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[{"key": "dc.contributor.advisor", "value": "Parkkonen, Jouni", "language": "", "element": "contributor", "qualifier": "advisor", "schema": "dc"}, {"key": "dc.contributor.author", "value": "R\u00e4ty, Joona", "language": "", "element": "contributor", "qualifier": "author", "schema": "dc"}, {"key": "dc.date.accessioned", "value": "2022-06-08T09:08:32Z", "language": null, "element": "date", "qualifier": "accessioned", "schema": "dc"}, {"key": "dc.date.available", "value": "2022-06-08T09:08:32Z", "language": null, "element": "date", "qualifier": "available", "schema": "dc"}, {"key": "dc.date.issued", "value": "2022", "language": "", "element": "date", "qualifier": "issued", "schema": "dc"}, {"key": "dc.identifier.uri", "value": "https://jyx.jyu.fi/handle/123456789/81567", "language": null, "element": "identifier", "qualifier": "uri", "schema": "dc"}, {"key": "dc.description.abstract", "value": "T\u00e4m\u00e4n tutkielman tavoitteena on tutkia ryhmi\u00e4 geometrian avulla. T\u00e4h\u00e4n\ntarkoitukseen esitell\u00e4\u00e4n ryhm\u00e4n Cayleyn kaavio, joka on jonkin ryhm\u00e4n viritt\u00e4j\u00e4joukon pohjalta muodostettu graafi. Cayleyn kaaviosta saadaan luonnollisella tavalla muodostettua geodeesinen metrinen avaruus, jonka pohjalta voidaan tehd\u00e4 p\u00e4\u00e4telmi\u00e4 my\u00f6s alkuper\u00e4isest\u00e4 ryhm\u00e4st\u00e4.\nVaikka Cayleyn kaavio on riippuvainen valitusta viritt\u00e4j\u00e4joukosta, osoittautuu ett\u00e4 jotkin Cayleyn kaavion geometriset piirteet s\u00e4ilyv\u00e4t kaikilla \u00e4\u00e4rellisill\u00e4\nviritt\u00e4j\u00e4joukoilla. T\u00e4m\u00e4n formalisoimista varten m\u00e4\u00e4ritell\u00e4\u00e4n kvasi-isometriat,\njotka ovat metristen avaruuksien v\u00e4lisi\u00e4 kuvauksia, jotka tietyss\u00e4 mieless\u00e4 pysyv\u00e4t l\u00e4hell\u00e4 isometrioita.\nTutkielmassa tarkastellaan ryhmi\u00e4 kolmen eri geometrisen piirteen kautta.\nN\u00e4m\u00e4 ovat hyperbolisuus, ryhm\u00e4n p\u00e4\u00e4tyjen lukum\u00e4\u00e4r\u00e4 ja ryhm\u00e4n kasvunopeus. N\u00e4iden avulla tutkielmassa karakterisoidaan ryhm\u00e4t, joilla on \u00e4\u00e4rett\u00f6m\u00e4n syklisen ryhm\u00e4n kanssa isomorfinen \u00e4\u00e4rellisen indeksin aliryhm\u00e4. Lis\u00e4ksi osoitetaan, ett\u00e4 hyperbolisuuden tai p\u00e4\u00e4tyjen lukum\u00e4\u00e4r\u00e4n avulla voidaan p\u00e4\u00e4tell\u00e4,\nett\u00e4 ryhm\u00e4ll\u00e4 on toisen asteen vapaan ryhm\u00e4n kanssa isomorfinen aliryhm\u00e4.\nTutkielman lopussa k\u00e4yd\u00e4\u00e4n viel\u00e4 l\u00e4pi joitain esimerkkej\u00e4, jotka havainnollistavat miss\u00e4 m\u00e4\u00e4rin hyperbolisuus, p\u00e4\u00e4tyjen lukum\u00e4\u00e4r\u00e4 ja kasvunopeus liittyv\u00e4t toisiinsa.", "language": "fi", "element": "description", "qualifier": "abstract", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Submitted by Miia Hakanen (mihakane@jyu.fi) on 2022-06-08T09:08:32Z\nNo. of bitstreams: 0", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Made available in DSpace on 2022-06-08T09:08:32Z (GMT). No. of bitstreams: 0\n Previous issue date: 2022", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.format.extent", "value": "59", "language": "", "element": "format", "qualifier": "extent", "schema": "dc"}, {"key": "dc.format.mimetype", "value": "application/pdf", "language": null, "element": "format", "qualifier": "mimetype", "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": "Cayleyn graafit", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "geometrinen ryhm\u00e4teoria", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "hyperboliset ryhm\u00e4t", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.title", "value": "\u00c4\u00e4rellisesti viritettyjen ryhmien kasvunopeus, hyperbolisuus ja p\u00e4\u00e4tyjen lukum\u00e4\u00e4r\u00e4", "language": "", "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-202206083180", "language": "", "element": "identifier", "qualifier": "urn", "schema": "dc"}, {"key": "dc.type.ontasot", "value": "Pro gradu -tutkielma", "language": "fi", "element": "type", "qualifier": "ontasot", "schema": "dc"}, {"key": "dc.type.ontasot", "value": "Master\u2019s 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": "Jyv\u00e4skyl\u00e4n yliopisto", "language": "fi", "element": "contributor", "qualifier": "organization", "schema": "dc"}, {"key": "dc.contributor.organization", "value": "University of Jyv\u00e4skyl\u00e4", "language": "en", "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": "yvv.contractresearch.funding", "value": "0", "language": "", "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": null, "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": "", "element": "subject", "qualifier": "oppiainekoodi", "schema": "dc"}, {"key": "dc.subject.yso", "value": "matematiikka", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "graafit", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "geometria", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "ryhm\u00e4teoria", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.format.content", "value": "fulltext", "language": null, "element": "format", "qualifier": "content", "schema": "dc"}, {"key": "dc.rights.url", "value": "https://rightsstatements.org/page/InC/1.0/", "language": null, "element": "rights", "qualifier": "url", "schema": "dc"}, {"key": "dc.type.okm", "value": "G2", "language": null, "element": "type", "qualifier": "okm", "schema": "dc"}]
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