Character簇的电流、收缩和紧化
Currents, Systoles, and Compactifications of Character Varieties
摘要：
我们研究了外尔室长度边界的希钦和最大的字符品种，并确定其中的一个开放的一组不连续的行动的映射类群。这一结果是作为一个典型的分解的结果，一个有限型表面上的测地线电流所产生的拓扑分解的表面沿特殊测地线。我们发现，每个组件要么是相关的测量层压或具有积极的收缩。对于一个电流与正收缩，我们表明，交叉功能的一组闭曲线是bi-Lipschitz等价的长度函数关于一个双曲度量。本文关于电流的结果将arXiv：1710.07060v1中的结果推广到具有测地线边界的有限面积表面的情况。关于外尔腔边界，我们改进了arXiv：1710.07060v1中的结果，表明对于更高秩的群，所述不连续的开集不是空的。我们也给出明确的例子的情况下，$SL(3,\mathbb R)$ -Hitchin组件。
英文摘要：
We study the Weyl chamber length boundary both of the Hitchin and of the maximal character varieties and determine therein an open set of discontinuity for the action of the mapping class group. This result is obtained as consequence of a canonical decomposition of a geodesic current on a surface of finite type arising from a topological decomposition of the surface along special geodesics. We show that each component either is associated to a measured lamination or has positive systole. For a current with positive systole, we show that the intersection function on the set of closed curves is bi-Lipschitz equivalent to the length function with respect to a hyperbolic metric. The results of this paper on currents generalise the ones in arXiv:1710.07060v1 to the case of surfaces of finite area with geodesic boundary. Concerning the Weyl chamber boundary we improve on the results in arXiv:1710.07060v1 by showing that for higher rank groups, said open set of discontinuity is not empty. We give also explicit examples in the case of the $SL(3,\mathbb R)$-Hitchin component.
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