We consider scenarios where a group of agents wish to simplify a given abstract argumentation framework—specifying a set of arguments and the attacks between them—by eliminating cycles in the attack-relation on the basis of their preferences over arguments. They do so by first aggregating their individual preferences into a collective preference order and then removing any attacks involved in a cycle that go against that order. Our analysis integrates insights from formal argumentation and social choice theory. We obtain sweeping impossibility results for essentially all standard methods of preference aggregation, showing that no Condorcet method and no positional scoring rule can uphold the fundamental principle expressing that views held by every single member of the group must be respected. But we also find that so-called representative-agent rules do offer this guarantee.