Goldman’s conjecture for type-preserving representations

Goldman’s conjecture predicts ergodicity of the mapping class group action on appropriate character-space components for type-preserving representations. The related Bowditch question asks whether non-Fuchsian representations must send some simple closed curve to a non-hyperbolic element.

Known results

• Four-punctured sphere: ergodicity holds on every non-extremal component (Marché–Wolff, 2014).
• For $e(\rho)=\pm1$, there are uncountably many non-Fuchsian examples with every non-peripheral simple closed curve hyperbolic (Marché–Wolff, 2014).
• For $e(\rho)=0$ on the four-punctured sphere, some non-peripheral simple closed curve is elliptic or parabolic (Marché–Wolff, 2014).
• Souto proved Bowditch’s question for Euler-class-zero representations of closed surfaces of genus $g>3$; nonzero non-extremal cases remain open.

Recent reduction

A new paper proves that an affirmative answer to Bowditch’s question implies Goldman’s conjecture on components with fixed peripheral signs and non-extremal relative Euler classes satisfying the generalized Milnor–Wood inequality. This is a substantial reduction, not a solution of either conjecture, and the paper’s claim has not been independently verified here.

Current status (as of September 2026): A substantial conditional reduction is reported, but Bowditch’s question and Goldman’s conjecture remain open in general.