A note on groups definable in difference fields

In this paper we record some observations around groups definable in difference fields. We were motivated by a question of Zoe Chatzidakis as to whether any group definable in a model of ACFA is virtually definably embeddable in an algebraic group. We give a positive answer. Among the possibly “new” ingredients is the (stable) group configuration theorem in the ∗-definable category. The embeddability result for groups definable in ACFA of finite SU -rank was already noted in [2], more or less by saying that the proof in [5] for groups definable in pseudofinite fields goes through. We also take the opportunity to give an improved treatment of the analogous theorem for differentially closed fields (avoiding the category of ∗-definable groups). Finally we adapt results from [6] and [1] about the unipotence of differential groups on affine spaces to the difference field context. ∗Supported by grant KBN 2 PO3A 020 18. †Supported by NSF grant.