论文标题
右角Artin和Coxeter组的稳定换向器长度
Stable commutator length in right-angled Artin and Coxeter groups
论文作者
论文摘要
我们建立了右角Artin组（RAAGS）中整体链稳定换向器长度（SCL）的光谱差距。我们表明，这个差距不是统一的，即有scl任意接近零的RAAG和积分链。我们根据定义图的相对路径长度确定此间隙的大小为乘法常数。该结果与RAAG中元素的已知均匀差距1/2形成鲜明对比。我们证明了右角的Coxeter组的类似结果。 在本文的第二部分中，我们将RAAG中的某些积分链与图形的分数稳定性数相关联。这有几个后果：首先，我们表明每个有理数q> = 1是在某些RAAG中积分链的稳定换向器长度出现的。其次，我们表明，RAAG中的计算元素和链条的计算很难。最后，我们将自由组中随机元素的SCL分布与随机图中的分数稳定数分布联系起来。 我们在图形产品的一般环境中证明了所有结果。特别是上述结果，右角高速公路群逐字化。
We establish a spectral gap for stable commutator length (scl) of integral chains in right-angled Artin groups (RAAGs). We show that this gap is not uniform, i.e. there are RAAGs and integral chains with scl arbitrarily close to zero. We determine the size of this gap up to a multiplicative constant in terms of the opposite path length of the defining graph. This result is in stark contrast with the known uniform gap 1/2 for elements in RAAGs. We prove an analogous result for right-angled Coxeter groups. In a second part of this paper we relate certain integral chains in RAAGs to the fractional stability number of graphs. This has several consequences: Firstly, we show that every rational number q>=1 arises as the stable commutator length of an integral chain in some RAAG. Secondly, we show that computing scl of elements and chains in RAAGs is NP hard. Finally, we heuristically relate the distribution of scl for random elements in the free group to the distribution of fractional stability number in random graphs. We prove all of our results in the general setting of graph products. In particular all above results hold verbatim for right-angled Coxeter groups.
