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 $m \geq 0$ such that $X$ supports an effective action of $(\mathbb{Z} / r)^m$ for arbitrarily big values of $r$. We prove that if $X$ is connected then disc-sym $(X) \leq$ $3 n / 2$. We propose the question of whether for every closed connected $n$-manifold $X$ the inequality disc-sym $(X) \leq n$ holds true, and whether the only closed connected $n$-manifold $X$ for which disc-sym $(X)=n$ is the torus $T^n$. We prove partial results providing evidence for an affirmative answer to this question.