Discrete Degree of Symmetry of Manifolds

Abstract

We define the discrete degree of symmetry disc-sym(X) of a closed n-manifold X as the biggest m≥0 such that X supports an effective action of (Z/r)m for arbitrarily big values of r. We prove that if X is connected then disc-sym(X)≤3n/2. We propose the question of whether for every closed connected n-manifold X the inequality disc-sym(X)≤n holds true, and whether the only closed connected n-manifold X for which disc-sym(X)=n is the torus Tn. We prove partial results providing evidence for an affirmative answer to this question. © The Author(s) 2024.