关于Bowditch定理的一个注记

相对双曲群的局部割点与分裂

Local cut points and splittings of relatively hyperbolic groups

本文证明了在相对双曲群$(G,\mathbb{P})$的Bowditch边界$\partial(G,\mathbb{P})$中存在非抛物局部割点，这意味着$G$在$2$ -终止子群上分裂.这个定理将Bowditch的一个定理从双曲群的集合推广到相对双曲群。因此，我们能够推广Kapovich和Kleiner的一个定理，通过分类同胚类型的$1$维Bowditch边界的相对双曲群，满足某些性质，如没有分裂的$2$结束的子群和没有周边分裂。为了证明边界分类结果，我们需要一个比标准概念更一般的群的端点的概念。证明了如果一个广义Peano连续统$X$和$Y$上的一个离散群正确作用且上紧，则$Ends(X)$同胚于$Ends(Y)$ .因此，我们提出了另一种定义的$Ends(G)$，增加了类的空间上，$G$可以采取行动。

在群复形上粘合相对双曲群

Combining relatively hyperbolic groups over a complex of groups

本研究针对满足特定条件的相对双曲群构成的群复形，通过构造其基本群的Bowditch边界模型并结合Yaman定理，证明该基本群是相对双曲的，还给出Bowditch边界的一般分解方法，为相关拓扑问题研究提供工具。

余维数为1的子群与双曲群的边界

Codimension one subgroups and boundaries of hyperbolic groups

我们构造了具有如下性质的双曲群：群的边界维数大，边界被一个康托尔集分隔，群不分裂.这表明，Bowditch的定理，其特征在于分裂的双曲群在2-端的群体在边界方面不能推广到更复杂的子群的分裂。

A combination theorem for relatively hyperbolic groups and finite relative height of splitting

In this paper, we prove a combination theorem for a relatively acylindrical graph of relatively hyperbolic groups (Theorem 1.1). Here, we are extending the technique of [Tom21] and constructing Bowditch boundary of the fundamental group of graph of groups. Suppose G(Y) is a graph of relatively hyperbolic groups such that edge groups are relatively quasi-convex in adjacent vertex groups. Also, assume that the fundamental group of G(Y) is relatively hyperbolic. Then we show that the edge groups of G(Y) have finite relative height (Definition 1.5) if and only if they are relatively quasi-convex (…

A combination theorem for combinatorially non-positively curved complexes of hyperbolic groups

We prove a combination theorem for hyperbolic groups, in the case of groups acting on complexes displaying combinatorial features reminiscent of non-positive curvature. Such complexes include for instance weakly systolic complexes and C'(1/6) small cancellation polygonal complexes. Our proof involves constructing a potential Gromov boundary for the resulting groups and analyzing the dynamics of the action on the boundary in order to use Bowditch's characterization of hyperbolicity. A key ingredient is the introduction of a combinatorial property that implies a weak form of non-positive curvatu…

Some groups with planar boundaries

In this expository note, we illustrate phenomena and conjectures about boundaries of hyperbolic groups by considering the special cases of certain amalgams of hyperbolic groups. While doing so, we describe fundamental results on hyperbolic groups and their boundaries by Bowditch and Haissinsky, along with special treatments for the boundaries of free groups by Otal and Cashen.

Boundary classification and 2-ended splittings of groups with isolated flats

In this paper we provide a classification theorem for 1-dimensional boundaries of groups with isolated flats. Given a group $Γ$ acting geometrically on a $CAT(0)$ space $X$ with isolated flats and 1-dimensional boundary, we show that if $Γ$ does not split over a virtually cyclic subgroup, then $\partial X$ is homeomorphic to a circle, a Sierpinski carpet, or a Menger curve. This theorem generalizes a theorem of Kapovich-Kleiner, and resolves a question due to Kim Ruane. We also study the relationship between local cut points in $\partial X$ and splittings of $Γ$ over $2$-ended subgroups. In pa…

关于一个Schur定理推广的注记

A note regarding some generalization of a Schur's theorem

M. de Falco，F. de Giovanni，C. Musella和Y. P. Sysak，Proceedings of the american mathematical society，Volume 139，Number 2，February 2011，385- 389包括一个主定理的相当长的证明。我们提供了一个简单的和非常简短的证明这个结果。

关于辫群的Artin-Markov normal form定理的一个注记

A note on Artin-Markov normal form theorem for braid groups

在L. A. Bokut、V.V. Chaynikov和K. P. Shum在2007年提出的，以Artin-Burau关系为代表的辫群$B_n$。对于这样的表示，它被告知可以以相同的方式检查所有其他组成。在本说明中，我们支持这一主张并检查所有成分。

弗洛伊德映射到相对双曲群的边界

Floyd maps to the boundaries of relatively hyperbolic groups

构造了一个从相对双曲群的Floyd边界到Bowditch边界的连续等变映射。这样的映射是唯一的，除非该组是两端的。为了优化映射定理的证明和使用，我们提出了两个新的相对双曲性定义，它们等价于其他已知的定义。在我们的方法中，图中的一些“可见性”条件是必不可少的。我们引入了一类“可见性行动”，其中包含的类相对双曲的行动。对于可见性操作，收敛属性仍然有效。本文描述并利用了局部紧群的两个作用的“吸引子和”的一个相当一般的构造。

曲线图的一致双曲性

Uniform Hyperbolicity of the Graphs of Curves

设$\mathcal{C}(S_{g,p})$表示亏格为$g$的闭可定向曲面的曲线复形，其中含有$p$穿孔. Masur-Minksy和随后的Bowditch证明了对于某些$δ=δ(g,p)$，$\mathcal{C}(S_{g,p})$是$δ$ -双曲的。本文证明了存在一些与$g,p$无关的$δ>0$使得曲线图$\mathcal{C}_{1}(S_{g,p})$是$δ$ -双曲的.此外，我们使用这个定理的证明中的主要工具，以显示一致的有界性的其他两个量的先验增长与$g$和$p$：曲线复距离之间的两个顶点循环的同一训练轨道，和Lipschitz常数的映射从Teichmüller空间到$\mathcal{C}(S)$发送黎曼曲面的曲线（S）的最短极值长度。

关于Bourbaki一个定理的注记

A note on a theorem of Bourbaki

最近我们证明了双曲李代数的Poincare级数具有所选有限李代数的Poincare级数与有限次多项式之比的形式。借助于适当选取的例子，我们对布尔巴基的一个有关定理给出一些注记。

A note on the Assmus--Mattson theorem for some binary codes II

Let $C$ be a four-weight binary code, which has all one vector. Furthermore, we assume that $C$ supports $t$-designs for all weights obtained from the Assmus--Mattson theorem. We previously showed that $t\leq 5$. In the present paper, we show an analogue of this result in the cases of five and six-weight codes.