Constructing highly symmetric compact manifolds and algebraic varieties

For every algebraically closed field k k and natural number r r , we construct several algebraic varieties (over k k ) whose birational automorphism group contains every finite nilpotent group of class at most 2 2 , rank at most r r whose order is coprime to the characteristic of k k . This construction is sharp in characteristic 0 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 r , we construct several compact manifolds whose diffeomorphism groups contain every finite nilpotent group of class at most 2 2 , rank at most r r . This result answers a question of Mundet i Riera affirmatively and is conjecturally sharp up to bounded extension.