Rank of mapping tori and companion matrices (Q353420)

The authors study the problem of computing the rank of certain semidirect products. They are mostly concerned with semidirect products of \(\mathbb{Z}^d\) by \(\mathbb{Z}\) and of \(\pi_1(S)\) by \(\mathbb{Z}\), where \(S=S_g\) is a closed surface of genus \(g\). The rank of such a group is the smallest number of generators for the group.NEWLINENEWLINEFor a semidirect product \(G\) of \(\mathbb{Z}^d\) by \(\mathbb{Z}\) with monodromy \(\phi\), they show that the rank of \(G\) is one plus the minimal number of \(\phi\)--orbits needed to generate \(\mathbb{Z}^d\). They show that \(\mathbb{Z}^d\) is generated by a single \(\phi\) orbit if and only if \(\phi\) is conjugate to a certain companion matrix having the same characteristic polynomial. Using the solvability of the conjugacy problem in \(GL_d(\mathbb{Z})\), they show that it is decidable whether or not \(G\) has rank two, which is the main result of the paper.NEWLINENEWLINEIf \(\phi\) has infinite order as an automorphism of \(\mathbb{Z}^d\), the authors show that for all sufficiently large \(n\), the semidirect product of \(\mathbb{Z}^d\) by \(\mathbb{Z}\) with monodromy \(\phi^n\) has rank at least three.NEWLINENEWLINEThe authors also consider the fundamental group of a closed fibered hyperbolic \(3\)--manifold \(M\) with monodromy \(\psi\) and fiber \(S=S_g\). They show that for any sufficiently large \(n\), the cover \(M_n\) of \(M\) given by replacing \(\psi\) by \(\psi^n\) has a fundamental group of rank \(2g+1\).