Arithmetic structure of generalized Inoue--Bombieri manifolds

TL;DR

This paper investigates the algebraic and geometric structure of generalized Inoue--Bombieri manifolds, focusing on their monodromy representations and how these relate to arithmetic groups, providing new examples and classification insights.

Contribution

It characterizes the monodromy groups of GIB manifolds as subgroups of cocompact arithmetic lattices and describes their structure in the case where the fiber is a symmetric space.

Findings

Monodromy groups are subgroups of cocompact arithmetic lattices.

When the fiber is a symmetric space, the monodromy is arithmetic.

Provides new examples and obstructions for GIB manifolds.

Abstract

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.