ON CONFORMALLY FLAT CIRCLE BUNDLES OVER SURFACES
| dc.contributor.advisor | Goldman, William M | en_US |
| dc.contributor.author | Ho, Son Lam | en_US |
| dc.contributor.department | Mathematics | 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.date.accessioned | 2014-10-11T05:58:43Z | |
| dc.date.available | 2014-10-11T05:58:43Z | |
| dc.date.issued | 2014 | en_US |
| 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.identifier | https://doi.org/10.13016/M23P40 | |
| dc.identifier.uri | http://hdl.handle.net/1903/15825 | |
| dc.language.iso | en | 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 |
| dc.title | ON CONFORMALLY FLAT CIRCLE BUNDLES OVER SURFACES | en_US |
| dc.type | Dissertation | en_US |
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