Diameters of 3-Sphere Quotients

Abstract

Let G, a subset of O(4), act isometrically on the 3-sphere. In this article we calculate a lower bound for the diameter of the quotient spaces $S^3/G$. We find it to be ${1/2}\arccos(\frac{\tan(\frac{3 \pi}{10})}{\sqrt3})$, which is exactly the value of the lower bound for diameters of the spherical space forms. In the process, we are also able to find a lower bound for diameters for the spherical Aleksandrov spaces, $S^n/G$, of cohomogeneities 1 and 2, as well as for cohomogeneity 3 (with some restrictions on the group type). This leads us to conjecture that the diameter of $S^n/G$ is increasing as the cohomogeneity of the group $G$ increases.