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 It was introduced in unpublished work of Eliyahu Rips in about 1991.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 ${\displaystyle \mathbb {R} }$-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 ${\displaystyle \mathbb {R} }$-trees machinery provides substantial shortcuts in modern proofs of Thurston's hyperbolization theorem for Haken 3-manifolds. ^{[5] [6]} Similarly, ${\displaystyle \mathbb {R} }$-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 ${\displaystyle \mathbb {R} }$-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. 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, MR   1098910
2. 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, MR   1346208, S2CID   122048815
3. 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
4. Bestvina, Mladen (1988), "Degenerations of the hyperbolic space", Duke Mathematical Journal, 56 (1): 143–161, doi:10.1215/S0012-7094-88-05607-4
5. Kapovich, Michael (2001), Hyperbolic manifolds and discrete groups, Progress in Mathematics, vol. 183, Birkhäuser, Boston, MA, doi: 10.1007/978-0-8176-4913-5 , ISBN   0-8176-3904-7
6. Otal, Jean-Pierre (2001), The hyperbolization theorem for fibered 3-manifolds, SMF/AMS Texts and Monographs, vol. 7, American Mathematical Society, Providence, RI and Société Mathématique de France, Paris, ISBN   0-8218-2153-9
7. Cohen, Marshall; Lustig, Martin (1995), "Very small group actions on ${\displaystyle \mathbb {R} }$-trees and Dehn twist automorphisms", Topology , 34 (3): 575–617, doi: 10.1016/0040-9383(94)00038-M
8. Levitt, Gilbert; Lustig, Martin (2003), "Irreducible automorphisms of F_n have north-south dynamics on compactified outer space", Journal de l'Institut de Mathématiques de Jussieu, 2 (1): 59–72, doi:10.1017/S1474748003000033, S2CID   120675231
9. Druţu, Cornelia; Sapir, Mark (2005), "Tree-graded spaces and asymptotic cones of groups (With an appendix by Denis Osin and Mark Sapir.)", Topology , 44 (5): 959–1058, arXiv: math/0405030 , doi: 10.1016/j.top.2005.03.003
10. Druţu, Cornelia; Sapir, Mark (2008), "Groups acting on tree-graded spaces and splittings of relatively hyperbolic groups", Advances in Mathematics , 217 (3): 1313–1367, doi: 10.1016/j.aim.2007.08.012
11. Sela, Zlil (2002), "Diophantine geometry over groups and the elementary theory of free and hyperbolic groups", Proceedings of the International Congress of Mathematicians, vol. II, Beijing: Higher Education Press, Beijing, pp. 87–92, ISBN   7-04-008690-5
12. 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, MR   1286836, S2CID   122353183
• Kapovich, Michael (2009) [2001], 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, MR   1792613
• Shalen, Peter B. (1987), "Dendrology of groups: an introduction", in Gersten, S. M. (ed.), Essays in group theory, Math. Sci. Res. Inst. Publ., vol. 8, Berlin, New York: Springer-Verlag, pp. 265–319, ISBN   978-0-387-96618-2, MR   0919830