(仸仸柤徧丗嬥戲戝妛丒妛廗堾戝妛崌摨僩億儘僕乕僙儈僫乕丗2018.7.乣乯

(仸仸偙偺儁乕僕偵 MathJax 傪摫擖偟傑偟偨. 2014.1.7)
(仸仸悽榖恖丗2015.8.乣丗拞懞埳撿嵐
2013.11.乣2015.7.: 拞懞怣桾
2012.4.乣2013.10.: 拞懞埳撿嵐)

$\Gamma$懡崁幃偼, 寢傃栚偺2曄悢懡崁幃晄曄検偱偁傞HOMFLYPT懡 崁幃偲Kauffman懡崁幃偺嫟捠偺楇斣學悢懡崁幃偱偁傞. 堦斒偵, 屳偄偵慺側惍悢p(>0), q偵懳偟偰, 梌偊傜傟偨寢傃栚晄曄検偐傜(p,q)働乕僽儖壔晄曄検偑摼傜傟傞. 杮島墘 偱偼, $\Gamma$懡崁幃偺(p,q)働乕僽儖壔晄曄検偵拝栚偟, 摿偵p=1,2,3偺応崌偵偮偄偰, 偙傟傑偱偵摼傜傟偨寢壥傪徯夘偡傞.

2019擭搙

If a discrete subgroup of ${\rm PSL}(2,{\mathbb C})$ contains a parabolic map then, via Shimizu's lemma, there is a precisely invariant horoball based at the parabolic fixed point. If this fixed point is infinity then the height of the horoball equals the Euclidean translation length of the parabolic map. In hyperbolic space of higher dimensions there are parabolic maps that are not translations, called screw-parabolic maps. If the rotational part of such a map has infinite order then there is no precisely invariant horoball. However, there are sub-horospherical regions that are precisely invariant. We consider discrete subgroups of four dimensional hyperbolic space with a screw-parabolic map with irrational rotational part. Waterman gave an example where the height is linear in the distance to the axis. Erlandsson and Zakeri, in a beautiful paper, gave a way to construct such a region where the height is proportional to the square root of the distance to the axis. However, their region depends heavily on the continued fraction expansion of the rotation angle and is hard to describe. Subsequently I gave a slightly worse region that is described by a simple function. By doing so I was able to improve on the asymptotics of Erlandsson and Zakeri. In this talk I will survey the problem as an example of how simple number theory influences hyperbolic geometry.

We will present a smooth version of the Gromov space of pointed proper metric spaces. The theory surrounding this space is related to the realization problem in foliation theory. Using a result about graph colorings, we obtain an answer to the aforementioned problem in the setting of laminations. We will also explore some applications to the theory of random Riemannian manifolds.

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Thurston偑桳柤側Haken懡條懱偺堦堄壔掕棟偺拞偱弎傋偨 "bounded image theorem" 偼尰嵼偵帄傞傑偱徹柧偑抦傜傟偰偄側偐偭偨丏偙偺島墘偱偼尰戙偺Klein孮榑偺抦幆傪巊偊偽偙偺掕棟偑徹柧偱偒傞偙偲傪帵偡丏Toulouse戝妛偺Cyril Lecuire偲偺嫟摨尋媶丏

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