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Spectrum of SL(2,R)-characters: the once-punctured torus case

Selim Ghazouani, Florestan Martin-Baillon

Abstract

Consider a topological surface $Σ$. We introduce the spectrum of a representation from the fundamental group of $Σ$ to SL(2,R), which is a subset of projective measured lamination on the surface, which captures the directions along which the representation fails to be Fuchsian, and which characterizes the action of the mapping class group on this representation. In the case of the once-punctured torus, we show that the spectrum of a generic representation is a Cantor set, and that it completely describes the dynamics of the familly of locally constant cocycles above interval exchange transformations associated to the representation.

Spectrum of SL(2,R)-characters: the once-punctured torus case

Abstract

Consider a topological surface . We introduce the spectrum of a representation from the fundamental group of to SL(2,R), which is a subset of projective measured lamination on the surface, which captures the directions along which the representation fails to be Fuchsian, and which characterizes the action of the mapping class group on this representation. In the case of the once-punctured torus, we show that the spectrum of a generic representation is a Cantor set, and that it completely describes the dynamics of the familly of locally constant cocycles above interval exchange transformations associated to the representation.

Paper Structure

This paper contains 46 sections, 28 theorems, 58 equations, 11 figures.

Table of Contents

  1. Spectrum of a representation
  2. Character variety
  3. Relative character varieties
  4. Space of measured laminations and measured foliations
  5. Measured foliations
  6. Measured laminations
  7. Spectrum
  8. The lifted dynamics
  9. Lyapunov exponent
  10. Spectrum
  11. Mapping class group action and the spectrum
  12. Reduction to interval exchange maps
  13. First-return map
  14. Cocycle over $I$
  15. Family of cocycles
  16. ...and 31 more sections

Key Result

Theorem 2

Let $\rho : \pi_1 ( \mathrm{T}^*) \longrightarrow \mathrm{SL}(2,\mathrm{R})$ be a non-simple Fuchsian representation which we further assume to be injective and that has no parabolic elements in its image. Then the spectrum of $\rho$ is a Cantor set.

Figures (11)

  • Figure 1: Moving from foliation to a map of the interval.
  • Figure 2: Examples of configurations where bottom or top wins
  • Figure 3: Transitions between types.
  • Figure 4: Product of a hyperbolic and an elliptic
  • Figure 5: The three different cases for $(H,H)^{-}$
  • ...and 6 more figures

Theorems & Definitions (48)

  • Conjecture 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Definition 1.1: Lyapunov exponent
  • Definition 1.2: Spectrum
  • Proposition 1.3
  • proof
  • Definition 2.1
  • ...and 38 more