Profinite genus of fundamental groups of torus bundles

Abstract: In this paper we establish lower and upper bounds for the cardinality of the profinite genus of the fundamental group $\pi_{1}(M_A)\cong (\mathbb{Z} \times \mathbb{Z})\rtimes_{A}\mathbb{Z}$ of a torus bundle $M_{A}$ in terms of the number of ideal classes of the order $\mathbb{Z}[\lambda]$, where $\lambda$ is an eigenvalue of the matrix $A$ in $\mathrm{GL}_{2}(\mathbb{Z})$.