Coxeter groups and nonpositive curvature

Coxeter群是球面上或欧氏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。我计划证明由伯恩哈德Muehlherr提出的扭曲猜想作为Coxeter群同构问题的假设解决方案。扭曲猜想预言不同的Coxeter生成集由一系列扭曲相关。扭曲是Coxeter生成集与有限子群中最长元素的部分共轭。** 我还打算证明Coxeter群的双自动性，说在群上有特定的路系统。策略是首先证明Coxeter群在弱模图上的几何作用-这是收缩和CAT（0）立方复形的1-β的一个广泛的普遍推广。第二步是证明几何作用在弱模图上的群是双自动的。第二个方向是研究各种非正曲群的Tits Alternative，即非正曲群生成的子群包含一个非阿贝尔自由群或实际上是可解的。我计划首先对收缩期组进行，然后对二维Artin组和FC型Artin组进行。 最后，我想使用CAT（0）空间作为k^3的驯服自同构群，以解决同样在这种情况下的乳房替代。最后，我想学习曲面上的圆弧和曲线系统，以及多面体的锐角三角剖分。

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.