论文标题
通用Coxeter组的差异，厚度和超图指数
Divergence, thickness and hypergraph index for general Coxeter groups
论文作者
论文摘要
我们研究一般Coxeter组$ W $的分歧和厚度。我们首先表征线性分歧，并表明如果$ w $具有超线性发散，那么它的差异至少是二次的。然后，我们为任意的Coxeter Systems $（W，S）$制定了可计算的组合不变型HyperGraph Index。这将列夫科维茨对右角案件的定义进行了概括。我们证明，如果$（w，s）$具有有限的超图指数$ h $，那么$ w $最多（强烈的代数）是$ h $的订单厚，因此，差异是在上面的差异$ h+1 $。我们猜想这些上限在厚度和差异的顺序上实际上是平等的，我们证明了我们对某些Coxeter群体的猜想。这些家族是通过新结构获得的，鉴于任何右角coxeter群，它会产生许多具有相同超刻录指数的非右角的coxeter系统的例子。最后，我们在任何coxeter系统$（W，s）$的超图指数上给出了上限，因此就其相关的dynkin图的拓扑而言，$ w $的差异。
We study divergence and thickness for general Coxeter groups $W$. We first characterise linear divergence, and show that if $W$ has superlinear divergence then its divergence is at least quadratic. We then formulate a computable combinatorial invariant, hypergraph index, for arbitrary Coxeter systems $(W,S)$. This generalises Levcovitz's definition for the right-angled case. We prove that if $(W,S)$ has finite hypergraph index $h$, then $W$ is (strongly algebraically) thick of order at most $h$, hence has divergence bounded above by a polynomial of degree $h+1$. We conjecture that these upper bounds on the order of thickness and divergence are in fact equalities, and we prove our conjecture for certain families of Coxeter groups. These families are obtained by a new construction which, given any right-angled Coxeter group, produces infinitely many examples of non-right-angled Coxeter systems with the same hypergraph index. Finally, we give an upper bound on the hypergraph index of any Coxeter system $(W,S)$, and hence on the divergence of $W$, in terms of, unexpectedly, the topology of its associated Dynkin diagram.
