论文标题
仅在环上定义的简单连接的分裂组的牢固界限
Strong boundedness of simply connected split Chevalley groups defined over rings
论文作者
论文摘要
本文涉及S-砷拆分雪佛莉组的某些单词规范的直径。众所周知，这些组是由根部元素产生的。我们证明，共轭类给出的单词指标在s- arithmetic split Chevalley群体上仅取决于共轭类的数量。假设r是主要的理想域和$ n \ geq 3 $，则该属性称为“强界”，是由Kedra，Libmann和Martin引入的，并以$ {\ rm sl} _n（r）$进行了证明。我们还提供了S-Arithmetic Split Chevalley组的正常生成集的示例，证明我们的边界在适当的意义上是尖锐的，并完整地说明了存在$ {\ rm sp} _4（r）$和$ g_2 $和$ g_2（r）$的小型正常生成的集合的障碍。例如，我们证明$ {\ rm sp} _4（\ mathbb {z} [\ frac {\ frac {1+ \ sqrt {-7}} {2}] $无法由单个共轭类生成。
This paper is concerned with the diameter of certain word norms on S-arithmetic split Chevalley groups. Such groups are well known to be boundedly generated by root elements. We prove that word metrics given by conjugacy classes on S-arithmetic split Chevalley groups have an upper bound only depending on the number of conjugacy classes. This property, called strong boundedness, was introduced by Kedra, Libmann and Martin and proven for ${\rm SL}_n(R)$, assuming R is a principal ideal domain and $n\geq 3$. We also provide examples of normal generating sets for S-arithmetic split Chevalley groups proving our bounds are sharp in an appropriate sense and give a complete account of obstructions to the existence of small normally generating sets of ${\rm Sp}_4(R)$ and $G_2(R)$. For instance, we prove that ${\rm Sp}_4(\mathbb{Z}[\frac{1+\sqrt{-7}}{2}])$ cannot be generated by a single conjugacy class.
