Nearly geodesic immersions and domains of discontinuity

Abstract

We study nearly geodesic immersions in higher rank symmetric spaces of non-compact type, which we define as immersions that satisfy a bound on their fundamental form, generalizing the notion of immersions in hyperbolic space with principal curvature in $(-1,1)$. This notion depends on the choice of a flag manifold embedded in the visual boundary, and immersions satisfying this bound admit a natural domain in this flag manifold that comes with a fibration. As an application we give an explicit fibration of some domains of discontinuity for some Anosov representations. Our method can be applied in particular to some $\Theta$-positive representations for each notion of $\Theta$-positivity.

67 pages, 7 figures. Comments are welcome

References in corpus (4)

• Anosov representations and dominated splittings
• The geometry of maximal representations of surface groups into SO(2,n)
• Higgs bundles and geometric structures on manifolds
• Geometry of compact complex manifolds associated to generalized quasi-Fuchsian representations

Cited by in corpus (1)

• Geometric structures for maximal representations and pencils