Non-Archimedean Hilbert geometry and degenerations of real Hilbert geometries

TL;DR

The article develops Hilbert geometry over general ordered valued fields and ties ultralimit methods to degenerations of convex real projective structures, revealing non-Archimedean Hilbert geometries (via Robinson fields) as limit objects. It shows that ultralimits of rescaled real Hilbert geometries are isometric to Hilbert geometries over Robinson fields, and uses this to realize boundary points of moduli spaces as actions on non-Archimedean spaces without fixed points. A detailed theory for non-Archimedean integral polytopes identifies their Hilbert geometry with the flag-complex realization modeled on a Weyl chamber, and provides a complete description of asymptotic cones and Gromov–Hausdorff limits for real polytopal Hilbert geometries. The results connect degenerations of convex real projective structures to non-Archimedean dynamics, offering tools to study higher-rank phenomena and representation varieties via ultralimit and valuation-theoretic methods.

Abstract

We develop a theory of Hilbert geometry over general ordered valued fields, associating with an open convex subset of the projective space a quotient Hilbert metric space. Under natural non-degeneracy assumptions, we prove that the ultralimit of a sequence of rescaled real Hilbert geometries is isometric to the Hilbert metric space of an open convex projective subset over a Robinson field. This result allows us to prove that ideal points of the space of convex real projective structures on a closed manifold arise from actions on non-Archimedean Hilbert geometries without global fixed point. We explicitly describe the Hilbert metric space of a non-Archimedean bounded polytope $P$ defined over a subfield of the valuation ring as the geometric realization of the flag complex of $P$ modeled on a Weyl chamber. As an application, we obtain a complete description of Gromov-Hausdorff limits of a real polytope with rescaled Hilbert metric.

Figures (13)

• Figure 1: The Hilbert pseudo-distance on $\Omega$.
• Figure 2: The model flat $(\mathbb{A},d_{\textnormal{H}})$ with its unit ball and the closed Weyl chamber $\mathbb{A}^+$ for $d=3$.
• Figure 3: The geometric realization of the flag complex of a 6-gon.
• Figure 4: Sandwiching $\overline{\Omega}$ between two simplices.
• Figure 5: A barycentric simplex (left) and a flag simplex that is not a barycentric simplex (right).

Theorems & Definitions (85)

• Definition 1.1: Hilbert pseudo-distance
• Definition 1.2: Associated Hilbert metric space
• Theorem 1: \ref{['thm:UltralimitsHilbertGeometries']}
• Theorem 2: \ref{['thm:DegenerationsConvProjStr']}
• Theorem 3: \ref{['thm:NAPolytopalHilbertMetricSpace']}
• Lemma 4: Flag simplex sandwich lemma, \ref{['lem:FlagSimplexSandwichLemma']}
• Theorem 5: \ref{['thm:AsymptoticConesPolytopalHilbertGeometries']}
• Corollary 6
• Example 2.1
• Definition 2.2