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 It was introduced in unpublished work of Eliyahu Rips in about 1991.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]

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.

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]

• 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