecture (AC), a question about the Murray-von Neumann dimension of kernels of certain operators. AC is the analytic, and more general, version of the Kaplansky's Zero-Divisors Conjecture. We will discuss the relationship among these questions and possible ways of approaching them by methods of combinatorial group theory.