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

Abstract We consider 3-manifolds admitting the action of an involution such that its space of orbits is homeomorphic to $$S^3.$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mi>S</mml:mi> <mml:mn>3</mml:mn> </mml:msup> <mml:mo>.</mml:mo> </mml:mrow> </mml:math> 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.