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

Finite abelian subgroups in the groups of birational and bimeromorphic selfmaps