CAT(0) Spaces of Higher Rank I

IF 2.4 1区数学 Q1 MATHEMATICS

Geometric and Functional Analysis Pub Date : 2024-02-02 DOI:10.1007/s00039-024-00661-2

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Abstract

A CAT(0) space has rank at least n if every geodesic lies in an n-flat. Ballmann’s Higher Rank Rigidity Conjecture predicts that a CAT(0) space of rank at least 2 with a geometric group action is rigid – isometric to a Riemannian symmetric space, a Euclidean building, or splits as a metric product. This paper is the first in a series motivated by Ballmann’s conjecture. Here we prove that a CAT(0) space of rank at least n≥2 is rigid if it contains a periodic n-flat and its Tits boundary has dimension (n−1). This does not require a geometric group action. The result relies essentially on the study of flats which do not bound flat half-spaces – so-called Morse flats. We show that the Tits boundary ∂_TF of a periodic Morse n-flat F contains a regular point – a point with a Tits-neighborhood entirely contained in ∂_TF. More precisely, we show that the set of singular points in ∂_TF can be covered by finitely many round spheres of positive codimension.

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CAT(0) I 类高级职位空缺

摘要如果每条测地线都位于一个 n 扁平中，则 CAT(0) 空间的秩至少为 n。鲍尔曼的高阶刚性猜想预言，具有几何群作用的至少 2 阶 CAT(0) 空间是刚性的--与黎曼对称空间、欧几里得建筑等距，或分裂为度量积。本文是鲍尔曼猜想系列的第一篇论文。我们在此证明，如果秩至少为 n≥2 的 CAT(0) 空间包含周期性 n 平面，且其 Tits 边界维数为 (n-1)，那么它就是刚性的。这并不需要几何群作用。这一结果主要依赖于对不以平面半空间为界的平面--即所谓的莫尔斯平面--的研究。我们证明了周期性莫尔斯 n 平面 F 的 Tits 边界 ∂TF 包含一个正则点--一个 Tits 邻域完全包含在 ∂TF 中的点。更确切地说，我们证明了 ∂TF 中的奇异点集合可以被有限多个正标度圆球覆盖。

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来源期刊

Geometric and Functional Analysis 数学-数学

CiteScore

3.70

自引率

4.50%

发文量

审稿时长

6-12 weeks

期刊介绍： Geometric And Functional Analysis (GAFA) publishes original research papers of the highest quality on a broad range of mathematical topics related to geometry and analysis. GAFA scored in Scopus as best journal in "Geometry and Topology" since 2014 and as best journal in "Analysis" since 2016. Publishes major results on topics in geometry and analysis. Features papers which make connections between relevant fields and their applications to other areas.