Discrete degree of symmetry of manifolds

TL;DR

This paper introduces the discrete degree of symmetry for manifolds, establishing an upper bound related to dimension, and explores whether the maximum symmetry characterizes tori among all manifolds.

Contribution

It defines a new measure of symmetry for manifolds and provides bounds and partial results towards classifying manifolds with maximal symmetry.

Findings

For connected manifolds, the discrete degree of symmetry is at most 3n/2.

Open questions about whether the bound n holds and if tori are uniquely maximally symmetric.

Partial results support the conjecture that tori are characterized by maximal symmetry.

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 $({\mathbf Z}/r)^m$ for arbitrarily big values of $r$. We prove that if $X$ is connected then $disc-sym(X)\leq 3n/2$. 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.