Discrete Degree of Symmetry of Manifolds

We define the discrete degree of symmetry disc-sym( X ) of a closed n -manifold X as the biggest m≥ 0 m ≥ 0 such that X supports an effective action of (Z/r)ᵐ ( Z / r ) m for arbitrarily big values of r . We prove that if X is connected then disc-sym (X)≤ 3n/2 ( X ) ≤ 3 n / 2 . We propose the question of whether for every closed connected n -manifold X the inequality disc-sym (X)≤ n ( X ) ≤ n holds true, and whether the only closed connected n -manifold X for which disc-sym(X) $$=n$$ = n is the torus Tⁿ T n . We prove partial results providing evidence for an affirmative answer to this question.