Rips machine

在几何群论中，Rips machine是研究R-树上的群作用的一个方法。这是Eliyahu Rips于1991年左右在未发表的工作中引入的。

一个R-树是唯一地弧连通的度量空间，内里每条弧都与一个实区间等距。Rips证明了Morgan & Shalen (1991)的猜想，就是每个自由作用在R-树上的有限生成群都是自由阿贝尔群和曲面群的自由积(Bestvina & Feighn 1995)。

曲面群在R-树上的作用

根据Bass–Serre理论，一个自由作用在单纯树上的群是自由的。这结果对R-树不成立：Morgan & Shalen (1991) 证明了欧拉示性数小于-1的曲面的基本群也自由作用在R-树上。他们证明了一个连通闭曲面S的基本群在R-树上自由作用，当且仅当S不是欧拉示性数≥-1的三个不可定向曲面之一。

应用

对一个有限生成群G的一个稳定等距作用，Rips machine赋予一个“正规形式”的近似，即G在一个单纯树上的稳定作用，因此有Bass–Serre理论所指的G的一个分裂。几何拓扑学中有数种情况，会自然地遇到在R-树上的群作用：如在泰赫米勒空间的边界点^[1]（泰赫米勒空间的瑟斯顿边界上的每个点，都表示为曲面上的一个measured geodesic lamination，这个lamination提升到曲面的泛覆盖，这个提升的一个自然对偶对象是一个R-树，带有曲面的基本群的等距作用），克莱因群作用经适当地重标后的格罗莫夫-豪斯多夫极限，^[2]^[3]等等。使用 $\mathbb {R}$ -树的这个machine，大幅缩短了哈肯3-流形的瑟斯顿双曲化定理的现代证明。^[3]^[4]R-树担当关键角色的还有Culler-Vogtmann外空间的研究，^[5]^[6]，及几何群论的其他领域；例如群的渐近锥面常常有像树的结构，生出了R-树上的群作用。^[7]^[8]R-树和Bass–Serre理论是Sela工作的关键工具，以解决（无扭）字双曲群的同构问题，建立Sela版本的JSJ-分解理论，对自由群的塔斯基猜想的工作，及极限群理论。^[9]^[10]

参考

^ 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. (编), 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

外部链接

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

[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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