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dc.contributor.advisorGoldman, William Men_US
dc.contributor.authorHo, Son Lamen_US
dc.date.accessioned2014-10-11T05:58:43Z
dc.date.available2014-10-11T05:58:43Z
dc.date.issued2014en_US
dc.identifierhttps://doi.org/10.13016/M23P40
dc.identifier.urihttp://hdl.handle.net/1903/15825
dc.description.abstractWe 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. We 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.en_US
dc.language.isoenen_US
dc.titleON CONFORMALLY FLAT CIRCLE BUNDLES OVER SURFACESen_US
dc.typeDissertationen_US
dc.contributor.publisherDigital Repository at the University of Marylanden_US
dc.contributor.publisherUniversity of Maryland (College Park, Md.)en_US
dc.contributor.departmentMathematicsen_US
dc.subject.pqcontrolledMathematicsen_US
dc.subject.pquncontrolledCircle Bundleen_US
dc.subject.pquncontrolledConformally Flaten_US
dc.subject.pquncontrolledEuler numberen_US
dc.subject.pquncontrolledHyperbolicen_US
dc.subject.pquncontrolledSurface groupen_US


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