Rips machine
In geometric group theory, the Rips machine is a method of studying the action of groups on R-trees. It was introduced in unpublished work of Eliyahu Rips in about 1991. An R-tree is a uniquely arcwise-connected metric space in which every arc is isometric to some real interval. Rips proved the conjecture of Morgan and Shalen[1] that any finitely generated group acting freely on an R-tree is a free product of free abelian and surface groups.[2]
Actions of surface groups on R-trees
By Bass–Serre theory, a group acting freely on a simplicial tree is free. This is no longer true for R-trees, as Morgan and Shalen showed that the fundamental groups of surfaces of Euler characteristic less than −1 also act freely on a R-trees.[1] They proved that the fundamental group of a connected closed surface S acts freely on an R-tree if and only if S is not one of the 3 nonorientable surfaces of Euler characteristic ≥−1.
Applications
The Rips machine assigns to a stable isometric action of a finitely generated group G a certain "normal form" approximation of that action by a stable action of G on a simplicial tree and hence a splitting of G in the sense of Bass–Serre theory. Group actions on real trees arise naturally in several contexts in geometric topology: for example as boundary points of the Teichmüller space[3] (every point in the Thurston boundary of the Teichmüller space is represented by a measured geodesic lamination on the surface; this lamination lifts to the universal cover of the surface and a naturally dual object to that lift is an [math]\displaystyle{ \mathbb R }[/math]-tree endowed with an isometric action of the fundamental group of the surface), as Gromov-Hausdorff limits of, appropriately rescaled, Kleinian group actions,[4][5] and so on. The use of [math]\displaystyle{ \mathbb R }[/math]-trees machinery provides substantial shortcuts in modern proofs of Thurston's Hyperbolization Theorem for Haken 3-manifolds.[5][6] Similarly, [math]\displaystyle{ \mathbb R }[/math]-trees play a key role in the study of Culler-Vogtmann's Outer space[7][8] as well as in other areas of geometric group theory; for example, asymptotic cones of groups often have a tree-like structure and give rise to group actions on real trees.[9][10] The use of [math]\displaystyle{ \mathbb R }[/math]-trees, together with Bass–Serre theory, is a key tool in the work of Sela on solving the isomorphism problem for (torsion-free) word-hyperbolic groups, Sela's version of the JSJ-decomposition theory and the work of Sela on the Tarski Conjecture for free groups and the theory of limit groups.[11][12]
References
- ↑ 1.0 1.1 Morgan, John W.; Shalen, Peter B. (1991), "Free actions of surface groups on R-trees", Topology 30 (2): 143–154, doi:10.1016/0040-9383(91)90002-L, ISSN 0040-9383
- ↑ Bestvina, Mladen; Feighn, Mark (1995), "Stable actions of groups on real trees", Inventiones Mathematicae 121 (2): 287–321, doi:10.1007/BF01884300, ISSN 0020-9910
- ↑ Skora, Richard (1990), "Splittings of surfaces", Bulletin of the American Mathematical Society, New Series 23 (1): 85–90, doi:10.1090/S0273-0979-1990-15907-5
- ↑ Bestvina, Mladen (1988), "Degenerations of the hyperbolic space", Duke Mathematical Journal 56 (1): 143–161, doi:10.1215/S0012-7094-88-05607-4
- ↑ 5.0 5.1 Kapovich, Michael (2001), Hyperbolic manifolds and discrete groups, Progress in Mathematics, 183, Birkhäuser, Boston, MA, doi:10.1007/978-0-8176-4913-5, ISBN 0-8176-3904-7
- ↑ Otal, Jean-Pierre (2001), The hyperbolization theorem for fibered 3-manifolds, SMF/AMS Texts and Monographs, 7, American Mathematical Society, Providence, RI and Société Mathématique de France, Paris, ISBN 0-8218-2153-9
- ↑ Cohen, Marshall; Lustig, Martin (1995), "Very small group actions on [math]\displaystyle{ \mathbb R }[/math]-trees and Dehn twist automorphisms", Topology 34 (3): 575–617, doi:10.1016/0040-9383(94)00038-M
- ↑ Levitt, Gilbert; Lustig, Martin (2003), "Irreducible automorphisms of Fn have north-south dynamics on compactified outer space", Journal de l'Institut de Mathématiques de Jussieu 2 (1): 59–72, doi:10.1017/S1474748003000033
- ↑ "Tree-graded spaces and asymptotic cones of groups (With an appendix by Denis Osin and Mark Sapir.)", Topology 44 (5): 959–1058, 2005, doi:10.1016/j.top.2005.03.003
- ↑ "Groups acting on tree-graded spaces and splittings of relatively hyperbolic groups", Advances in Mathematics 217 (3): 1313–1367, 2008, doi:10.1016/j.aim.2007.08.012
- ↑ Sela, Zlil (2002), "Diophantine geometry over groups and the elementary theory of free and hyperbolic groups", Proceedings of the International Congress of Mathematicians, II, Beijing: Higher Education Press, Beijing, pp. 87–92, ISBN 7-04-008690-5
- ↑ Sela, Zlil (2001), "Diophantine geometry over groups. I. Makanin-Razborov diagrams", Publications Mathématiques de l'Institut des Hautes Études Scientifiques 93: 31–105, doi:10.1007/s10240-001-8188-y
Further reading
- Gaboriau, D.; Levitt, G.; Paulin, F. (1994), "Pseudogroups of isometries of R and Rips' theorem on free actions on R-trees", Israel Journal of Mathematics 87 (1): 403–428, doi:10.1007/BF02773004, ISSN 0021-2172
- Kapovich, Michael (2009), Hyperbolic manifolds and discrete groups, Modern Birkhäuser Classics, Boston, MA: Birkhäuser Boston, doi:10.1007/978-0-8176-4913-5, ISBN 978-0-8176-4912-8
- Shalen, Peter B. (1987), "Dendrology of groups: an introduction", in Gersten, S. M., Essays in group theory, Math. Sci. Res. Inst. Publ., 8, Berlin, New York: Springer-Verlag, pp. 265–319, ISBN 978-0-387-96618-2
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