On totally hyperbolic non-Fuchsian type-preserving representations

We identify type-preserving representations $\phi :{\pi }_1(\mathrm{\Sigma })\to \mathrm{P}\mathrm{S}\mathrm{L}(2,\mathbb{R})$ of the fundamental group of every punctured surface $\mathrm{\Sigma }={\mathrm{\Sigma }}_{g,p}$ 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 $e(\phi )=\pm (\chi (\mathrm{\Sigma })+1)$, and their $\mathrm{P}\mathrm{S}\mathrm{L}(2,\mathbb{R})$-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 $\mathrm{\Sigma }$ are Fuchsian.