Traces in Complex Hyperbolic Triangle Groups
Traces in Complex Hyperbolic Triangle Groups
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DOI：
10.1007/s10711-004-1493-0
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发表时间：
2004-02
影响因子：
0.5
通讯作者：
A. Pratoussevitch
中科院分区：
文献类型：
--
作者：
A. Pratoussevitch
本文给出了复反射产生的复双曲三角形群中元素迹的几个公式。这种固定签名群的空间是真实的一维的。我们用复双曲平面上三角形的一个真实的不变量α来参数化这个空间。本文的主要结果是一个公式，它将群中元素的迹表示为在eiα中的Laurent多项式，其系数与α无关，并且可以使用一定的组合缠绕数来计算。我们还给出了这些Laurent多项式的递推公式，并推广了复μ-反射生成群的迹公式。我们应用这些公式证明了复双曲三角形群的一些离散性和非离散性结果。
We present several formulas for the traces of elements in complex hyperbolic triangle groups generated by complex reflections. The space of such groups of fixed signature is of real dimension one. We parameterise this space by a real invariant α of triangles in the complex hyperbolic plane. The main result of the paper is a formula, which expresses the trace of an element of the group as a Laurent polynomial in eiαwith coefficients independent of α and computable using a certain combinatorial winding number. We also give a recursion formula for these Laurent polynomials and generalise the trace formulas for the groups generated by complex μ-reflections. We apply these formulas to prove some discreteness and some non-discreteness results for complex hyperbolic triangle groups.