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$Θ$-positive representations over real closed fields

Xenia Flamm, Nicolas Tholozan, Tianqi Wang, Tengren Zhang

TL;DR

This work unifies and extends the theory of $\Theta$-positive representations to semisimple groups over real closed fields, including non-Archimedean settings, by developing a semi-algebraic framework for flag-positivity and boundary data. It introduces a general notion of $\Theta$-positivity that does not presuppose boundary maps, proves equivalences with weak proximality and positive proximal limit maps, and constructs positive boundary extensions along with maximal framed variants. The authors show that positivity is open and closed in representation varieties and analyze the real spectrum compactification to relate positivity to semi-algebraic data across fields; they also provide a Cantor-complete-field surjectivity result for boundary data and furnish explicit examples (including non-frameable cases) to delineate the limits of framing. Collectively, these results yield a robust, field-agnostic theory connecting hyperbolic geometry, higher Teichmüller theory, and real algebraic geometry, with consequences for discrete, faithful representations and their boundary dynamics across diverse algebraic settings.

Abstract

We develop the theory of $Θ$-positive representations from general Fuchsian groups to linear groups over real closed fields. Our definition, which does not assume the boundary map to be continuous, encompasses many generalizations of positive or Anosov representations that have been considered in the literature.

$Θ$-positive representations over real closed fields

TL;DR

This work unifies and extends the theory of -positive representations to semisimple groups over real closed fields, including non-Archimedean settings, by developing a semi-algebraic framework for flag-positivity and boundary data. It introduces a general notion of -positivity that does not presuppose boundary maps, proves equivalences with weak proximality and positive proximal limit maps, and constructs positive boundary extensions along with maximal framed variants. The authors show that positivity is open and closed in representation varieties and analyze the real spectrum compactification to relate positivity to semi-algebraic data across fields; they also provide a Cantor-complete-field surjectivity result for boundary data and furnish explicit examples (including non-frameable cases) to delineate the limits of framing. Collectively, these results yield a robust, field-agnostic theory connecting hyperbolic geometry, higher Teichmüller theory, and real algebraic geometry, with consequences for discrete, faithful representations and their boundary dynamics across diverse algebraic settings.

Abstract

We develop the theory of -positive representations from general Fuchsian groups to linear groups over real closed fields. Our definition, which does not assume the boundary map to be continuous, encompasses many generalizations of positive or Anosov representations that have been considered in the literature.
Paper Structure (65 sections, 127 theorems, 318 equations, 4 figures)

This paper contains 65 sections, 127 theorems, 318 equations, 4 figures.

Table of Contents

  1. Introduction
  2. A general notion of Theta-positive representation
  3. Examples
  4. Continuity of the boundary map
  5. Positivity and extended geometric finiteness
  6. Theta-positivity is open and closed
  7. Theta-positivity and the real spectrum compactification
  8. Related works
  9. Structure of the paper
  10. Preliminaries
  11. Fuchsian groups
  12. Fuchsian groups acting on their limit sets
  13. Conical limit points
  14. The geometrically finite case
  15. Ordered fields
  16. ...and 50 more sections

Key Result

Theorem 1.1

The set of $\Theta$-positive representations of a closed surface group into a real linear group $G$ forms a union of connected components.

Figures (4)

  • Figure 1: A $\Theta$-positively rotating element (left) and a $\Theta$-positively translating element (right)
  • Figure 2: The orbit of $g$
  • Figure 3: A $\Theta$-positive representation that is not frameable.
  • Figure 4: A positive representation that is not frameable.

Theorems & Definitions (271)

  • Theorem 1.1: GLWBeyrerGuichardLabouriePozzettiWienhard_PositivityCrossRatiosCollarLemma
  • Definition 1.2
  • Theorem A
  • Definition 1.3
  • Theorem B
  • Theorem C
  • Remark 1.4
  • Theorem D
  • Theorem E
  • Theorem F
  • ...and 261 more