查看“︁Rips machine”︁的源代码

在[[幾何群論]]中，'''Rips machine'''是研究[[實樹|'''R'''-樹]]上的[[群作用]]的一個方法。這是[[Eliyahu Rips]]於1991年左右在未發表的工作中引入的。

一個'''R'''-樹是唯一地[[弧連通]]的[[度量空間]]，內裏每條弧都與一個實區間等距。Rips證明了{{harvtxt|Morgan|Shalen|1991}}的猜想，就是每個自由作用在'''R'''-樹上的[[有限生成群]]都是自由阿貝爾群和曲面群的[[自由積]]{{harv|Bestvina|Feighn|1995}}。

==曲面群在R-樹上的作用==

根據[[Bass–Serre理論]]，一個自由作用在單純樹上的群是自由的。這結果對'''R'''-樹不成立：{{harvtxt|Morgan|Shalen|1991}} 證明了[[歐拉示性數]]小於-1的曲面的基本群也自由作用在'''R'''-樹上。他們證明了一個連通閉曲面''S''的基本群在'''R'''-樹上自由作用，當且僅當''S''不是歐拉示性數≥-1的三個不可定向曲面之一。

==應用==

對一個有限生成群''G''的一個穩定等距作用，Rips machine賦予一個「正規形式」的近似，即''G''在一個單純樹上的穩定作用，因此有Bass–Serre理論所指的''G''的一個分裂。[[幾何拓撲學]]中有數種情況，會自然地遇到在'''R'''-樹上的群作用：如在[[泰赫米勒空間]]的邊界點<ref>Richard Skora. ''Splittings of surfaces.'' Bulletin of the American Mathematical Societ (N.S.), vol. 23 (1990), no. 1, pp. 85–90</ref>（[[泰赫米勒空間]]的瑟斯頓邊界上的每個點，都表示為曲面上的一個measured geodesic lamination，這個lamination提升到曲面的泛覆蓋，這個提升的一個自然對偶對象是一個'''R'''-樹，帶有曲面的基本群的等距作用），克萊因群作用經適當地重標後的[[Gromov-Hausdorff convergence|格羅莫夫-豪斯多夫極限]]，<ref>Mladen Bestvina. ''Degenerations of the hyperbolic space.'' Duke Mathematical Journal. vol. 56 (1988), no. 1, pp. 143–161</ref><ref name="k"/>等等。使用<math>\mathbb R</math>-樹的這個machine，大幅縮短了[[哈肯流形|哈肯3-流形]]的[[幾何化猜想|瑟斯頓雙曲化定理]]的現代證明。<ref name="k">M. Kapovich. ''Hyperbolic manifolds and discrete groups.''
Progress in Mathematics, 183. Birkhäuser. Boston, MA, 2001. ISBN 0-8176-3904-7</ref><ref>J.-P. Otal. ''The hyperbolization theorem for fibered 3-manifolds.''
Translated from the 1996 French original by Leslie D. Kay. SMF/AMS Texts and Monographs, 7. American Mathematical Society, Providence, RI; Société Mathématique de France, Paris. ISBN 0-8218-2153-9</ref>'''R'''-樹擔當關鍵角色的還有Culler-Vogtmann外空間的研究，<ref>Marshall Cohen, and Martin Lustig. ''Very small group actions on <math>\mathbb R</math>-trees and Dehn twist automorphisms.'' Topology, vol. 34 (1995), no. 3, pp. 575–617</ref><ref>Gilbert Levitt and Martin Lustig. ''Irreducible automorphisms of F<sub>n</sub> have north-south dynamics on compactified outer space.'' Journal de l'Institut de Mathématiques de Jussieu, vol. 2 (2003), no. 1, pp. 59–72</ref>，及幾何群論的其他領域；例如群的[[超極限|漸近錐面]]常常有像樹的結構，生出了'''R'''-樹上的群作用。<ref>[[Cornelia Druţu]] and Mark Sapir. ''Tree-graded spaces and asymptotic cones of groups.'' (With an appendix by Denis Osin and Sapir.) Topology, vol. 44 (2005), no. 5, pp. 959–1058</ref><ref>Cornelia Drutu, and Mark Sapir. ''Groups acting on tree-graded spaces and splittings of relatively hyperbolic groups.'' [[Advances in Mathematics]], vol. 217 (2008), no. 3, pp. 1313–1367</ref>'''R'''-樹和Bass–Serre理論是Sela工作的關鍵工具，以解決（無扭）[[字雙曲群]]的同構問題，建立Sela版本的JSJ-分解理論，對自由群的塔斯基猜想的工作，及[[極限群]]理論。<ref>Zlil Sela. ''Diophantine geometry over groups and the elementary theory of free and hyperbolic groups.'' Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002), pp. 87–92, Higher Ed. Press, Beijing, 2002; ISBN 7-04-008690-5</ref><ref>Zlil Sela.  ''Diophantine geometry over groups. I. Makanin-Razborov diagrams.'' Publications Mathématiques. Institut de Hautes Études Scientifiques, No. 93 (2001), pp. 31–105</ref>

==參考==
<references />
*{{Citation | last1=Bestvina | first1=Mladen | last2=Feighn | first2=Mark | title=Stable actions of groups on real trees | doi=10.1007/BF01884300 | mr=1346208 | year=1995 | journal=[[Inventiones Mathematicae]] | issn=0020-9910 | volume=121 | issue=2 | pages=287–321}}
*{{Citation | last1=Gaboriau | first1=D. | last2=Levitt | first2=G. | last3=Paulin | first3=F. | title=Pseudogroups of isometries of '''R''' and Rips' theorem on free actions on '''R'''-trees | doi=10.1007/BF02773004 | mr=1286836 | year=1994 | journal=Israel Journal of Mathematics | issn=0021-2172 | volume=87 | issue=1 | pages=403–428}}
*{{Citation | last1=Kapovich | first1=Michael | title=Hyperbolic manifolds and discrete groups | origyear=2001 | publisher=Birkhäuser Boston | location=Boston, MA | series=Modern Birkhäuser Classics | isbn=978-0-8176-4912-8 | doi=10.1007/978-0-8176-4913-5 | mr=1792613 | year=2009}}
*{{Citation | last1=Morgan | first1=John W. | last2=Shalen | first2=Peter B. | title=Free actions of surface groups on '''R'''-trees | doi=10.1016/0040-9383(91)90002-L | mr=1098910 | year=1991 | journal=[[Topology (journal)|Topology. An International Journal of Mathematics]] | issn=0040-9383 | volume=30 | issue=2 | pages=143–154}}
*{{Citation | last1=Shalen | first1=Peter B. | editor1-last=Gersten | editor1-first=S. M. | title=Essays in group theory | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Math. Sci. Res. Inst. Publ. | isbn=978-0-387-96618-2 | mr=919830 | year=1987 | volume=8 | chapter=Dendrology of groups: an introduction | pages=265–319}}

==外部連結==
*{{citation|url=http://www.homepages.ucl.ac.uk/~ucahhjr/Notes/rips.pdf|title=Rips theory|first=Henry|last=Wilton|year=2003}}{{dead link|date=2017年11月 |bot=InternetArchiveBot |fix-attempted=yes }}

{{DEFAULTSORT:Rips Machine}}
[[分類:雙曲幾何]]
[[分類:幾何群論]]