Constructing Highly Symmetric Compact Manifolds and Algebraic Varieties

For every algebraically closed field $k$ and natural number $r$, we construct several algebraic varieties (over $k$) whose birational automorphism group contains every finite nilpotent group of class at most $2$, rank at most $r$ whose order is coprime to the characteristic of $k$. This construction is sharp in characteristic $0$, i.e. up to bounded extension, the set of groups from the statement cannot be replaced by a larger one. Using similar main ideas (with different technical details), for every $r$, we construct several compact manifolds whose diffeomorphism groups contain every finite nilpotent group of class at most $2$, rank at most $r$. This result answers a question of Mundet~i~Riera affirmatively and is conjecturally sharp up to bounded extension.