AI 中文总结
研究辫群$\text{Br}_4(d)$到$\text{Br}_3(d)$的特殊商群，用纯群论方法找到截断商映射核的表示形式为“本原”伯恩赛德群，并给出其有限性结果，还对相关解释作了评论。
AI 中文摘要
辫群$\text{Br}_4$到$\text{Br}_3$存在一个特殊商群，它与代数、拓扑和几何中的许多有趣构造相关。该商映射也可降至“截断”辫群$\text{Br}_4(d)$到$\text{Br}_3(d)$的商群，其半扭转生成元上有额外的挠关系。本文找到了这个截断商映射核的一个表示形式，即我们所谓的“本原”伯恩赛德群，并给出了这些本原伯恩赛德群的一些有限性结果。方法纯为群论方法，文末对涉及莱夫谢茨纤维化的解释作了评论。
英文摘要
There exists an exceptional quotient of braid groups $\text{Br}_4 \twoheadrightarrow \text{Br}_3$ that is related to many interesting constructions in algebra, topology, and geometry. This quotient map also descends to a quotient of "truncated" braid groups $\text{Br}_4(d) \twoheadrightarrow \text{Br}_3(d)$, which have an added torsion relation on their half twist generators. In this article, we find a presentation for the kernel of this truncated quotient map that takes the form of what we deem a "primitive" Burnside group. We give a few finiteness results on these primitive Burnside groups. Our methods are purely group theoretic, but we comment on an interpretation involving Lefschetz fibrations at the end.
Comments10 pages, 2 figures
