  We give a formula relating the topological rank of the full group of an
aperiodic pmp equivalence relation to the cost of its ergodic components.
Furthermore, we obtain examples of full groups having a dense free subgroup
whose rank is equal to the topological rank of the full group, using a Baire
category argument. We then study the automatic continuity property for full
groups of aperiodic equivalence relations, and find a connected metric for
which they have the automatic continuity property. This allows us to give an
algebraic characterization of aperiodicity for pmp equivalence relations,
namely the non-existence of homomorphisms from their full groups into totally
disconnected separable groups. A simple proof of the extreme amenability of
full groups of hyperfinite pmp equivalence relations is also given,
generalizing to the non ergodic case a result of Giordano and Pestov.
