Finding generators and relations for groups acting on the hyperbolic ball
Finding generators and relations for groups acting on the hyperbolic ball
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寻找作用于双曲球的群的生成元和关系
DOI：
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发表时间：
2017
期刊：
影响因子：
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通讯作者：
T. Steger
中科院分区：
文献类型：
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作者：
D. I. Cartwright;T. Steger
为了枚举伪射影平面，正如在~cite{CS}中所宣布的，我们找到了每个极大算术子群$的显式生成元和表示。对于某个整数~$Nge 1 $，（适当归一化的）余体积等于~$1/N$的~$PU（2，1）$的arGamma$。Prasad和Yeung引用{PY 1，PY 2}给出了所有此类$arGamma$（直到等价）。
生成元是通过计算机搜索找到的，该搜索使用了$PU（2，1）$对单位球$B（C^2）$在~$C^2$中的自然作用。我们的主要结果给出了保证计算机搜索能找到足够多的~$元素的准则arGamma$生成$arGamma$，并描述了一个家庭的关系之间的发电机集足以给一个表示~$arGamma$.
我们给出了一个例子，详细说明了在特定的~$情况下如何做到这一点。arGamma$（其中$N=864$）。虽然在这种情况下不存在伪投影平面，但我们在~$中证明了一个指数为~$N$的无挠子群~$Pi$。arGamma$，并给出曲面~$Pi的一些性质ackslash B（C^2）$。
In order to enumerate the fake projective planes, as announced in~cite{CS}, we found explicit generators and a presentation for each maximal arithmetic subgroup $arGamma$ of~$PU(2,1)$ for which the (appropriately normalized) covolume equals~$1/N$ for some integer~$Nge1$. Prasad and Yeung cite{PY1,PY2} had given a list of all such $arGamma$ (up to equivalence).
The generators were found by a computer search which uses the natural action of $PU(2,1)$ on the unit ball $B(C^2)$ in~$C^2$. Our main results here give criteria which ensure that the computer search has found sufficiently many elements of~$arGamma$ to generate $arGamma$, and describes a family of relations amongst the generating set sufficient to give a presentation of~$arGamma$.
We give an example illustrating details of how this was done in the case of a particular~$arGamma$ (for which $N=864$). While there are no fake projective planes in this case, we exhibit a torsion-free subgroup~$Pi$ of index~$N$ in~$arGamma$, and give some properties of the surface~$Piackslash B(C^2)$.