论文标题
小型coxeter组的一致性亚组和晶体学商
Congruence subgroups and crystallographic quotients of small Coxeter groups
论文作者
论文摘要
小型代码组恰恰是山雀表示是不可或缺的，这使得其一致性亚组的研究相关。对称组$ s_n $具有三个天然扩展名，即编织组$ b_n $，Twin Group $ t_n $和Triplet Group $ l_n $。后两个组是小型coxeter群，并在适当的结理论的亚历山大·马尔科夫（Alexander-Markov）信件下扮演辫子基团的作用，其纯粹的亚组将合适的超平面布置作为Eilenberg-Maclane空间。在本文中，我们证明了几乎不是阿贝利亚人的无限小型库集团的一致性亚组属性失败。作为一个应用程序，我们推断出一致性子组属性在$ t_n $和$ l_n $中失败，当时$ n \ ge 4 $。我们还确定了$ t_n $的主要一致性子组的子粒子，并确定纯双胞胎组$ pt_n $和纯的三重态组$ pl_n $，以及合适的主要一致性子组。此外，我们研究了这两个小型Coxeter组的晶体学商，并证明$ t_n /pt_n^{'} $，$ t_n /t_n /t_n^{''} $和$ l_n /pl_n /pl_n^{'} $是晶体学组。我们还确定了这些组的晶体学维度，并确定$ t_n/t_n^{''} $的整体表示。
Small Coxeter groups are precisely the ones for which the Tits representation is integral, which makes the study of their congruence subgroups relevant. The symmetric group $S_n$ has three natural extensions, namely, the braid group $B_n$, the twin group $T_n$ and the triplet group $L_n$. The latter two groups are small Coxeter groups, and play the role of braid groups under the Alexander-Markov correspondence for appropriate knot theories, with their pure subgroups admitting suitable hyperplane arrangements as Eilenberg-MacLane spaces. In this paper, we prove that the congruence subgroup property fails for infinite small Coxeter groups which are not virtually abelian. As an application, we deduce that the congruence subgroup property fails for both $T_n$ and $L_n$ when $n \ge 4$. We also determine subquotients of principal congruence subgroups of $T_n$, and identify the pure twin group $PT_n$ and the pure triplet group $PL_n$ with suitable principal congruence subgroups. Further, we investigate crystallographic quotients of these two families of small Coxeter groups, and prove that $T_n /PT_n^{'}$, $T_n/T_n^{''}$ and $L_n /PL_n^{'}$ are crystallographic groups. We also determine crystallographic dimensions of these groups and identify the holonomy representation of $T_n/T_n^{''}$.
