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Nearly geodesic immersions and domains of discontinuity

Colin Davalo

Geometry & Topology 29 (2025) 2391–2461
Abstract

We study nearly geodesic immersions in higher-rank symmetric spaces of noncompact 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 Θ-positive representations for each notion of Θ-positivity.

Keywords
domains of discontinuity, Anosov representations, symmetric spaces of noncompact type
Mathematical Subject Classification
Primary: 53C35
Secondary: 22E40
References
Publication
Received: 31 March 2023
Revised: 7 May 2024
Accepted: 30 August 2024
Published: 14 August 2025
Proposed: Anna Wienhard
Seconded: David Fisher, Ian Agol
Authors
Colin Davalo
Mathematisches Institut
Ruprecht-Karls Universität Heidelberg
Heidelberg
Germany

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