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Bernhard Muehlherr提出的Twist猜想是Coxeter群同构问题的假设解。扭曲猜想预测，不同的Coxeter生成集由一系列扭曲联系在一起。扭曲是有限子群中最长元素对Coxeter生成集的部分共轭。 我还计划证明Coxeter群的双自动机，即群上有特定的路径系统。我们的策略是首先证明Coxeter群在弱模图上起几何作用--这是收缩复形和CAT(0)立方复形的1-骨架的一个深远的常见推广。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曲群的Tits替代，即有限生成的子群包含一个非交换自由群或实际上是可解的。我计划首先为收缩组做这件事，然后为二维Artin组和FC类型的Artin组做。最后，我想使用k^3的驯服自同构群的cat(0)空间来解决同样在那个上下文中的Tits选择。 最后，我想研究曲面上的圆弧和曲线系统，以及多面体的锐化三角剖分。

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.