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。

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.