Coxeter groups and nonpositive curvature

Coxeter群是反射群在球面或欧几里德空间中的推广。我的第一个方向是研究Coxeter群的组合学。我打算证明Bernhard Muehlherr提出的Twist猜想作为Coxeter群同构问题的假设解。扭转猜想预测不同的Coxeter发电机组是由一系列扭转联系在一起的。扭转是有限子群中最长元对Coxeter生成集的部分共轭。**我还计划证明Coxeter群的双自动性，说群上有特定的路径系统。策略是首先显示Coxeter群在弱模图上的几何作用——收缩和CAT(0)立方复合体的1-骨架的一种深远的共同推广。第二步是证明在弱模图上几何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，说明有限生成的子群包含一个非阿贝尔自由群或实际上是可解的。我计划先对收缩期组进行实验，然后对二维Artin组和FC型Artin组进行实验。最后，我想对k^3的驯服自同构群使用CAT(0)空间来解决在这种情况下的Tits替代。******最后，我想学习曲面上的弧和曲线系统，以及多面体的锐角三角形。

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.