On hierarchical hyperbolicity of cubical groups
Let χ be a proper CAT(0) cube complex admitting a proper cocompact action by a group G . We give three conditions on the action, any one of which ensures that χ has a factor system in the sense of [BHS17]. We also prove that one of these conditions is necessary. This combines with [BHS17] to show th...
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| Veröffentlicht in: | Israel journal of mathematics 2020-03, Vol.236 (1), p.45-89 |
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| container_title | Israel journal of mathematics |
| container_volume | 236 |
| creator | Hagen, Mark F. Susse, Tim |
| description | Let
χ
be a proper CAT(0) cube complex admitting a proper cocompact action by a group
G
. We give three conditions on the action, any one of which ensures that
χ
has a factor system in the sense of [BHS17]. We also prove that one of these conditions is necessary. This combines with [BHS17] to show that
G
is a hierarchically hyperbolic group; this partially answers questions raised in [BHS17, BHS19]. Under any of these conditions, our results also affirm a conjecture of Behrstock-Hagen on boundaries of cube complexes, which implies that
χ
cannot contain a convex staircase. The necessary conditions on the action are all strictly weaker than virtual cospecialness, and we are not aware of a cocompactly cubulated group that does not satisfy at least one of the conditions. |
| doi_str_mv | 10.1007/s11856-020-1967-2 |
| format | Article |
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χ
be a proper CAT(0) cube complex admitting a proper cocompact action by a group
G
. We give three conditions on the action, any one of which ensures that
χ
has a factor system in the sense of [BHS17]. We also prove that one of these conditions is necessary. This combines with [BHS17] to show that
G
is a hierarchically hyperbolic group; this partially answers questions raised in [BHS17, BHS19]. Under any of these conditions, our results also affirm a conjecture of Behrstock-Hagen on boundaries of cube complexes, which implies that
χ
cannot contain a convex staircase. The necessary conditions on the action are all strictly weaker than virtual cospecialness, and we are not aware of a cocompactly cubulated group that does not satisfy at least one of the conditions.</description><identifier>ISSN: 0021-2172</identifier><identifier>EISSN: 1565-8511</identifier><identifier>DOI: 10.1007/s11856-020-1967-2</identifier><language>eng</language><publisher>Jerusalem: The Hebrew University Magnes Press</publisher><subject>Algebra ; Analysis ; Applications of Mathematics ; Group Theory and Generalizations ; Mathematical and Computational Physics ; Mathematics ; Mathematics and Statistics ; Theoretical</subject><ispartof>Israel journal of mathematics, 2020-03, Vol.236 (1), p.45-89</ispartof><rights>The Hebrew University of Jerusalem 2020</rights><rights>The Hebrew University of Jerusalem 2020.</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c316t-4bd483c6313a923708b733f8666135e2db975f2d1a81d36382744bb9a988e0293</citedby><cites>FETCH-LOGICAL-c316t-4bd483c6313a923708b733f8666135e2db975f2d1a81d36382744bb9a988e0293</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/s11856-020-1967-2$$EPDF$$P50$$Gspringer$$H</linktopdf><linktohtml>$$Uhttps://link.springer.com/10.1007/s11856-020-1967-2$$EHTML$$P50$$Gspringer$$H</linktohtml><link.rule.ids>314,780,784,27923,27924,41487,42556,51318</link.rule.ids></links><search><creatorcontrib>Hagen, Mark F.</creatorcontrib><creatorcontrib>Susse, Tim</creatorcontrib><title>On hierarchical hyperbolicity of cubical groups</title><title>Israel journal of mathematics</title><addtitle>Isr. J. Math</addtitle><description>Let
χ
be a proper CAT(0) cube complex admitting a proper cocompact action by a group
G
. We give three conditions on the action, any one of which ensures that
χ
has a factor system in the sense of [BHS17]. We also prove that one of these conditions is necessary. This combines with [BHS17] to show that
G
