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On totally hyperbolic non-Fuchsian type-preserving representations

Inyoung Ryu

TL;DR

The paper constructs explicit counterexamples to Bowditch's question by exhibiting uncountably many totally hyperbolic but non-Fuchsian type-preserving representations of punctured surfaces. These representations have relative Euler class $e(φ)=n$ with $n$ in the Milnor–Wood-satisfying range and, for certain sign data $s$, their PSL$(2,ℝ)$-conjugacy classes occupy a full-measure subset of $2p$ components of the relative character variety, demonstrating that total hyperbolicity does not force Fuchsian behavior on punctured surfaces. A key feature is that, although globally non-Fuchsian, these representations restrict to Fuchsian (holonomy) representations on suitable subsurfaces, revealing an almost-Fuchsian structure on pieces of $Σ$. The results illuminate the dynamics of the mapping class group action on relative character varieties and provide new insights into Goldman-type conjectures in the punctured setting.

Abstract

We identify type-preserving representations $φ: π_1(Σ)\to \mathrm{PSL}(2,\mathbb{R})$ of the fundamental group of every punctured surface $Σ= Σ_{g,p}$ that are not Fuchsian yet send all non-peripheral simple closed curves to hyperbolic elements, which give a negative answer to a question of Bowditch. These representations have relative Euler class $e(φ) = \pm (χ(Σ) + 1)$, and their $\mathrm{PSL}(2,\mathbb{R})$-conjugacy classes form a full-measure subset of $2p$ connected components of the relative character variety. We further show that, while these representations are not Fuchsian, their restrictions to certain subsurfaces of $Σ$ are Fuchsian.

On totally hyperbolic non-Fuchsian type-preserving representations

TL;DR

The paper constructs explicit counterexamples to Bowditch's question by exhibiting uncountably many totally hyperbolic but non-Fuchsian type-preserving representations of punctured surfaces. These representations have relative Euler class with in the Milnor–Wood-satisfying range and, for certain sign data , their PSL-conjugacy classes occupy a full-measure subset of components of the relative character variety, demonstrating that total hyperbolicity does not force Fuchsian behavior on punctured surfaces. A key feature is that, although globally non-Fuchsian, these representations restrict to Fuchsian (holonomy) representations on suitable subsurfaces, revealing an almost-Fuchsian structure on pieces of . The results illuminate the dynamics of the mapping class group action on relative character varieties and provide new insights into Goldman-type conjectures in the punctured setting.

Abstract

We identify type-preserving representations of the fundamental group of every punctured surface that are not Fuchsian yet send all non-peripheral simple closed curves to hyperbolic elements, which give a negative answer to a question of Bowditch. These representations have relative Euler class , and their -conjugacy classes form a full-measure subset of connected components of the relative character variety. We further show that, while these representations are not Fuchsian, their restrictions to certain subsurfaces of are Fuchsian.

Paper Structure

This paper contains 9 sections, 17 theorems, 36 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminary
  3. The relative Euler classes
  4. The signs
  5. Images of commutator and product maps
  6. Proof of the main results
  7. Proof of Proposition \ref{['prop_nonpara']}: $\mathrm{NP}^s_n(\Sigma)$ has full measure in $\mathcal{R}_n^s(\Sigma)$
  8. Proof of Theorem \ref{['thm_tothyp']}: total hyperbolicity of representations in $\mathrm{NP}^s_n(\Sigma)$
  9. Proof of Corollary \ref{['cor_almost_fuchsian']}

Key Result

Theorem 1.3

Let $\Sigma = \Sigma_{g,p}$ with $\chi(\Sigma)\leqslant -2$ and $p\geqslant 1$, and assume $n \in \mathbb{Z}$ and $s\in \{\pm 1\}^p$ satisfy one of the following: Then there exist uncountably many non-Fuchsian type-preserving representations $\phi:\pi_1(\Sigma)\to \mathrm{PSL}(2,\mathbb{R})$ of relative Euler class $e(\phi) = n$ and sign $s(\phi) = s$ that are totally hyperbolic. Furthermore, the

Figures (4)

  • Figure 1: The universal covering $\widetilde{\mathrm{SL}(2,\mathbb{R})}$
  • Figure 2: The image of the commutator map $\widetilde{R}: \widetilde{\mathrm{SL}(2,\mathbb{R})}\times \widetilde{\mathrm{SL}(2,\mathbb{R})}\to \widetilde{\mathrm{SL}(2,\mathbb{R})}$
  • Figure 3: $\mathrm{Ell}_{-1}$ and $\mathrm{Ell}_{2}$
  • Figure 4: Decomposition of $\Sigma$ for the induction.

Theorems & Definitions (25)

  • Definition 1.1
  • Theorem 1.3
  • Remark 1.4
  • Corollary 1.5
  • Proposition 2.1
  • Proposition 2.2
  • Proposition 2.3
  • Theorem 2.4
  • Theorem 2.5
  • Theorem 2.6
  • ...and 15 more