arXiv 0901.14042009-01-11math.GTmath.AC

一类简单双曲曲面的迹坐标

Trace Coordinates on Fricke spaces of some simple hyperbolic surfaces

William M. Goldman

备注：: To appear in "Handbook of Teichmueller theory, volume II". The paper substantially expands, and extends, math.GM/0402103 An exposition of results of Fricke
发表信息：: Handbook of Teichmüller theory. Vol. II, 611-684, IRMA Lect. Math. Theor. Phys., 13, Eur. Math. Soc., Zürich, 2009

摘要：

一般么模2 × 2复矩阵的共轭类由它的迹决定，迹可以是任意的复数。在19世纪，人们知道这样的对的一般对（X，Y）由迹的三元组（tr（X），tr（Y），tr（XY））确定直到共轭，这可能是C^3的任意元素。本文给出了Vogt于1889年发表的关于这一事实的初步而详细的证明。民间定理描述的一个代表性的一个代表性的指数为2的超群，这是一个自由积的三个群体的秩序二，详细介绍，并涉及到双曲几何。当n > 2时，SL（2，C）中n元组的共轭类的分类更为复杂。当n= 3时，我们详细描述它。在迹坐标下计算了基本群不含二秩或三秩的简单曲面S上的双曲结构的形变空间。(We仅考虑其基本群具有秩3的两个可定向曲面。

英文摘要：

The conjugacy class of a generic unimodular 2 by 2 complex matrix is determined by its trace, which may be an arbitrary complex number. In the nineteenth century, it was known that a generic pair (X,Y) of such pairs is determined up to conjugacy by the triple of traces (tr(X),tr(Y),tr(XY), which may be an arbitary element of C^3. This paper gives an elementary and detailed proof of this fact, which was published by Vogt in 1889. The folk theorem describing the extension of a representation to a representation of the index-two supergroup which is a free product of three groups of order two, is described in detail, and related to hyperbolic geometry. When n > 2, the classification of conjugacy-classes of n-tuples in SL(2,C) is more complicated. We describe it in detail when n= 3. The deformation spaces of hyperbolic structures on some simple surfaces S whose fundamental group is free of rank two or three are computed in trace coordinates. (We only consider the two orientable surfaces whose fundamental group has rank 3.)

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