Rips machine

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title: "Rips machine" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["hyperbolic-geometry", "geometric-group-theory", "trees-(topology)"] topic_path: "science/mathematics" source: "https://en.wikipedia.org/wiki/Rips_machine" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0

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 It was introduced in unpublished work of Eliyahu Rips in about 1991.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

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. 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{{citation | last=Skora | first=Richard | title=Splittings of surfaces | journal=Bulletin of the American Mathematical Society |series=New Series | volume=23 | date=1990 | issue=1 | pages=85–90 | doi=10.1090/S0273-0979-1990-15907-5 | doi-access=free}} (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 \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,{{citation | first=Mladen | last=Bestvina | title=Degenerations of the hyperbolic space | journal=Duke Mathematical Journal | volume=56 | date=1988 | issue=1 | pages=143–161 | doi=10.1215/S0012-7094-88-05607-4}} and so on. The use of \mathbb R-trees machinery provides substantial shortcuts in modern proofs of Thurston's hyperbolization theorem for Haken 3-manifolds.{{citation | last=Kapovich | first=Michael | title=Hyperbolic manifolds and discrete groups | series=Progress in Mathematics | volume=183 | publisher=Birkhäuser, Boston, MA | date=2001 | isbn=0-8176-3904-7 | doi=10.1007/978-0-8176-4913-5 | doi-access=free}}{{citation | last=Otal | first=Jean-Pierre | title=The hyperbolization theorem for fibered 3-manifolds | series=SMF/AMS Texts and Monographs | year=2001 | volume=7 | publisher=American Mathematical Society, Providence, RI and Société Mathématique de France, Paris | isbn=0-8218-2153-9}} Similarly, \mathbb R-trees play a key role in the study of Culler-Vogtmann's Outer space{{citation | last1=Cohen | first1=Marshall | last2=Lustig | first2=Martin | title=Very small group actions on \mathbb R-trees and Dehn twist automorphisms | journal=Topology | volume=34 | date=1995 | issue=3 | pages=575–617 | doi=10.1016/0040-9383(94)00038-M | doi-access=free}}{{citation | last1=Levitt | first1=Gilbert | last2=Lustig | first2=Martin | title=Irreducible automorphisms of Fn have north-south dynamics on compactified outer space | journal=Journal de l'Institut de Mathématiques de Jussieu | volume=2 | date=2003 | issue=1 | pages=59–72 | doi=10.1017/S1474748003000033| s2cid=120675231 | last1=Druţu | first1=Cornelia | authorlink1=Cornelia Druţu | last2=Sapir | first2=Mark | title=Tree-graded spaces and asymptotic cones of groups (With an appendix by Denis Osin and Mark Sapir.) | journal=Topology | volume=44 | year=2005 | issue=5 |pages=959–1058 | doi=10.1016/j.top.2005.03.003 | doi-access=free| arxiv=math/0405030 | last1=Druţu | first1=Cornelia | authorlink1=Cornelia Druţu | last2=Sapir | first2=Mark | title=Groups acting on tree-graded spaces and splittings of relatively hyperbolic groups | journal=Advances in Mathematics | volume=217 | date=2008 | issue=3 | pages=1313–1367 | doi=10.1016/j.aim.2007.08.012 | doi-access=free}} The use of \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.{{citation | last=Sela | first=Zlil | chapter=Diophantine geometry over groups and the elementary theory of free and hyperbolic groups | title=Proceedings of the International Congress of Mathematicians | volume=II | place=Beijing | date=2002 | pages=87–92 | publisher=Higher Education Press, Beijing | isbn=7-04-008690-5}}{{citation | last1=Sela | first1=Zlil | title=Diophantine geometry over groups. I. Makanin-Razborov diagrams | journal=Publications Mathématiques de l'Institut des Hautes Études Scientifiques | volume=93 | date=2001 | pages=31–105 | doi=10.1007/s10240-001-8188-y | doi-access=free}}

References

(1991). "Free actions of surface groups on '''R'''-trees". [[Topology (journal).
(1995). "Stable actions of groups on real trees". [[Inventiones Mathematicae]].

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