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 It was introduced in unpublished work of Eliyahu Rips in about 1991.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 & Shalen (1991) that any finitely generated group acting freely on an R-tree is a free product of free abelian and surface groups ( Bestvina & Feighn 1995 ).

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 & Shalen (1991) showed that the fundamental groups of surfaces of Euler characteristic less than −1 also act freely on a R-trees. 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 ^[1] (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, ^{[2] [3]} 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. ^{[3] [4]} Similarly, ${\displaystyle \mathbb {R} }$-trees play a key role in the study of Culler-Vogtmann's Outer space ^{[5] [6]} 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. ^{[7] [8]} 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. ^{[9] [10]}

References

1. Richard Skora. Splittings of surfaces. Bulletin of the American Mathematical Society (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
3. 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 ${\displaystyle \mathbb {R} }$-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 Mark 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

• 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
• 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
• 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
• 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
• Shalen, Peter B. (1987), "Dendrology of groups: an introduction", in Gersten, S. M. (ed.), Essays in group theory, Math. Sci. Res. Inst. Publ., 8, Berlin, New York: Springer-Verlag, pp. 265–319, ISBN   978-0-387-96618-2, MR   0919830