Higher rank hyperbolicity
The large-scale geometry of hyperbolic metric spaces exhibits many distinctive features, such as the stability of quasi-geodesics (the Morse Lemma), the visibility property, and the homeomorphism between visual boundaries induced by a quasi-isometry. We prove a number of closely analogous results fo...
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| Veröffentlicht in: | Inventiones mathematicae 2020-08, Vol.221 (2), p.597-664 |
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| container_title | Inventiones mathematicae |
| container_volume | 221 |
| creator | Kleiner, Bruce Lang, Urs |
| description | The large-scale geometry of hyperbolic metric spaces exhibits many distinctive features, such as the stability of quasi-geodesics (the Morse Lemma), the visibility property, and the homeomorphism between visual boundaries induced by a quasi-isometry. We prove a number of closely analogous results for spaces of rank
n
≥
2
in an asymptotic sense, under some weak assumptions reminiscent of nonpositive curvature. For this purpose we replace quasi-geodesic lines with quasi-minimizing (locally finite)
n
-cycles of
r
n
volume growth; prime examples include
n
-cycles associated with
n
-quasiflats. Solving an asymptotic Plateau problem and producing unique tangent cones at infinity for such cycles, we show in particular that every quasi-isometry between two proper
CAT
(
0
)
spaces of asymptotic rank
n
extends to a class of
(
n
-
1
)
-cycles in the Tits boundaries. |
| doi_str_mv | 10.1007/s00222-020-00955-w |
| format | Article |
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n
≥
2
in an asymptotic sense, under some weak assumptions reminiscent of nonpositive curvature. For this purpose we replace quasi-geodesic lines with quasi-minimizing (locally finite)
n
-cycles of
r
n
volume growth; prime examples include
n
-cycles associated with
n
-quasiflats. Solving an asymptotic Plateau problem and producing unique tangent cones at infinity for such cycles, we show in particular that every quasi-isometry between two proper
CAT
(
0
)
spaces of asymptotic rank
n
extends to a class of
(
n
-
1
)
-cycles in the Tits boundaries.</description><identifier>ISSN: 0020-9910</identifier><identifier>EISSN: 1432-1297</identifier><identifier>DOI: 10.1007/s00222-020-00955-w</identifier><language>eng</language><publisher>Berlin/Heidelberg: Springer Berlin Heidelberg</publisher><subject>Asymptotic properties ; Boundaries ; Cones ; Geodesic lines ; Geodesy ; Mathematics ; Mathematics and Statistics ; Metric space ; Topology ; Visibility</subject><ispartof>Inventiones mathematicae, 2020-08, Vol.221 (2), p.597-664</ispartof><rights>Springer-Verlag GmbH Germany, part of Springer Nature 2020</rights><rights>Springer-Verlag GmbH Germany, part of Springer Nature 2020.</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c319t-a6eff741a33532090fe2b88521ee86220d7ac4358bb71ff70f595008bfc7e1b13</citedby><cites>FETCH-LOGICAL-c319t-a6eff741a33532090fe2b88521ee86220d7ac4358bb71ff70f595008bfc7e1b13</cites></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktopdf>$$Uhttps://link.springer.com/content/pdf/10.1007/s00222-020-00955-w$$EPDF$$P50$$Gspringer$$H</linktopdf><linktohtml>$$Uhttps://link.springer.com/10.1007/s00222-020-00955-w$$EHTML$$P50$$Gspringer$$H</linktohtml><link.rule.ids>314,776,780,27901,27902,41464,42533,51294</link.rule.ids></links><search><creatorcontrib>Kleiner, Bruce</creatorcontrib><creatorcontrib>Lang, Urs</creatorcontrib><title>Higher rank hyperbolicity</title><title>Inventiones mathematicae</title><addtitle>Invent. math</addtitle><description>The large-scale geometry of hyperbolic metric spaces exhibits many distinctive features, such as the stability of quasi-geodesics (the Morse Lemma), the visibility property, and the homeomorphism between visual boundaries induced by a quasi-isometry. We prove a number of closely analogous results for spaces of rank
