AI 中文总结
该研究证明PSL(2,C)及PO(n,1)中算术格的子格正规化子性质，推导投射有限柔性格的集合特征，还证实有限群均可作为算术双曲流形的等距群，证明依赖格正规化子与子群增长理论。
AI 中文摘要
我们证明，PSL$(2,\mathbb{C})$中的每个算术格，以及PO$(n,1)$（$n\ge 2$）中每个最简型算术格，都是其任意多个子格的正规化子。结合此前研究，该结果表明PSL$(2,\mathbb{C})$中的所有格都具备这一性质。由此我们证明，PSL$(2,\mathbb{C})$中的投射有限柔性格集合要么为空，要么是可数无穷的。另一项结果为，每个有限群都可实现为某个算术双曲$n$维流形的完全等距群。该定理的证明基于对格的正规化子的研究以及子群增长理论。
英文摘要
We prove that every arithmetic lattice in PSL$(2,\mathbb{C})$ and every arithmetic lattice of the simplest type in PO$(n,1)$, $n\ge 2$, is the normalizer of arbitrarily many of its sublattices. Combined with previous work, this result implies that every lattice in PSL$(2,\mathbb{C})$ has this property. In this way, we prove that the set of profinitely flexible lattices in PSL$(2,\mathbb{C})$ is either empty or countably infinite. Another result is that every finite group is realized as the full isometry group of an arithmetic hyperbolic $n$-manifold. The proof of this theorem is based on study of normalizers of lattices and subgroup growth theory.
Comments12 pages. Comments welcome
