论文标题
亚组嵌入和渐近锥的几何形状
The geometry of subgroup embeddings and asymptotic cones
论文作者
论文摘要
给定有限生成的亚组$ h $的有限生成的集团$ g $和非主要的超级滤波器$ω$，我们认为自然子空间，$ cone^ω_{g}（g}（h）$，$ g $的渐近锥的$ g $的渐近锥。非正式地，该子空间由$ g $的渐近锥的点组成，由超能$ h^ω$的元素表示。我们表明，$ cone^ω_{g}（h）$检测$ h $ $ g $中嵌入的自然属性的连接性和凸度。我们首先定义失真函数的概括，并证明此功能确定是否连接了$ cone^ω_{g}（h）$。然后，我们表明，$ h $在$ g $中是否强烈地是由$ g $的$ cone^ω_{g}（h）$的自然凸属性检测到的$ g $。
Given a finitely generated subgroup $H$ of a finitely generated group $G$ and a non-principal ultrafilter $ω$, we consider a natural subspace, $Cone^ω_{G}(H)$, of the asymptotic cone of $G$ corresponding to $H$. Informally, this subspace consists of the points of the asymptotic cone of $G$ represented by elements of the ultrapower $H^ω$. We show that the connectedness and convexity of $Cone^ω_{G}(H)$ detect natural properties of the embedding of $H$ in $G$. We begin by defining a generalization of the distortion function and show that this function determines whether $Cone^ω_{G}(H)$ is connected. We then show that whether $H$ is strongly quasi-convex in $G$ is detected by a natural convexity property of $Cone^ω_{G}(H)$ in the asymptotic cone of $G$.
