Coxeter groups and nonpositive curvature

Coxeter 群和非正曲率

• 批准号: RGPIN-2019-04458
• 负责人: Przytycki, Piotr
• 金额: $ 2.33万
• 依托单位: McGill University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2021
• 资助国家: 加拿大
• 起止时间: 2021-01-01 至 2022-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=737642
• 关键词: Coxeter; groups; nonpositive; curvature

Coxeter groups are generalisations of reflection group on the sphere or in the Euclidean space. My first direction is to study the combinatorics of Coxeter groups. I plan to prove the Twist Conjecture proposed by Bernhard Muehlherr as a hypothetical solution to the Isomorphism Problem for Coxeter groups. Twist Conjecture predicts that different Coxeter generating sets are related by a sequence of twists. A twist is a partial conjugation of the Coxeter generating set by a longest element in a finite subgroup. I also plan to prove biautomaticity of Coxeter groups, saying that that there is particular system of paths on the group. The strategy is to show first that Coxeter groups act geometrically on weakly modular graphs - a far reaching common generalization of the 1-skeleta of systolic and CAT(0) cubical complexes. The second step will be to prove that groups acting geometrically on weakly modular graphs are biautomatic. The second direction is to study the Tits Alternative for various nonpositively curved groups, saying that finitely generated subgroups contain a non-abelian free group or are virtually solvable. I plan to do it first for systolic groups, and next for 2-dimensional Artin groups and for Artin groups of FC type.  Finally I want to use the CAT(0) space for the tame automorphism group for k^3 to solve the Tits Alternative also in that context. Finally, I want to study arc and curve systems on surfaces, and acute triangulations of polyhedra.

考克塞特群是球体或欧几里德空间中反射群的推广。我的第一个方向是研究 Coxeter 群的组合学。我计划证明 Bernhard Muehlherr 提出的扭转猜想作为 Coxeter 群同构问题的假设解决方案。扭曲猜想预测不同的 Coxeter 发电机组通过一系列扭曲而相关。扭曲是 Coxeter 生成集与有限子群中最长元素的部分共轭。 我还计划证明 Coxeter 群的双自动性，即群上存在特定的路径系统。该策略首先是要证明 Coxeter 群在弱模图上的几何作用 - 收缩和 CAT(0) 立方复合体的 1 骨架的广泛普遍推广。第二步是证明在弱模图上以几何方式作用的群是双自动的。第二个方向是研究各种非正弯曲群的 Tits Alternative，认为有限生成子群包含非阿贝尔自由群或者实际上是可解的。我计划首先针对收缩组进行此操作，然后针对二维 Artin 组和 FC 类型的 Artin 组进行此操作。  最后，我想使用 k^3 的驯服自同构群的 CAT(0) 空间来解决该上下文中的 Tits Alternative 问题。最后，我想研究曲面上的圆弧和曲线系统以及多面体的锐角三角剖分。