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Virtual Betti numbers of genus $$2$$ bundles. (English) Zbl 1009.57023
It is part of a general conjecture for closed 3-manifolds that the virtual first Betti number of a surface bundle over the circle is infinite (i.e., there are finite covers with arbitrarily large first Betti numbers). In the present paper, this conjecture is proved for surface bundles with fibres of genus 2. The crucial fact is that every homeomorphism of a genus 2 surface commutes, up to isotopy, with a hyperelliptic involution. In fact, the main result of the paper implies that the virtual first Betti number of a surface bundle $$M_f$$ associated to a surface homeomorphism $$f:F \to F$$ is infinite if $$f$$ commutes with a hyperelliptic involution on $$F$$ (or, more generally, with every element of a finite group $$G$$ of automorphisms of $$F$$ such that the quotient $$F/G$$ is a torus with at least one branch point). As a corollary it is obtained that a genus 2 surface bundle can be geometrized using only the non-fiber case of Thurston’s Geometrization Theorem for Haken 3-manifolds (for this, a cover with first Betti number at least two is needed because such a cover has a non-separating incompressible surface which is not a fiber).

##### MSC:
 57N10 Topology of general $$3$$-manifolds (MSC2010) 57M50 General geometric structures on low-dimensional manifolds 57M10 Covering spaces and low-dimensional topology
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