is a hierarchically hyperbolic group; this partially answers questions raised in [BHS17, BHS19]. Under any of these conditions, our results also affirm a conjecture of Behrstock-Hagen on boundaries of cube complexes, which implies that
χ
cannot contain a convex staircase. The necessary conditions on the action are all strictly weaker than virtual cospecialness, and we are not aware of a cocompactly cubulated group that does not satisfy at least one of the conditions.</description><subject>Algebra</subject><subject>Analysis</subject><subject>Applications of Mathematics</subject><subject>Group Theory and Generalizations</subject><subject>Mathematical and Computational Physics</subject><subject>Mathematics</subject><subject>Mathematics and Statistics</subject><subject>Theoretical</subject><issn>0021-2172</issn><issn>1565-8511</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2020</creationdate><recordtype>article</recordtype><recordid>eNp1kD1PwzAURS0EEqXwA9giMZv6-cVfI6qAIlXqArNlO06bqjTBTob-e1KCxMT0hnvPfdIh5B7YIzCmFhlAC0kZZxSMVJRfkBkIKagWAJdkxhgHykHxa3KT854xgQpwRhabY7FrYnIp7JrgDsXu1MXk20MTmv5UtHURBv8TbFM7dPmWXNXukOPd752Tj5fn9-WKrjevb8unNQ0Isqelr0qNQSKgMxwV014h1lpKCSgir7xRouYVOA0VStRclaX3xhmtI-MG5-Rh2u1S-zXE3Nt9O6Tj-NJyNJqDVEaOLZhaIbU5p1jbLjWfLp0sMHv2YicvdvRiz15GeE74xOSxe9zG9Lf8P_QNtP9jCg</recordid><startdate>20200301</startdate><enddate>20200301</enddate><creator>Hagen, Mark F.</creator><creator>Susse, Tim</creator><general>The Hebrew University Magnes Press</general><general>Springer Nature B.V</general><scope>AAYXX</scope><scope>CITATION</scope></search><sort><creationdate>20200301</creationdate><title>On hierarchical hyperbolicity of cubical groups</title><author>Hagen, Mark F. ; Susse, Tim</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c316t-4bd483c6313a923708b733f8666135e2db975f2d1a81d36382744bb9a988e0293</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2020</creationdate><topic>Algebra</topic><topic>Analysis</topic><topic>Applications of Mathematics</topic><topic>Group Theory and Generalizations</topic><topic>Mathematical and Computational Physics</topic><topic>Mathematics</topic><topic>Mathematics and Statistics</topic><topic>Theoretical</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Hagen, Mark F.</creatorcontrib><creatorcontrib>Susse, Tim</creatorcontrib><collection>CrossRef</collection><jtitle>Israel journal of mathematics</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Hagen, Mark F.</au><au>Susse, Tim</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>On hierarchical hyperbolicity of cubical groups</atitle><jtitle>Israel journal of mathematics</jtitle><stitle>Isr. J. Math</stitle><date>2020-03-01</date><risdate>2020</risdate><volume>236</volume><issue>1</issue><spage>45</spage><epage>89</epage><pages>45-89</pages><issn>0021-2172</issn><eissn>1565-8511</eissn><abstract>Let
χ
be a proper CAT(0) cube complex admitting a proper cocompact action by a group
G
. We give three conditions on the action, any one of which ensures that
χ
has a factor system in the sense of [BHS17]. We also prove that one of these conditions is necessary. This combines with [BHS17] to show that
G
is a hierarchically hyperbolic group; this partially answers questions raised in [BHS17, BHS19]. Under any of these conditions, our results also affirm a conjecture of Behrstock-Hagen on boundaries of cube complexes, which implies that
χ
cannot contain a convex staircase. The necessary conditions on the action are all strictly weaker than virtual cospecialness, and we are not aware of a cocompactly cubulated group that does not satisfy at least one of the conditions.</abstract><cop>Jerusalem</cop><pub>The Hebrew University Magnes Press</pub><doi>10.1007/s11856-020-1967-2</doi><tpages>45</tpages></addata></record> |
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| language | eng |
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| subjects | Algebra Analysis Applications of Mathematics Group Theory and Generalizations Mathematical and Computational Physics Mathematics Mathematics and Statistics Theoretical |
| title | On hierarchical hyperbolicity of cubical groups |
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