Arithmetic structure of generalized Inoue--Bombieri manifolds

A Generalized Inoue--Bombieri (GIB) manifold $M$ is a compact quotient of a connected Riemannian product $\mathbb{R}^q \times (N,g _N)$ by a discrete subgroup of $\mathrm{Sim}(\mathbb{R}^q) \times \mathrm{Isom}(N,g_N)$. The flat factor induces a transversely Riemannian foliation whose leaf closures determine, up to a natural geometric modification, a torus fibration $M \to X$. The main goal of this article is to study the associated monodromy representation $\rho: \pi_1(X) \to \mathrm{GL}(n,\mathbb{Z})$. We prove that the image of $\rho$ is a subgroup of a cocompact arithmetic lattice of a reductive group, and we discuss which groups may be realized as monodromy groups of GIB manifolds. When $(N,g_N)$ is a symmetric space of non-compact type, the monodromy itself is arithmetic. Moreover, one may describe the fibration and the monodromy in terms of parabolic subgroups of the isometry group of $(N,g_N)$. This yields new examples of GIB manifolds, as well as obstructions, and opens the way toward a complete classification in this particular case.