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[{"key": "dc.contributor.advisor", "value": "Rajala, Kai", "language": null, "element": "contributor", "qualifier": "advisor", "schema": "dc"}, {"key": "dc.contributor.author", "value": "Martimo, Meeri", "language": null, "element": "contributor", "qualifier": "author", "schema": "dc"}, {"key": "dc.date.accessioned", "value": "2017-03-24T15:07:27Z", "language": null, "element": "date", "qualifier": "accessioned", "schema": "dc"}, {"key": "dc.date.available", "value": "2017-03-24T15:07:27Z", "language": null, "element": "date", "qualifier": "available", "schema": "dc"}, {"key": "dc.date.issued", "value": "2016", "language": null, "element": "date", "qualifier": "issued", "schema": "dc"}, {"key": "dc.identifier.other", "value": "oai:jykdok.linneanet.fi:1690427", "language": null, "element": "identifier", "qualifier": "other", "schema": "dc"}, {"key": "dc.identifier.uri", "value": "https://jyx.jyu.fi/handle/123456789/53366", "language": null, "element": "identifier", "qualifier": "uri", "schema": "dc"}, {"key": "dc.description.abstract", "value": "Meeri Martimo, Gromov-hyperboliset ryhm\u00e4t (engl. Gromov-hyperbolic\ngroups), matematiikan pro gradu -tutkielma, 51 s., Jyv\u00e4skyl\u00e4n yliopisto, Matematiikan\nja tilastotieteen laitos, syksy 2016.\n\nT\u00e4ss\u00e4 tutkielmassa k\u00e4sitell\u00e4\u00e4n Gromov-hyperbolisia ryhmi\u00e4, jotka ovat geometrisen\nryhm\u00e4teorian tutkimuskohde. Geometrinen ryhm\u00e4teoria on melko uusi matematiikan\nsuuntaus, ja 1980-luvulla Gromov-hyperboliset ryhm\u00e4t kehitt\u00e4nyt ranskalaisven\u00e4l\u00e4inen\nmatemaatikko Mikhail Gromov yksi sen uranuurtajista.\nGromov-hyperbolisuus m\u00e4\u00e4ritell\u00e4\u00e4n ensin metrisille avaruuksille tietyll\u00e4 tavalla ohuiden\nkolmioiden avulla. Kolmiot ovat vaaditulla tavalla ohuita esimerkiksi hyperbolisissa\navaruuksissa, mutta eiv\u00e4t reaaliakselia korkeampiulotteisissa euklidisissa\navaruuksissa. Gromov-hyperbolisen metrisen avaruuden reuna voidaan m\u00e4\u00e4ritell\u00e4\nkiinnitetyst\u00e4 pisteest\u00e4 alkavien puolisuorien ekvivalenssiluokkien joukkona. N\u00e4in\nm\u00e4\u00e4riteltyyn reunaan ja sill\u00e4 t\u00e4ydennettyyn alkuper\u00e4iseen metriseen avaruuteen voidaan\nedelleen m\u00e4\u00e4ritell\u00e4 topologiat. T\u00e4ll\u00f6in osoittautuu, ett\u00e4 reunalla t\u00e4ydennetty\nmetrinen avaruus on alkuper\u00e4isen avaruuden kompaktisointi.\nGromov-hyperbolisuuden ja reunan k\u00e4sitteet yleistet\u00e4\u00e4n metrisilt\u00e4 avaruuksilta\nryhmille samaistamalla ryhm\u00e4 ja sen Cayleyn graafi kvasi-isometrian avulla. T\u00e4ll\u00f6in\nvoidaan m\u00e4\u00e4ritell\u00e4, ett\u00e4 ryhm\u00e4 on Gromov-hyperbolinen, jos sen Cayleyn graafi\non metrisen\u00e4 avaruutena Gromov-hyperbolinen, ja ett\u00e4 ryhm\u00e4n reuna on sen Cayleyn\ngraafin reuna. N\u00e4m\u00e4 m\u00e4\u00e4ritelm\u00e4t ovat mielekk\u00e4it\u00e4, sill\u00e4 ryhm\u00e4n ja sen Cayleyn graafin\nkvasi-isometrisuus takaa, ett\u00e4 monet niiden suuren mittakaavan ominaisuuksista\novat samanlaisia.\nGromov-hyperbolisten ryhmien erilaisten reunojen kirjo on mielenkiintoinen: esimerkiksi\ntyhj\u00e4 joukko, kaksi pistett\u00e4, ympyr\u00e4 tai sen osa, Cantorin joukko ja Sierpinskin\nmatto ovat t\u00e4ss\u00e4 tutkielmassa esitelt\u00e4vi\u00e4 hyperbolisten ryhmien reunoja.\nToinen hyperbolisten ryhmien mielenkiintoinen ominaisuus on ryhm\u00e4teorian klassisten\np\u00e4\u00e4t\u00f6songelmien ratkeavuus. Tutkielman p\u00e4\u00e4tuloksena osoitetaan, ett\u00e4 Gromov-hyperbolisen\nryhm\u00e4n sanaongelmalla on ratkaisu. Toisin sanoen jokaiselle Gromov-hyperboliselle\nryhm\u00e4lle on olemassa ratkaisualgoritmi, joka kertoo \u00e4\u00e4rellisen monen\nvaiheen j\u00e4lkeen, onko mielivaltainen ryhm\u00e4n viritt\u00e4j\u00e4joukon alkioiden avulla kirjoitettu\nsana ryhm\u00e4n yksikk\u00f6alkion esitys vai ei. T\u00e4m\u00e4 tulos ei ole triviaali, sill\u00e4 on\nolemassa \u00e4\u00e4rellisesti esitettyj\u00e4 ryhmi\u00e4, joiden sanaongelmalla ei ole ratkaisua.", "language": "fi", "element": "description", "qualifier": "abstract", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Submitted using Plone Publishing form by Meeri Martimo (mehetuma) on 2017-03-24 15:07:26.854318. Form: Pro gradu -lomake (https://kirjasto.jyu.fi/julkaisut/julkaisulomakkeet/pro-gradu-lomake). JyX data: [jyx_publishing-allowed (fi) =True]", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Submitted