论文标题
部分可观测时空混沌系统的无模型预测
Weak commutativity, virtually nilpotent groups, and Dehn functions
论文作者
论文摘要
组$ \ mathfrak {x}（g）$是通过在第一个自由因素中强迫每个元素$ g $在第二自由因素中使用$ g $的副本来从$ g \ ast g $获得的。我们对函数$ \ mathfrak {x} $的属性列表进行了重大添加。我们还研究了$ \ mathfrak {x}（g）$的单词问题的几何形状和复杂性。 $ \ mathfrak {x} $的微妙特征用普通的Abelian子组$ W <\ Mathfrak {x}（g）$，它是$ \ Mathbb {z} Q $上的模块，其中$ q = h_1（g，\ sathbb {z}）$。我们为该模块建立了一个结构性结果，并通过证明$ \ mathfrak {x} $保留虚拟nilpotence，Engel条件和增长类型 - 多项式，指数，指数或中间体来说明其实用性。我们还使用它来建立$ \ mathfrak {x}（g）$的等等不平等现象时，当$ g $放在包括Thompson的Group $ f $和所有非纤维化的Kähler组的班级中。当$ \ mathfrak {x}（g）$中，只有在$ g $中可解决的时，就可以在$ \ mathfrak {x}（g）$中解决这个词。 $ \ mathfrak {x}（g）$的dehn函数在下面由立方多项式（如果$ g $地图）限制到非阿布莱安免费组。
The group $\mathfrak{X}(G)$ is obtained from $G\ast G$ by forcing each element $g$ in the first free factor to commute with the copy of $g$ in the second free factor. We make significant additions to the list of properties that the functor $\mathfrak{X}$ is known to preserve. We also investigate the geometry and complexity of the word problem for $\mathfrak{X}(G)$. Subtle features of $\mathfrak{X}$ are encoded in a normal abelian subgroup $W<\mathfrak{X}(G)$ that is a module over $\mathbb{Z} Q$, where $Q= H_1(G,\mathbb{Z})$. We establish a structural result for this module and illustrate its utility by proving that $\mathfrak{X}$ preserves virtual nilpotence, the Engel condition, and growth type -- polynomial, exponential, or intermediate. We also use it to establish isoperimetric inequalities for $\mathfrak{X}(G)$ when $G$ lies in a class that includes Thompson's group $F$ and all non-fibered Kähler groups. The word problem is solvable in $\mathfrak{X}(G)$ if and only if it is solvable in $G$. The Dehn function of $\mathfrak{X}(G)$ is bounded below by a cubic polynomial if $G$ maps onto a non-abelian free group.
