The 2-rank of finite groups acting on hyperelliptic 3-manifolds

We consider 3-manifolds admitting the action of an involution such that its space of orbits is homeomorphic to S3. Such involutions are called hyperelliptic as the manifolds admitting such an action. We prove that the sectional 2-rank of a finite group acting on a 3-manifold and containing a hyperelliptic involution whose fixed-point set has two components has sectional 2-rank at most four; this upper bound is sharp. The cases where the hyperelliptic involution has a fixed-point set with a number of components different from 2 have been already considered in the literature. Our result completes the analysis and we obtain general results where the number of the components of the fixed-point set is not fixed. In particular, we obtain that a finite group acting on a 3-manifold and containing a hyperelliptic involution has 2-rank at most four, and four is the best possible upper bound. Finally, we restrict to the basic case of simple groups acting on hyperelliptic 3-manifolds: we use our result about the sectional 2-rank to prove that a simple group containing a hyperelliptic involution is isomorphic to PSL(2, q) for some odd prime power q, or to one of four other small simple groups.