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Positive surface group representations in PO(p,q)

Jonas Beyrer, Beatrice Pozzetti

Abstract

We show that $Θ$-positive Anosov representations $ρ:Γ\to{\sf PO}(p,q)$ of a surface group $Γ$ satisfy root versus weight collar lemmas for all the Anosov roots, and are positively ratioed with respect to all such roots. We deduce from this, using a result of Beyrer-Pozzetti (2024), that $Θ$-positive Anosov representations $ρ:Γ\to{\sf PO}(p,q)$ form connected components of character varieties.

Positive surface group representations in PO(p,q)

Abstract

We show that -positive Anosov representations of a surface group satisfy root versus weight collar lemmas for all the Anosov roots, and are positively ratioed with respect to all such roots. We deduce from this, using a result of Beyrer-Pozzetti (2024), that -positive Anosov representations form connected components of character varieties.

Paper Structure

This paper contains 22 sections, 45 theorems, 130 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Anosov representations
  4. Cross ratios
  5. Property $H_k$
  6. $\Theta$-Positive structure for $\mathsf{PO}(p,q)$
  7. $\Theta$-Positive elements
  8. $\Theta$-positivity of triples of partial flags
  9. $\Theta$-Positive triples in $\mathsf{PO}(2,q)$
  10. $\Theta$-Positive representations and $\Theta$-positive curves
  11. The boundary map of a $\Theta$-positive representation
  12. $\Theta$-positive representations are positively ratioed
  13. Tangent cones
  14. The derivative of the cross ratio
  15. Collar lemmas for $\Theta$-positive representations
  16. ...and 7 more sections

Key Result

Theorem 1

(Theorem thm.pos ratioed) Let $\rho:\Gamma\to\mathsf{PO}(p,q)$ be a $\Theta$-positive Anosov representation, then $\rho$ is $k$-positively ratioed for all $k< p$.

Figures (3)

  • Figure 1: The proof of Theorem \ref{['t.collarmax']}.
  • Figure 2: The order of $x,y,y_0,y_t,z,w,u$ for $t>0$.
  • Figure 3: The proof of Theorem \ref{['t.collar']}.

Theorems & Definitions (92)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Corollary 1.1
  • Remark 1.2
  • Theorem : cfr. ABC-IM
  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Proposition 2.4
  • ...and 82 more