We introduce a new class of jump operators on Borel equivalence relations, associated to countable groups. For each countable group Gamma, we define the Gamma-jump of an equivalence relation E and produce an analysis of these jumps analogous to the situation of the Friedman--Stanley jump with respect to actions of S_infty. In particular, we show that for many (but not all) groups the Gamma-jump of E is strictly above E and iterates of the Gamma-jump produce a hierarchy of equivalence relations cofinal in terms of potential Borel complexity. We also produce new examples of equivalence relations strictly between E_0^\omega and F_2, and give an application to the complexity of the isomorphism problem for countable scattered linear orders. This is joint work with Sam Coskey.