A decision dated December 9, 2014 of the Court of Rome states that Facebook groups names should be considered atypical distinctive signs if used for business purposes. Therefore the appropriation and exploitation of Facebook groups can represent a competitive offence and a trademark rights infringement.
The decision of the Court of Rome confirms the principle stated by the Court of Turin. Here you can find a brief comment on the decision.
According to the Court of Rome it’s the holder of the IP rights on the name of the group that can dispose of it, not the group administrator. For this reason the administrator had to give the password and to allow the access to the Facebook page to the claimant company.