Let G ⊂ O(4) act isometrically on S 3 . 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( tan( 3π 10 ) √ 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, 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.

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Consider a compact, connected Lie groupGacting isometrically on a sphereSnof radius1. Two-dimensional quotient spaces of the typeSn/Ghave been investigated extensively. This paper provides an elementary introduction, for nonspecialists, to this important field by way of several classical examples and supplies an explicit list of all the isotropy subgroups involved in these examples.

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