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PreprintCC BY 4.0arXiv

From Bowditch question to Goldman conjecture for type-preserving representations

  • Viraj Joshi
  • Inyoung Ryu
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Abstract

For punctured surfaces Σg,p of genus ggeqslant 2, we explore the dynamics of the mapping class group action on the relative PSL(2,ℝ)-character varieties of type-preserving representations. For such relative character varieties, Goldman's conjecture predicts that the mapping class group acts ergodically on their non-Teichmüller components. A related question of Bowditch asks whether every non-elementary type-preserving representation that is non-Fuchsian sends some non-peripheral simple closed curve to a non-hyperbolic element. We show that, on the components of the relative character variety indexed by fixed signs of images of peripheral elements and relative Euler classes non-extremal in the generalized Milnor-Wood inequality, an affirmative answer to Bowditch's question implies Goldman's conjecture.

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Cite

BibTeXRIS
BibTeX
@misc{joshi2026from,
  title = {{From Bowditch question to Goldman conjecture for type-preserving representations}},
  author = {Viraj Joshi and Inyoung Ryu},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.02065},
  eprint = {2609.02065},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.02065},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - From Bowditch question to Goldman conjecture for type-preserving representations
AU  - Viraj Joshi
AU  - Inyoung Ryu
PY  - 2026
DA  - 2026/09/27
PB  - arXiv
DO  - 10.48550/arxiv.2609.02065
UR  - https://arxiv.org/abs/2609.02065
AB  - For punctured surfaces Σ_g,p of genus ggeqslant 2, we explore the dynamics of the mapping class group action on the relative PSL(2,ℝ)-character varieties of type-preserving representations. For such relative character varieties, Goldman's conjecture predicts that the mapping class group acts ergodically on their non-Teichmüller components. A related question of Bowditch asks whether every non-elementary type-preserving representation that is non-Fuchsian sends some non-peripheral simple closed curve to a non-hyperbolic element. We show that, on the components of the relative character variety indexed by fixed signs of images of peripheral elements and relative Euler classes non-extremal in the generalized Milnor-Wood inequality, an affirmative answer to Bowditch's question implies Goldman's conjecture.
ER  -