  We introduce a new invariant of $G$-varieties, the dual complex, which
roughly measures how divisors in the complement of the free locus intersect. We
show that the top homology group of this complex is an equivariant birational
invariant of $G$-varieties. As an application, we demonstrate the
non-linearizability of certain large abelian group actions on smooth
hypersurfaces in projective space of any dimension and degree at least $3$.
