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Degenerations of k-positive surface group representations

Jonas Beyrer, Beatrice Pozzetti

Abstract

We introduce \emph{k-positive representations}, a large class of $\{1,\ldots,k\}$--Anosov surface group representations into PGL(E) that share many features with Hitchin representations, and we study their degenerations: unless they are Hitchin, they can be deformed to non-discrete representations, but any limit is at least (k-3)-positive and irreducible limits are (k-1)-positive. A major ingredient, of independent interest, is a general limit theorem for positively ratioed representations.

Degenerations of k-positive surface group representations

Abstract

We introduce \emph{k-positive representations}, a large class of --Anosov surface group representations into PGL(E) that share many features with Hitchin representations, and we study their degenerations: unless they are Hitchin, they can be deformed to non-discrete representations, but any limit is at least (k-3)-positive and irreducible limits are (k-1)-positive. A major ingredient, of independent interest, is a general limit theorem for positively ratioed representations.

Paper Structure

This paper contains 32 sections, 76 theorems, 95 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. Anosov representations
  5. Strong irreducibility
  6. Cross ratios
  7. Property $H_k$
  8. Property $C_k$
  9. Semi-simplification
  10. Exterior power of representations
  11. Boundary maps for limit representations
  12. Condition (Tr) and transversality
  13. Boundary map for the limit representation
  14. Limits of positively ratioed representations
  15. Zariski dense limits
  16. ...and 17 more sections

Key Result

Theorem 1

Let $\{\rho_n:\Gamma\to \mathsf{PGL}(E)\}_{n\in \mathbb N}$ be $k$--positive and converge to $\rho_0$ in ${\rm Hom}(\Gamma, \mathsf{PGL}(E))$. Then the following hold:

Figures (4)

  • Figure 1: The three possible configuration of points in the proof of Proposition \ref{['prop.limits of spr: transverse orbits']}. In all cases the 4-tuples $(s,\gamma t,h\gamma t,hs), (\gamma^{-1} s, t,g\gamma t,gs), (s,h\gamma t, t,gs)$ are cyclically ordered
  • Figure 2: The configuration of points in the proof of Lemma \ref{['lem.cro left limit exists']}
  • Figure 3: The configuration of points in Lemma \ref{['lem.strict convex for mixed quotient']}
  • Figure 4: The points $p_n(h_i^+)$ for $i=1,2,3$ can not be joined by a convex curve which is at $p_n(h_2^+)$ tangent to $p_n^{(2)}(h_2^+)$.

Theorems & Definitions (140)

  • Definition 1.1
  • Theorem 1
  • Theorem 2
  • Proposition 1.2: Proposition \ref{['lem.dynamics preserving for strongly irred']}
  • Definition 2.1
  • Remark 2.2
  • Proposition 2.3: Guichard-Wienhard-IM
  • Example 2.4: Fuchsian Loci
  • Proposition 2.5: Labourie-IM
  • proof
  • ...and 130 more