{"data":{"id":"10.13016/m23p40","type":"dois","attributes":{"doi":"10.13016/m23p40","prefix":"10.13016","suffix":"m23p40","identifiers":[],"alternateIdentifiers":[],"creators":[{"name":"Ho, Son Lam","nameType":"Personal","givenName":"Son Lam","familyName":"Ho","affiliation":[],"nameIdentifiers":[]}],"titles":[{"title":"ON CONFORMALLY FLAT CIRCLE BUNDLES OVER SURFACES"}],"publisher":"Digital Repository at the University of Maryland","container":{},"publicationYear":2014,"subjects":[{"subject":"Mathematics","subjectScheme":"pqcontrolled"},{"subject":"FOS: Mathematics","schemeUri":"http://www.oecd.org/science/inno/38235147.pdf","subjectScheme":"Fields of Science and Technology (FOS)"},{"subject":"Circle Bundle","subjectScheme":"pquncontrolled"},{"subject":"Conformally Flat","subjectScheme":"pquncontrolled"},{"subject":"Euler number","subjectScheme":"pquncontrolled"},{"subject":"Hyperbolic","subjectScheme":"pquncontrolled"},{"subject":"Surface group","subjectScheme":"pquncontrolled"}],"contributors":[],"dates":[{"date":"2014","dateType":"Issued"}],"language":"en","types":{"ris":"THES","bibtex":"phdthesis","citeproc":"thesis","schemaOrg":"Thesis","resourceType":"Dissertation","resourceTypeGeneral":"Collection"},"relatedIdentifiers":[],"relatedItems":[],"sizes":[],"formats":[],"version":null,"rightsList":[],"descriptions":[{"description":"We study surface groups $\\Gamma$ in $SO(4,1)$, which is the group of conformal automorphisms of $S^3$, and also the group of isometries of $\\mathbb{H}^4$. We consider such $\\Gamma$ so that its limit set $\\Lambda_\\Gamma$ is a quasi-circle in $S^3$, and so that the quotient $(S^3 - \\Lambda_\\Gamma) / \\Gamma$ is a circle bundle over a surface. This circle bundle is said to be conformally flat, and our main goal is to discover how twisted such bundle may be by establishing a bound on its Euler number.\n\nWe have two results in this direction. First, given a surface group $\\Gamma$ which admits a nice fundamental domain with $n$ sides, we show that  $(S^3 - \\Lambda_\\Gamma) / \\Gamma$ has Euler number bounded by $n^2$. Second, if $\\Gamma$ is purely loxodromic acting properly discontinuously on $\\mathbb{H}^4$, and $\\Gamma$ satisfies a mild technical condition, then the disc bundle quotient $\\mathbb{H}^4/\\Gamma$ has Euler number bounded by $(4g-2)(36g-23)$ where $g$ is the genus of the underlying surface. Both results are proven using a direct combinatorial approach. The above are not tight bounds, improvements are possible in future research.","descriptionType":"Abstract"}],"geoLocations":[],"fundingReferences":[],"xml":"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","url":"http://hdl.handle.net/1903/15825","contentUrl":null,"metadataVersion":0,"schemaVersion":"http://datacite.org/schema/kernel-2.2","source":null,"isActive":true,"state":"findable","reason":null,"viewCount":0,"viewsOverTime":[],"downloadCount":0,"downloadsOverTime":[],"referenceCount":0,"citationCount":0,"citationsOverTime":[],"partCount":0,"partOfCount":0,"versionCount":0,"versionOfCount":0,"created":"2014-10-11T06:00:55.000Z","registered":"2014-10-11T06:00:56.000Z","published":"2014","updated":"2020-08-19T16:55:26.000Z"},"relationships":{"client":{"data":{"id":"umd.lib","type":"clients"}},"provider":{"data":{"id":"umd","type":"providers"}},"media":{"data":{"id":"10.13016/m23p40","type":"media"}},"references":{"data":[]},"citations":{"data":[]},"parts":{"data":[]},"partOf":{"data":[]},"versions":{"data":[]},"versionOf":{"data":[]}}}}