标题： 冯·诺伊曼连续环的单位群的几何性质 显示英文标题

标题： Geometric properties of unit groups of von Neumann's continuous rings

摘要： 我们证明，如果$R$是一个不可约的、连续的环，那么其单位群$\mathrm{GL}(R)$在由秩度量生成的拓扑下，模其中心是拓扑简单的，路径连通的，局部路径连通的，在Bourbaki的意义下是有界的，并且不承认任何非零逃逸函数。 所有这些拓扑见解都是关于秩度量的更精细几何结果的结论，特别是针对代数元素的集合。 由于自动连续现象，我们的结果也对底层抽象群产生了非平凡的影响。 显示英文摘要

摘要： We prove that, if $R$ is an irreducible, continuous ring, then its unit group $\mathrm{GL}(R)$, equipped with the topology generated by the rank metric, is topologically simple modulo its center, path-connected, locally path-connected, bounded in the sense of Bourbaki, and not admitting any non-zero escape function. All these topological insights are consequences of more refined geometric results concerning the rank metric, in particular with regard to the set of algebraic elements. Thanks to the phenomenon of automatic continuity, our results also have non-trivial ramifications for the underlying abstract groups.