不含非平凡等变紧化的拓扑变换群

Maximal equivariant compactification of the Urysohn spaces and other metric structures

We study isometric $G$-spaces and the question of when their maximal equivariant compactification is the Gromov compactification (meaning that it coincides with the compactification generated by the distance functions to points). Answering questions of Pestov, we show that this is the case for the Urysohn sphere and related spaces, but not for the unit sphere of the Gurarij space. We show that the maximal equivariant compactification of a separably categorical metric structure $M$ under the action of its automorphism group can be identified with the space $S_1(M)$ of 1-types over $M$, and is i…

群紧化上的收缩定理

Retraction Theorems for Group Compactifications

本文刻画了存在等变收缩到边界的离散群紧化，并证明了布劳威尔无收缩定理的等变类比，涵盖双曲群在格罗莫夫边界上的作用。

齐次空间的整点渐近性、等变紧化与等分布

Asymptotics of integral points, equivariant compactifications and equidistributions for homogeneous spaces

针对线性代数群在Q上的齐次簇整点渐近计数问题，借助幂零流测度刚性将其约化为等变双有理几何问题，证明了特定条件下齐次空间的强Hardy-Littlewood性质及整点与体积渐近性的关系，并给出具体例子说明方法。

Maximal equivariant compactifications

Let $G$ be a locally compact group. Then for every $G$-space $X$ the maximal $G$-proximity $β_G$ can be characterized by the maximal topological proximity $β$ as follows: $$ A \ \overline{β_G} \ B \Leftrightarrow \exists V \in N_e \ \ \ VA \ \overlineβ \ VB. $$ Here, $β_G \colon X \to β_G X$ is the maximal $G$-compactification of $X$ (which is an embedding for locally compact $G$), $V$ is a neighborhood of $e$ and $A \ \overline{β_G} \ B$ means that the closures of $A$ and $B$ do not meet in $β_G X$. Note that the local compactness of $G$ is essential. This theorem comes as a corollary of a ge…

超重力流和球体填充曲线

Quasigeodesic flows and sphere-filling curves

给定一个具有拟短程流的闭双曲3-流形M，在双曲空间的边界上构造了一条π_1-等变球填充曲线.具体地说，我们证明了H^3上升力流的任何完全横截P都具有作为继承π_1作用的闭盘的自然紧化。P在H^3中的嵌入连续地扩展到紧化，并且对边界的限制是从S^1到S^2_ \infty的满射π_1-等变映射.这推广了Cannon和Thurston关于纤维双曲3-流形的结果。

Adele群的多重混合与有理点

Multiple Mixing for adele groups and rational points

证明了$H\G$的射影等变紧化上有界高的有理点个数的一个渐近公式，其中$H$是对角嵌入到$G := H^n$中的连通单代数群.