{"title":"CAT(0) Spaces of Higher Rank I","authors":"","doi":"10.1007/s00039-024-00661-2","DOIUrl":null,"url":null,"abstract":"<h3>Abstract</h3> <p>A CAT(0) space has rank at least <em>n</em> if every geodesic lies in an <em>n</em>-flat. Ballmann’s Higher Rank Rigidity Conjecture predicts that a CAT(0) space of rank at least 2 with a geometric group action is <em>rigid</em> – 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 <em>n</em>≥2 is rigid if it contains a periodic <em>n</em>-flat and its Tits boundary has dimension (<em>n</em>−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 <em>Morse flats</em>. We show that the Tits boundary <em>∂</em><sub><em>T</em></sub><em>F</em> of a periodic Morse <em>n</em>-flat <em>F</em> contains a <em>regular point</em> – a point with a Tits-neighborhood entirely contained in <em>∂</em><sub><em>T</em></sub><em>F</em>. More precisely, we show that the set of singular points in <em>∂</em><sub><em>T</em></sub><em>F</em> can be covered by finitely many round spheres of positive codimension.</p>","PeriodicalId":2,"journal":{"name":"ACS Applied Bio Materials","volume":null,"pages":null},"PeriodicalIF":4.6000,"publicationDate":"2024-02-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"ACS Applied Bio Materials","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s00039-024-00661-2","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATERIALS SCIENCE, BIOMATERIALS","Score":null,"Total":0}
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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