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Geometrization of almost extremal representations in PSL$_2\Bbb R$. (arXiv:1802.00755v1 [math.GT])

来源于:arXiv
Let $S$ be a closed surface of genus $g$. In this paper, we investigate the relationship between hyperbolic cone-structure on $S$ and representations of the fundamental group into PSL$_2\Bbb R$. We prove that, if $S$ has genus $2$, then every representation $\rho$ with Euler number $\mathcal{E}(\rho)=\pm1$ sends a simple non-separating curve to an elliptic element. After that, we use this result to derive that every representation with $\mathcal{E}(\rho)=\pm1$ arises as the holonomy of some hyperbolic cone-structure. Finally we consider surfaces of genus greater than $2$ and we show that, under suitable condition, every representation $\rho:\pi_1 S\longrightarrow $ PSL$_2\Bbb R$ with Euler number $\mathcal{E}(\rho)=\pm\big(\chi(S)+1\big)$ arises as holonomy of a hyperbolic cone-structure $\sigma$ on $S$ with a single cone point of angle $4\pi$. 查看全文>>