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Arithmetic structure of generalized Inoue--Bombieri manifolds

Brice Flamencourt, Abdelghani Zeghib

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 $ρ: π_1(X) \to \mathrm{GL}(n,\mathbb{Z})$. We prove that the image of $ρ$ 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.

Arithmetic structure of generalized Inoue--Bombieri manifolds

Abstract

A Generalized Inoue--Bombieri (GIB) manifold is a compact quotient of a connected Riemannian product by a discrete subgroup of . The flat factor induces a transversely Riemannian foliation whose leaf closures determine, up to a natural geometric modification, a torus fibration . The main goal of this article is to study the associated monodromy representation . We prove that the image of 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 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 . This yields new examples of GIB manifolds, as well as obstructions, and opens the way toward a complete classification in this particular case.
Paper Structure (27 sections, 20 theorems, 71 equations)

This paper contains 27 sections, 20 theorems, 71 equations.

Table of Contents

  1. Introduction
  2. GIB manifolds
  3. Monodromy group
  4. GIB arithmetic data
  5. Topology of GIB manifolds with a given monodromy group
  6. The case of symmetric spaces
  7. Organization of the paper
  8. Notations and preliminaries
  9. Notations
  10. Arithmetic groups and lattices
  11. The restriction of scalars
  12. Arithmetic data
  13. Monodromy group of a GIB structure
  14. The arithmetic data of a GIB manifold
  15. The Zariski closure of the automorphism group: proof of Theorem \ref{['structureThm']}
  16. ...and 12 more sections

Key Result

Theorem 1.2

The automorphism group ${\bf D}= \mathrm{Aut}(n, H, V, b)$ of GIB arithmetic data $(n, H, V, b)$ is a cocompact arithmetic group. More precisely let $G$ be this Zariski closure. Then, $G$ is a $\mathbb{Q}$-anisotropic reductive group. In particular, $G$ is the almost-direct product of its center $T$

Theorems & Definitions (47)

  • Definition 1.1
  • Theorem 1.2
  • Remark 1.3
  • Theorem 1.4
  • Proposition 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • ...and 37 more