On Super-rigidity of Gromov's Random Monster Group

On Super-rigidity of Gromov's Random Monster Group | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article On Super-rigidity of Gromov's Random Monster Group Kajal Das This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-3936519/v1 This work is licensed under a CC BY 4.0 License Status: Published Journal Publication published 14 Nov, 2024 Read the published version in Geometriae Dedicata → Version 1 posted 7 You are reading this latest preprint version Abstract In this article, we show super-rigidity of Gromov’s random monster group. It is known from a paper of Assaf Naor and Lior Silberman that any homomorphic image of Gromov’s random monster group into a linear group is finite. It can be also derived from the previously known results that the same result is true for a-Lp-menable groups and K-amenable groups. We extend these results and prove that any morphism ϕα from Gromov’s random monster group Γα to a countable discrete group G has finite image for almost all α, where G is any of the following types of groups: mapping class group MCG(Sg,b), braid group Bn, outer automorphism group of a free group Out(FN), automorphism group of a free group Aut(FN) and hierarchically hyperbolic group. For acylindrically hyperbolic groups, we deduce that the homomorphic image is absolutely elliptic. We introduce another property called hereditary super-rigidity. It immediately follows from the literature that Γα has hereditary super-rigidity with respect to an a-Lp-menable group or a K-amenable group for a.e. α. In this article, we establish a stability result for the groups with respect to which Γα has super-rigidity and hereditary super-rigidity. Mathematics Subject Classification (2020): 20F65, 20F67, 20P05, 53C24, 57K20. Gromov’s random monster group elementary and non-elementary action Property (T) Property FLp mapping class group braid group automorphism and outer automorphism group of a free group hierarchically hyperbolic group acylindrically hyperbolic group. Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Published Journal Publication published 14 Nov, 2024 Read the published version in Geometriae Dedicata → Version 1 posted Editorial decision: Revision requested 25 May, 2024 Reviews received at journal 09 May, 2024 Reviewers agreed at journal 26 Feb, 2024 Reviewers invited by journal 19 Feb, 2024 Editor assigned by journal 10 Feb, 2024 Submission checks completed at journal 07 Feb, 2024 First submitted to journal 07 Feb, 2024 You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. 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Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-3936519","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":271689075,"identity":"05f8f110-5889-4e70-b8c0-1b00ecc92072","order_by":0,"name":"Kajal Das","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA2klEQVRIiWNgGAWjYBACCYYECMOAvQFEWhCvRcKA5wBIiwQpWiQSIHyCQLI9+fEHxhy7OnPJ51c3/CiQYOBv707Aq0Wa55mZBOO2ZAnL2TllN3uADpM4c3YDXi1yEglmDIzbmCUMbuek3eABajGQyCWkJf3zB8Zt9RIGN8+k3fxDjBZpiRwDoMMOSxjcYD92myhbJHvelEkkbjsuueFMDtttGQMJHoJ+kTievvnDx23V/AbHjz+7+eaPjRx/ey9+LWCQACZ5DMAkYeUIwP6AFNWjYBSMglEwggAA0MZFLWwD6IQAAAAASUVORK5CYII=","orcid":"","institution":"Indian Statistical Institute","correspondingAuthor":true,"prefix":"","firstName":"Kajal","middleName":"","lastName":"Das","suffix":""}],"badges":[],"createdAt":"2024-02-07 10:35:27","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-3936519/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-3936519/v1","draftVersion":[],"editorialEvents":[{"content":"https://doi.org/10.1007/s10711-024-00950-y","type":"published","date":"2024-11-14T15:56:58+00:00"}],"editorialNote":"","failedWorkflow":false,"files":[{"id":69274777,"identity":"2b9e7d38-b59a-49a2-9163-70af252debbb","added_by":"auto","created_at":"2024-11-18 16:25:43","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":260550,"visible":true,"origin":"","legend":"","description":"","filename":"GromovrandommonsterGD.tex.pdf","url":"https://assets-eu.researchsquare.com/files/rs-3936519/v1_covered_93e99193-631e-45a6-8ef9-8881c2df0d89.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"\u003cp\u003eOn Super-rigidity of Gromov's Random Monster Group\u003c/p\u003e","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":false,"highlight":"","institution":"","isAcceptedByJournal":true,"isAuthorSuppliedPdf":true,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":true,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"geometriae-dedicata","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"geom","sideBox":"Learn more about [Geometriae Dedicata](http://link.springer.com/journal/10711)","snPcode":"10711","submissionUrl":"https://submission.nature.com/new-submission/10711/3","title":"Geometriae Dedicata","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"em","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":false},"keywords":"Gromov’s random monster group, elementary and non-elementary action, Property (T), Property FLp , mapping class group, braid group, automorphism and outer automorphism group of a free group, hierarchically hyperbolic group, acylindrically hyperbolic group.","lastPublishedDoi":"10.21203/rs.3.rs-3936519/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-3936519/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eIn this article, we show super-rigidity of Gromov’s random monster group. 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