n
≥
2
in an asymptotic sense, under some weak assumptions reminiscent of nonpositive curvature. For this purpose we replace quasi-geodesic lines with quasi-minimizing (locally finite)
n
-cycles of
r
n
volume growth; prime examples include
n
-cycles associated with
n
-quasiflats. Solving an asymptotic Plateau problem and producing unique tangent cones at infinity for such cycles, we show in particular that every quasi-isometry between two proper
CAT
(
0
)
spaces of asymptotic rank
n
extends to a class of
(
n
-
1
)
-cycles in the Tits boundaries.</description><subject>Asymptotic properties</subject><subject>Boundaries</subject><subject>Cones</subject><subject>Geodesic lines</subject><subject>Geodesy</subject><subject>Mathematics</subject><subject>Mathematics and Statistics</subject><subject>Metric space</subject><subject>Topology</subject><subject>Visibility</subject><issn>0020-9910</issn><issn>1432-1297</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2020</creationdate><recordtype>article</recordtype><sourceid>BENPR</sourceid><recordid>eNp9kDFPwzAQhS0EEqHwA2CqxGy4O8d1PKIKWqRKLDBbcbDblJIEO1WVf49LkNiYbnjfe3f3GLtBuEMAdR8BiIgDAQfQUvLDCcswF8SRtDplGRwlrRHO2UWMW4AkKsrY9bJeb1yYhrL5mG6GzgXb7uqq7odLdubLXXRXv3PC3p4eX-dLvnpZPM8fVrwSqHtezpz3KsdSCCkINHhHtigkoXPFjAjeVVnlQhbWKkwkeKklQGF9pRxaFBN2O-Z2of3au9ibbbsPTVppKCekdGdBiaKRqkIbY3DedKH-LMNgEMyxAjNWYNKf5qcCc0gmMZpigpu1C3_R_7i-AVj2XNU</recordid><startdate>20200801</startdate><enddate>20200801</enddate><creator>Kleiner, Bruce</creator><creator>Lang, Urs</creator><general>Springer Berlin Heidelberg</general><general>Springer Nature B.V</general><scope>AAYXX</scope><scope>CITATION</scope><scope>3V.</scope><scope>7SC</scope><scope>7TB</scope><scope>7XB</scope><scope>8AL</scope><scope>8FD</scope><scope>8FE</scope><scope>8FG</scope><scope>8FK</scope><scope>ABJCF</scope><scope>ABUWG</scope><scope>AFKRA</scope><scope>ARAPS</scope><scope>AZQEC</scope><scope>BENPR</scope><scope>BGLVJ</scope><scope>CCPQU</scope><scope>DWQXO</scope><scope>FR3</scope><scope>GNUQQ</scope><scope>HCIFZ</scope><scope>JQ2</scope><scope>K7-</scope><scope>KR7</scope><scope>L6V</scope><scope>L7M</scope><scope>L~C</scope><scope>L~D</scope><scope>M0N</scope><scope>M7S</scope><scope>P5Z</scope><scope>P62</scope><scope>PQEST</scope><scope>PQQKQ</scope><scope>PQUKI</scope><scope>PRINS</scope><scope>PTHSS</scope><scope>Q9U</scope></search><sort><creationdate>20200801</creationdate><title>Higher rank hyperbolicity</title><author>Kleiner, Bruce ; Lang, Urs</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c319t-a6eff741a33532090fe2b88521ee86220d7ac4358bb71ff70f595008bfc7e1b13</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2020</creationdate><topic>Asymptotic properties</topic><topic>Boundaries</topic><topic>Cones</topic><topic>Geodesic lines</topic><topic>Geodesy</topic><topic>Mathematics</topic><topic>Mathematics and Statistics</topic><topic>Metric space</topic><topic>Topology</topic><topic>Visibility</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Kleiner, Bruce</creatorcontrib><creatorcontrib>Lang, Urs</creatorcontrib><collection>CrossRef</collection><collection>ProQuest Central (Corporate)</collection><collection>Computer and Information Systems Abstracts</collection><collection>Mechanical & Transportation Engineering Abstracts</collection><collection>ProQuest Central (purchase pre-March 2016)</collection><collection>Computing Database (Alumni Edition)</collection><collection>Technology Research Database</collection><collection>ProQuest SciTech Collection</collection><collection>ProQuest Technology