  We give a Tannakian description for categories of l-adic perverse sheaves on
semiabelian varieties which combines a construction of Gabber and Loeser for
algebraic tori with a generic vanishing theorem for the cohomology of
constructible sheaves on abelian varieties. As an application we explain how
the arising Tannaka groups on abelian varieties can be studied in terms of
semiabelian degenerations via the functor of nearby cycles.
