Diameters of 3-sphere quotients

Let G⊂O(4)G⊂O(4) act isometrically on S3S3. In this article we calculate a lower bound for the diameter of the quotient spaces S3/GS3/G. We find it to be 12arccos(tan(3π10)3), 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, Sn/GSn/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 Sn/GSn/G is increasing as the cohomogeneity of the group G increases.