Connected components of the space of type-preserving representations

Inyoung Ryu; Tian Yang

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arXiv · 2507.07391

Connected components of the space of type-preserving representations

Inyoung Ryu·Tian Yang

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Abstract

We complete the characterization of the connected components of the space of type-preserving representations of a punctured surface group into $\mathrm{PSL}(2,\mathbb{R})$. We show that the connected components are indexed by the relative Euler classes and the signs of the images of the peripheral elements satisfying a generalized Milnor-Wood inequality; and when the surface is a punctured sphere, there are additional connected components consisting of ``totally non-hyperbolic" representations. As a consequence, we count the total number of the connected components of the space of type-preserving representations.

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Inyoung Ryu, Tian Yang. 2025-07-10. Connected components of the space of type-preserving representations. https://arxiv.org/abs/2507.07391

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