Moduli via double pants decompositions

详细信息    查看全文

• 作者：Anna Felikson ; Sergey Natanzon
• 关键词：32G15 ; 57M50
• 刊名：Differential Geometry and its Applications
• 出版年：2012
• 出版时间：October, 2012
• 年：2012
• 卷：30
• 期：5
• 页码：490-508
• 全文大小：439 K

文摘

We consider (local) parameterizations of Teichm¨¹ller space (of genus g hyperbolic surfaces with n boundary components) by lengths of geodesics. We find a large family of suitable sets of geodesics, each set forming a special structure called ¡°admissible double pants decomposition¡±. For admissible double pants decompositions containing no double curves we show that the lengths of curves contained in the decomposition determine the point of up to finitely many choices. Moreover, these lengths provide a local coordinate in a neighborhood of all points of where X is a union of hypersurfaces. Furthermore, there exists a groupoid acting transitively on admissible double pants decompositions and generated by transformations exchanging only one curve of the decomposition. The local charts arising from different double pants decompositions compose an atlas covering the Teichm¨¹ller space. The gluings of the adjacent charts are coming from the elementary transformations of the decompositions, the gluing functions are algebraic. The same charts provide an atlas for a large part of the boundary strata in Deligne-Mumford compactification of the moduli space .