Non-injective representations of a closed surface group into PSL(2, ${mathbb R}$ )
Non-injective representations of a closed surface group into PSL(2, ${mathbb R}$ )
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封闭曲面群的非内射表示为 PSL(2, ${mathbb R}$ )
DOI：
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发表时间：
2005
影响因子：
0.8
通讯作者：
M. Wolff
中科院分区：
文献类型：
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作者：
L. Funar;M. Wolff
抽象让e表示空间上的欧拉类$ {ext {m Hom} (Gamma_g, {PSL (2, {mathbb R})}}美元表示的基本群Γg属的封闭曲面Σg g。高盛显示连接组件的$ {ext {m Hom} (Gamma_g, {PSL (2, {mathbb R})}}正是美元逆图片e−1 (k), 2−2 g≤k≤2 g−2,和欧拉的组件类2−2 g和2 g−2包含图像是一个离散的子群的单射表示美元的{PSL (2, {mathbb R})} $。我们证明了非忠实表示在所有其他分量中都是密集的。我们证明了离散表示的象本质上决定了它的欧拉类。此外，我们证明了对于每一个格和可能对应的欧拉类，都存在离散表示。
Abstract Let e denote the Euler class on the space ${ ext{
m Hom}(Gamma_g,{PSL(2,{mathbb R})}}$ of representations of the fundamental group Γg of the closed surface Σg of genus g. Goldman showed that the connected components of ${ ext{
m Hom}(Gamma_g,{PSL(2,{mathbb R})}}$ are precisely the inverse images e −1(k), for 2−2g≤ k≤ 2g−2, and that the components of Euler class 2−2g and 2g−2 consist of the injective representations whose image is a discrete subgroup of ${PSL(2,{mathbb R})}$ . We prove that non-faithful representations are dense in all the other components. We show that the image of a discrete representation essentially determines its Euler class. Moreover, we show that for every genus and possible corresponding Euler class, there exist discrete representations.