论文标题
在曾经是符合的圆环的Teichmüller空间上的无限产品
An infinite product on the Teichmüller space of the once-punctured torus
论文作者
论文摘要
我们证明了身份$$ \prod_γ\ left（\ frac {e^{l（γ）}+1} {e^{e^{l（γ）} -1} \ right）^{2H} = \ exp \ exp \ left（ \prod_γ\ left（\ frac {t（γ）^2} {t（γ）^2-4} \ right）^h = \ frac {t_1+\ sqrt {t_1^2-4} } {2} \ cdot \ frac {t_2+\ sqrt {t_2^2-4}}} {2} {2} \ cdot \ frac {t_3+\ sqrt {t_3^2-4}}} $$在痕量坐标中），其中产品在所有简单的封闭地球学上都在曾经是由曾经被调用的圆环上，$ l（γ）= 2 \ perperatorName {arccosh}（t（γ）/2）$是地球的长度，是$ l_i $（$ t_i $）的长度，而$ l_i $（$ t_i $）是任何trace $ geodes $ $ geodes $。在一个点相交。指数$ h = h（γ; \ {γ_i\}）$是一个积极的整数“高度”，随着我们从其$ sl_2（\ mathbb {z}）下的轨道中脱离所选的三重$ \ {γ_i\} $时，它会增加。为了进行比较，简短证明了麦克沙恩的身份$ \sum_γ\ frac {1} {1+e^{l（γ）}} = \ frac {1} {2} = \sum_γ\ frac \ frac {1- \ sqrt {1- \ sqrt {1-4/t {1-4/t {1-4/t {γ2}}}这两个证明都是基本的，并通过其TeichMüller轨道中的所选三重$ \ {γ_i\} $进行“集成”。
We prove the identity $$ \prod_γ\left(\frac{e^{l(γ)}+1}{e^{l(γ)}-1}\right)^{2h}=\exp\left(\frac{l_1+l_2+l_3}{2}\right), $$ (or $$ \prod_γ\left(\frac{t(γ)^2}{t(γ)^2-4}\right)^h=\frac{t_1+\sqrt{t_1^2-4}}{2}\cdot\frac{t_2+\sqrt{t_2^2-4}}{2}\cdot\frac{t_3+\sqrt{t_3^2-4}}{2} $$ in trace coordinates), where the product is over all simple closed geodesics on the once-punctured torus, $l(γ)=2\operatorname{arccosh}(t(γ)/2)$ is the length of the geodesic, and $l_i$ ($t_i$) are the lengths (traces) of any triple of simple geodesics $\{γ_i\}$ intersecting at a single point. The exponent $h=h(γ;\{γ_i\})$ is a positive integer "height" which increases as we move away from the chosen triple $\{γ_i\}$ in its orbit under $SL_2(\mathbb{Z})$ (see Figure 1 for the "definition by picture"). For comparison, a short proof of McShane's identity $$ \sum_γ\frac{1}{1+e^{l(γ)}}=\frac{1}{2}=\sum_γ\frac{1-\sqrt{1-4/t(γ)^2}}{2} $$ in the same spirit is given in an appendix. Both proofs are elementary and proceed by "integrating" around the chosen triple $\{γ_i\}$ in its Teichmüller orbit.
