Rips machine

在幾何群論中，Rips machine是研究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(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：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^[1]（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格羅莫夫-豪斯多夫極限，^[2]^[3]等等。使用 $\mathbb {R}$ -樹的這個machine，大幅縮短了哈肯3-流形的瑟斯頓雙曲化定理的現代證明。^[3]^[4]R-樹擔當關鍵角色的還有Culler-Vogtmann外空間的研究，^[5]^[6]，及幾何群論的其他領域；例如群的漸近錐面常常有像樹的結構，生出了R-樹上的群作用。^[7]^[8]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。^[9]^[10]

參考

^ Richard Skora. Splittings of surfaces. Bulletin of the American Mathematical Societ (N.S.), vol. 23 (1990), no. 1, pp. 85–90
^ Mladen Bestvina. Degenerations of the hyperbolic space. Duke Mathematical Journal. vol. 56 (1988), no. 1, pp. 143–161
^ ^3.0 ^3.1 M. Kapovich. Hyperbolic manifolds and discrete groups. Progress in Mathematics, 183. Birkhäuser. Boston, MA, 2001. ISBN 0-8176-3904-7
^ 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
^ Marshall Cohen, and Martin Lustig. Very small group actions on $\mathbb {R}$ -trees and Dehn twist automorphisms. Topology, vol. 34 (1995), no. 3, pp. 575–617
^ Gilbert Levitt and Martin Lustig. Irreducible automorphisms of F_n have north-south dynamics on compactified outer space. Journal de l'Institut de Mathématiques de Jussieu, vol. 2 (2003), no. 1, pp. 59–72
^ 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
^ 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
^ 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
^ Zlil Sela. Diophantine geometry over groups. I. Makanin-Razborov diagrams. Publications Mathématiques. Institut de Hautes Études Scientifiques, No. 93 (2001), pp. 31–105

Bestvina, Mladen; Feighn, Mark, Stable actions of groups on real trees, Inventiones Mathematicae, 1995, 121 (2): 287–321, ISSN 0020-9910, MR 1346208, doi:10.1007/BF01884300
Gaboriau, D.; Levitt, G.; Paulin, F., Pseudogroups of isometries of R and Rips' theorem on free actions on R-trees, Israel Journal of Mathematics, 1994, 87 (1): 403–428, ISSN 0021-2172, MR 1286836, doi:10.1007/BF02773004
Kapovich, Michael, Hyperbolic manifolds and discrete groups, Modern Birkhäuser Classics, Boston, MA: Birkhäuser Boston, 2009 [2001], ISBN 978-0-8176-4912-8, MR 1792613, doi:10.1007/978-0-8176-4913-5
Morgan, John W.; Shalen, Peter B., Free actions of surface groups on R-trees, Topology. An International Journal of Mathematics, 1991, 30 (2): 143–154, ISSN 0040-9383, MR 1098910, doi:10.1016/0040-9383(91)90002-L
Shalen, Peter B., Dendrology of groups: an introduction, Gersten, S. M. (編), Essays in group theory, Math. Sci. Res. Inst. Publ. 8, Berlin, New York: Springer-Verlag: 265–319, 1987, ISBN 978-0-387-96618-2, MR 0919830

外部連結

Wilton, Henry, Rips theory (PDF), 2003 ^{[永久失效連結]}

[1] Richard Skora. Splittings of surfaces. Bulletin of the American Mathematical Societ (N.S.), vol. 23 (1990), no. 1, pp. 85–90

[2] Mladen Bestvina. Degenerations of the hyperbolic space. Duke Mathematical Journal. vol. 56 (1988), no. 1, pp. 143–161

[k-3] 3.0 ^3.1 M. Kapovich. Hyperbolic manifolds and discrete groups. Progress in Mathematics, 183. Birkhäuser. Boston, MA, 2001. ISBN 0-8176-3904-7

[4] 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

[5] Marshall Cohen, and Martin Lustig. Very small group actions on $\mathbb {R}$ -trees and Dehn twist automorphisms. Topology, vol. 34 (1995), no. 3, pp. 575–617

[6] Gilbert Levitt and Martin Lustig. Irreducible automorphisms of F_n have north-south dynamics on compactified outer space. Journal de l'Institut de Mathématiques de Jussieu, vol. 2 (2003), no. 1, pp. 59–72

[7] 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

[8] 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

[9] 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

[10] Zlil Sela. Diophantine geometry over groups. I. Makanin-Razborov diagrams. Publications Mathématiques. Institut de Hautes Études Scientifiques, No. 93 (2001), pp. 31–105

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