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

On full groups of non ergodic probability measure preserving equivalence relations