  Let $X$ be a complex projective variety. Suppose that the group of birational
automorphisms of $X$ contains finite subgroups isomorphic to
$(\mathbb{Z}/N_i\mathbb{Z})^r$ for $r$ fixed and $N_i$ arbitrarily large. We
show that $r$ does not exceed $2\dim(X)$. We also show that the same result
holds for groups of bimeromorphic automorphisms of compact K\"ahler threefolds.
