dc.contributor.advisor Goldman, William M en_US dc.contributor.author Ho, Son Lam en_US dc.date.accessioned 2014-10-11T05:58:43Z dc.date.available 2014-10-11T05:58:43Z dc.date.issued 2014 en_US dc.identifier https://doi.org/10.13016/M23P40 dc.identifier.uri http://hdl.handle.net/1903/15825 dc.description.abstract 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. 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.iso en en_US dc.title ON CONFORMALLY FLAT CIRCLE BUNDLES OVER SURFACES en_US dc.type Dissertation en_US dc.contributor.publisher Digital Repository at the University of Maryland en_US dc.contributor.publisher University of Maryland (College Park, Md.) en_US dc.contributor.department Mathematics en_US dc.subject.pqcontrolled Mathematics en_US dc.subject.pquncontrolled Circle Bundle en_US dc.subject.pquncontrolled Conformally Flat en_US dc.subject.pquncontrolled Euler number en_US dc.subject.pquncontrolled Hyperbolic en_US dc.subject.pquncontrolled Surface group en_US
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