射影初等群的基本多面体
Fundamental polyhedra of projective elementary groups
摘要：
对于虚二次环$O$，我们计算了$\text{PSL}_2(O)$的射影初等子群$\text{PE}_2(O)$的一个基本多面体.这使得新的，简化证明定理的科恩，尼卡，罚款，和弗罗曼。也就是说，我们得到了一个介绍$\text{PE}_2(O)$，表明它有无穷指数，是自己的正规化$\text{PSL}_2(O)$，并分裂$\text{PSL}_2(O)$到一个自由积与融合，有$\text{PE}_2(O)$作为其因素之一。
英文摘要：
For $O$ an imaginary quadratic ring, we compute a fundamental polyhedron of $\text{PE}_2(O)$, the projective elementary subgroup of $\text{PSL}_2(O)$. This allows for new, simplified proofs of theorems of Cohn, Nica, Fine, and Frohman. Namely, we obtain a presentation for $\text{PE}_2(O)$, show that it has infinite-index and is its own normalizer in $\text{PSL}_2(O)$, and split $\text{PSL}_2(O)$ into a free product with amalgamation that has $\text{PE}_2(O)$ as one of its factors.
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