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The Margulis Region in Hyperbolic 4-...

Erlandsson, Viveka.

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The Margulis Region in Hyperbolic 4-Space.

紀錄類型:	書目-語言資料,印刷品 : Monograph/item
正題名/作者:	The Margulis Region in Hyperbolic 4-Space./
作者:	Erlandsson, Viveka.
面頁冊數:	62 p.
附註:	Source: Dissertation Abstracts International, Volume: 74-09(E), Section: B.
Contained By:	Dissertation Abstracts International74-09B(E).
標題:	Mathematics. -
電子資源:	http://pqdd.sinica.edu.tw/twdaoapp/servlet/advanced?query=3561584
ISBN:	9781303087851

The Margulis Region in Hyperbolic 4-Space.
Erlandsson, Viveka.

The Margulis Region in Hyperbolic 4-Space. - 62 p.

Source: Dissertation Abstracts International, Volume: 74-09(E), Section: B.

Thesis (Ph.D.)--City University of New York, 2013.

Given a discrete subgroup Gamma of the isometries of n-dimensional hyperbolic space there is always a region kept precisely invariant under the stabilizer of a parabolic fixed point, called the Margulis region. This region corresponds to thin pieces in Thurston's thick-thin decomposition of the quotient manifold (or orbifold) M = Hn/G . In particular, the components of the Margulis region given by parabolic fixed points are related to the cusps of M. In dimensions 2 and 3 the Margulis region and the corresponding cusps are well-understood. In these dimensions parabolic isometries are conjugate to Euclidean translations and it follows that the Margulis region corresponding to a parabolic fixed point in dimensions 2 and 3 is always a horoball. In higher dimensions the region has in general a more complicated shape. This is due to the fact that parabolic isometries in dimensions 4 and higher can have a rotational part, which are called screw parabolic elements. There are examples due to Ohtake and Apanasov of discrete groups containing screw parabolic elements for which there is no precisely invariant horoball. Hence the corresponding Margulis region cannot be a horoball.

ISBN: 9781303087851Subjects--Topical Terms:

515831
Mathematics.

The Margulis Region in Hyperbolic 4-Space.
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500 $a Source: Dissertation Abstracts International, Volume: 74-09(E), Section: B.
500 $a Adviser: Ara Basmajian.
502 $a Thesis (Ph.D.)--City University of New York, 2013.
520 $a Given a discrete subgroup Gamma of the isometries of n-dimensional hyperbolic space there is always a region kept precisely invariant under the stabilizer of a parabolic fixed point, called the Margulis region. This region corresponds to thin pieces in Thurston's thick-thin decomposition of the quotient manifold (or orbifold) M = Hn/G . In particular, the components of the Margulis region given by parabolic fixed points are related to the cusps of M. In dimensions 2 and 3 the Margulis region and the corresponding cusps are well-understood. In these dimensions parabolic isometries are conjugate to Euclidean translations and it follows that the Margulis region corresponding to a parabolic fixed point in dimensions 2 and 3 is always a horoball. In higher dimensions the region has in general a more complicated shape. This is due to the fact that parabolic isometries in dimensions 4 and higher can have a rotational part, which are called screw parabolic elements. There are examples due to Ohtake and Apanasov of discrete groups containing screw parabolic elements for which there is no precisely invariant horoball. Hence the corresponding Margulis region cannot be a horoball.
520 $a It is natural to wonder about the shape of the Margulis region corresponding to a screw parabolic fixed point, and how it differs from that of a horoball. We describe the asymptotic behavior of the boundary of the Margulis region in hyperbolic 4-space corresponding to the fixed point of a screw parabolic isometry with an irrational rotation of bounded type. As a corollary we show that the region is quasi-isometric to a horoball. That is, there is a quasi-isometry of hyperbolic 4-space that maps the Margulis region to a horoball. Although it is known that two screw parabolic isometries with distinct irrational rotational parts are not conjugate by any quasi-isometry of H4 , this corollary implies that their corresponding Margulis regions (in the bounded type case) are quasi-isometric.
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