CAT(0)空间的渐近锥与边界

IF 1.2 2区数学 Q1 MATHEMATICS

Indiana University Mathematics Journal Pub Date : 2021-01-01 DOI:10.1512/IUMJ.2021.70.8552

Curtis Kent, Russell Ricks

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引用次数: 3

摘要

我们研究了CAT(0)空间在标准视觉(即锥)拓扑和Tits度量下的渐近锥与边界之间的关系。证明了固有紧CAT(0)空间的渐近锥集允许正则连接映射，其直接极限与Tits边界上的欧几里得锥等距。我们还证明了渐近锥之间的映射如何诱导Tits边界之间的映射，我们用它来证明几乎自由的阿贝尔群正是具有紧Tits边界的CAT(0)群。

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Asymptotic cones and boundaries of CAT(0) spaces

We explore the relationships between the asymptotic cones of a CAT(0) space and its boundary under both the standard visual (i.e. cone) topology and the Tits metric. We show that the set of asymptotic cones of a proper cocompact CAT(0) space admits canonical connecting maps under which the direct limit is isometric to the Euclidean cone on the Tits boundary. We also demonstrate how maps between asymptotic cones induce maps between Tits boundaries, which we use to show that virtually free Abelian groups are exactly the CAT(0) groups with compact Tits boundary.

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