Collection</collection><collection>ProQuest Central (Alumni) (purchase pre-March 2016)</collection><collection>Materials Science & Engineering Collection</collection><collection>ProQuest Central (Alumni Edition)</collection><collection>ProQuest Central UK/Ireland</collection><collection>Advanced Technologies & Aerospace Collection</collection><collection>ProQuest Central Essentials</collection><collection>ProQuest Central</collection><collection>Technology Collection (ProQuest)</collection><collection>ProQuest One Community College</collection><collection>ProQuest Central Korea</collection><collection>Engineering Research Database</collection><collection>ProQuest Central Student</collection><collection>SciTech Premium Collection</collection><collection>ProQuest Computer Science Collection</collection><collection>Computer Science Database</collection><collection>Civil Engineering Abstracts</collection><collection>ProQuest Engineering Collection</collection><collection>Advanced Technologies Database with Aerospace</collection><collection>Computer and Information Systems Abstracts Academic</collection><collection>Computer and Information Systems Abstracts Professional</collection><collection>Computing Database</collection><collection>Engineering Database</collection><collection>Advanced Technologies & Aerospace Database</collection><collection>ProQuest Advanced Technologies & Aerospace Collection</collection><collection>ProQuest One Academic Eastern Edition (DO NOT USE)</collection><collection>ProQuest One Academic</collection><collection>ProQuest One Academic UKI Edition</collection><collection>ProQuest Central China</collection><collection>Engineering Collection</collection><collection>ProQuest Central Basic</collection><jtitle>Inventiones mathematicae</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Kleiner, Bruce</au><au>Lang, Urs</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Higher rank hyperbolicity</atitle><jtitle>Inventiones mathematicae</jtitle><stitle>Invent. math</stitle><date>2020-08-01</date><risdate>2020</risdate><volume>221</volume><issue>2</issue><spage>597</spage><epage>664</epage><pages>597-664</pages><issn>0020-9910</issn><eissn>1432-1297</eissn><abstract>The large-scale geometry of hyperbolic metric spaces exhibits many distinctive features, such as the stability of quasi-geodesics (the Morse Lemma), the visibility property, and the homeomorphism between visual boundaries induced by a quasi-isometry. We prove a number of closely analogous results for spaces of rank
n
≥
2
in an asymptotic sense, under some weak assumptions reminiscent of nonpositive curvature. For this purpose we replace quasi-geodesic lines with quasi-minimizing (locally finite)
n
-cycles of
r
n
volume growth; prime examples include
n
-cycles associated with
n
-quasiflats. Solving an asymptotic Plateau problem and producing unique tangent cones at infinity for such cycles, we show in particular that every quasi-isometry between two proper
CAT
(
0
)
spaces of asymptotic rank
n
extends to a class of
(
n
-
1
)
-cycles in the Tits boundaries.</abstract><cop>Berlin/Heidelberg</cop><pub>Springer Berlin Heidelberg</pub><doi>10.1007/s00222-020-00955-w</doi><tpages>68</tpages></addata></record> |
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| ispartof | Inventiones mathematicae, 2020-08, Vol.221 (2), p.597-664 |
| issn | 0020-9910 1432-1297 |
| language | eng |
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| source | SpringerLink Journals - AutoHoldings |
| subjects | Asymptotic properties Boundaries Cones Geodesic lines Geodesy Mathematics Mathematics and Statistics Metric space Topology Visibility |
| title | Higher rank hyperbolicity |
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