Rank gradient, cost of groups and the rank vs Heegaard genus
In the talk I will tell about two famous problems, one in 3-manifold theory and the other in ergodic theory. The problems seem to be completely unrelated, but when one puts profinite actions (or, on another language, graph limits) in the middle, it turns out that one of them (or maybe both) necessarily has a negative solution. The talk will be self-contained, I will carefully explain all the notions. This is joint work with Nikolay Nikolov (Imperial College).