CAT(0) Spaces of Higher Rank I

IF 4.6 Q2 MATERIALS SCIENCE, BIOMATERIALS

ACS Applied Bio Materials Pub Date : 2024-02-02 DOI:10.1007/s00039-024-00661-2

引用次数: 0

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。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

ACS Applied Bio Materials Chemistry-Chemistry (all)

CiteScore

9.40

自引率

2.10%

发文量

464