The Atiyah Conjecture and submultiplicativity

The Strengthened Hanna Neumann Conjecture (SHNC), a question about free
groups and graphs, can be stated in analytic terms using l^2 Betti
numbers. This gives a generalization of the statement of SHNC from graphs
to comlexes: 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 conjectures and possible ways of approaching
them.