Affine manifolds with diagonal holonomy

Abstract

As first defined by Smillie, an affine manifold with diagonal holonomy is a manifold equipped with an atlas such that the changes of charts are restrictions of elements of the subgroup of Aff (\({\mathbb{R}^n}\)) formed by diagonal matrices. Refining Smillie’s theorem, Benoist proved that if a compact manifold M is split into manifolds with corners corresponding to complete simplicial fans of a fixed frame by its hypersurfaces with normal crossing, then the product of M and a torus of suitable dimension is a finite cover of an affine manifold with diagonal holonomy, and conversely. Motivated by the result of Benoist, we introduce a “Benoist manifold” and a natural definition of complexity for them. In particular, we study some properties of “Benoist manifolds”.