Real spectrum compactification of Hitchin components, Weyl chamber valued lengths, and dual spaces

TL;DR

This work characterizes Hitchin representations over real closed fields by a positivity condition, showing that a representation into PSL(n, F) is conjugate to an F-Hitchin representation if and only if it is F-positive, using a real-logic transfer (Tarski–Seidenberg) and a multiplicative Bonahon–Dreyer coordinate system. It develops a semi-algebraic framework for the F-extension of Hitchin components, proves positive hyperbolicity and weak dynamics-preservation for F-positive representations, and constructs geodesic currents associated to these representations via positive cross-ratios. The results yield a semi-algebraic description of boundary points in the real spectrum compactification and enable applications to Weyl chamber length compactifications and dual spaces of geodesic currents, including piecewise semi-algebraic degeneration paths. Overall, the paper extends higher Teichmüller theory to arbitrary real closed fields, linking algebraic geometry, flag positivity, and geometric structures on surfaces.

Abstract

The main result of this article is that Hitchin representations over real closed field extensions $\mathbb{F}$ of $\mathbb{R}$ correspond precisely to those representations of the fundamental group of a closed surface into $\textrm{PSL}(n,\mathbb{F})$ that are conjugate to $\mathbb{F}$-positive representations, i.e. representations that admit an equivariant limit map from the set of fixed points in the boundary of the universal cover of the surface into the set of full flags in $\mathbb{F}^n$ satisfying specific positivity properties. As the theorem treats general real closed fields, and not only the reals, the tools of analysis are not available. Instead, our proof is based on the Tarski-Seidenberg transfer principle and a multiplicative version of the Bonahon-Dreyer coordinates. We use this result to prove that $\mathbb{F}$-positive representations form semi-algebraically connected components of the space of all representations, that consist entirely of injective and discrete representations, which are positively hyperbolic and weakly dynamics-preserving over $\mathbb{F}$. Furthermore, we show how to associate intersection geodesic currents to $\mathbb{F}$-positive representations, and conclude with applications to the Weyl chamber length compactification and to dual spaces of geodesic currents.

Figures (4)

• Figure 1: Monotonicity of the double ratios.
• Figure 2: The cyclically ordered tuple $(\gamma^+,x,\gamma^-,y,\gamma y)$.
• Figure 3: A maximal geodesic lamination of $S$ (left) and of a pair of pants (right). The white and shaded regions are ideal triangles.
• Figure 6: The component $\widetilde{t}$ and its $\gamma$-translate.

Theorems & Definitions (153)

• Theorem 1.0
• Definition 1.1
• Definition 1.2
• Definition 1.3
• Corollary 1.4
• Theorem 1.4
• Proposition 1.5: \ref{['lem_hitchinReprAreDiagonalizable']}, \ref{['propo_positiveReprHaveDistinctEigenvalues']}, \ref{['propo_FHitchinDiscrete']}
• Definition 1.6
• Theorem 1.7
• Corollary 1.8