Ergodicity and the promotion of quasiconformality

There comes a point in the proof of Mostow rigidity in which you have a quasiconformal map from the sphere to itself which is equivariant with respect to certain group actions, and thanks to the magic of "ergodicity" you discover that this map is actually 1-quasiconformal. The goal is to unpack this statement a bit: what is ergodicity, where does it come from in this context, and how does it allow you to strengthen conformality? The talk will be very informal.