by jyx lomake-julkaisija (jyx-julkaisija.group@korppi.jyu.fi) on 2017-03-24T15:07:27Z\nNo. of bitstreams: 2\nURN:NBN:fi:jyu-201703241753.pdf: 693368 bytes, checksum: e29a9b3d5552e7a5966d3d2592b9128e (MD5)\nlicense.html: 4783 bytes, checksum: 3612fca917cc5b38ea144001dcbc9637 (MD5)", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Made available in DSpace on 2017-03-24T15:07:27Z (GMT). No. of bitstreams: 2\nURN:NBN:fi:jyu-201703241753.pdf: 693368 bytes, checksum: e29a9b3d5552e7a5966d3d2592b9128e (MD5)\nlicense.html: 4783 bytes, checksum: 3612fca917cc5b38ea144001dcbc9637 (MD5)\n Previous issue date: 2016", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.format.extent", "value": "1 verkkoaineisto (57 sivua)", "language": null, "element": "format", "qualifier": "extent", "schema": "dc"}, {"key": "dc.language.iso", "value": "fin", "language": null, "element": "language", "qualifier": "iso", "schema": "dc"}, {"key": "dc.rights", "value": "In Copyright", "language": "en", "element": "rights", "qualifier": null, "schema": "dc"}, {"key": "dc.subject.other", "value": "Gromov-hyperbolisuus", "language": null, "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "hyperbolinen ryhm\u00e4", "language": null, "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.title", "value": "Gromov-hyperboliset ryhm\u00e4t", "language": null, "element": "title", "qualifier": null, "schema": "dc"}, {"key": "dc.type", "value": "master thesis", "language": null, "element": "type", "qualifier": null, "schema": "dc"}, {"key": "dc.identifier.urn", "value": "URN:NBN:fi:jyu-201703241753", "language": null, "element": "identifier", "qualifier": "urn", "schema": "dc"}, {"key": "dc.type.ontasot", "value": "Pro gradu", "language": "fi", "element": "type", "qualifier": "ontasot", "schema": "dc"}, {"key": "dc.type.ontasot", "value": "Master's thesis", "language": "en", "element": "type", "qualifier": "ontasot", "schema": "dc"}, {"key": "dc.contributor.faculty", "value": "Matemaattis-luonnontieteellinen tiedekunta", "language": "fi", "element": "contributor", "qualifier": "faculty", "schema": "dc"}, {"key": "dc.contributor.faculty", "value": "Faculty of Sciences", "language": "en", "element": "contributor", "qualifier": "faculty", "schema": "dc"}, {"key": "dc.contributor.department", "value": "Matematiikan ja tilastotieteen laitos", "language": "fi", "element": "contributor", "qualifier": "department", "schema": "dc"}, {"key": "dc.contributor.department", "value": "Department of Mathematics and Statistics", "language": "en", "element": "contributor", "qualifier": "department", "schema": "dc"}, {"key": "dc.contributor.organization", "value": "University of Jyv\u00e4skyl\u00e4", "language": "en", "element": "contributor", "qualifier": "organization", "schema": "dc"}, {"key": "dc.contributor.organization", "value": "Jyv\u00e4skyl\u00e4n yliopisto", "language": "fi", "element": "contributor", "qualifier": "organization", "schema": "dc"}, {"key": "dc.subject.discipline", "value": "Matematiikka", "language": "fi", "element": "subject", "qualifier": "discipline", "schema": "dc"}, {"key": "dc.subject.discipline", "value": "Mathematics", "language": "en", "element": "subject", "qualifier": "discipline", "schema": "dc"}, {"key": "dc.date.updated", "value": "2017-03-24T15:07:28Z", "language": null, "element": "date", "qualifier": "updated", "schema": "dc"}, {"key": "yvv.contractresearch.funding", "value": "0", "language": null, "element": "contractresearch", "qualifier": "funding", "schema": "yvv"}, {"key": "dc.type.coar", "value": "http://purl.org/coar/resource_type/c_bdcc", "language": null, "element": "type", "qualifier": "coar", "schema": "dc"}, {"key": "dc.rights.accesslevel", "value": "openAccess", "language": "fi", "element": "rights", "qualifier": "accesslevel", "schema": "dc"}, {"key": "dc.type.publication", "value": "masterThesis", "language": null, "element": "type", "qualifier": "publication", "schema": "dc"}, {"key": "dc.subject.oppiainekoodi", "value": "4041", "language": null, "element": "subject", "qualifier": "oppiainekoodi", "schema": "dc"}, {"key": "dc.subject.yso", "value": "ryhm\u00e4teoria", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "geometria", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.rights.url", "value": "https://rightsstatements.org/page/InC/1.0/", "language": null, "element": "rights", "qualifier": "url", "schema": "dc"}]
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