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 It was introduced in unpublished work of Eliyahu Rips in about 1991.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<ref name="MS1991">{{citation | last1=Morgan | first1=John W. | last2=Shalen | first2=Peter B. | title=Free actions of surface groups on '''R'''-trees | doi=10.1016/0040-9383(91)90002-L | mr=1098910 | year=1991 | journal=Topology | issn=0040-9383 | volume=30 | issue=2 | pages=143–154| doi-access=free }}</ref> that any finitely generated group acting freely on an '''R'''-tree is a free product of free abelian and surface groups.<ref>{{citation | last1=Bestvina | first1=Mladen | last2=Feighn | first2=Mark | title=Stable actions of groups on real trees | doi=10.1007/BF01884300 | doi-access=free | mr=1346208 | year=1995 | journal=Inventiones Mathematicae | issn=0020-9910 | volume=121 | issue=2 | pages=287–321| s2cid=122048815 }}</ref>
==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.<ref name="MS1991"/> 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<ref>{{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}}</ref> (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>\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,<ref>{{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}}</ref><ref name="k"/> and so on. The use of <math>\mathbb R</math>-trees machinery provides substantial shortcuts in modern proofs of Thurston's hyperbolization theorem for Haken 3-manifolds.<ref name="k">{{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}}</ref><ref>{{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}}</ref> Similarly, <math>\mathbb R</math>-trees play a key role in the study of Culler-Vogtmann's Outer space<ref>{{citation | last1=Cohen | first1=Marshall | last2=Lustig | first2=Martin | title=Very small group actions on <math>\mathbb R</math>-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}}</ref><ref>{{citation | last1=Levitt | first1=Gilbert | last2=Lustig | first2=Martin | title=Irreducible automorphisms of F<sub>n</sub> 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 }}</ref> 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.<ref>{{citation | 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 }}</ref><ref>{{citation | 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}}</ref> The use of <math>\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.<ref>{{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}}</ref><ref>{{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}}</ref>
==References== {{reflist}}
==Further reading== *{{Citation | last1=Gaboriau | first1=D. | last2=Levitt | first2=G. | last3=Paulin | first3=F. | title=Pseudogroups of isometries of '''R''' and Rips' theorem on free actions on '''R'''-trees | doi=10.1007/BF02773004 | doi-access=free | mr=1286836 | year=1994 | journal=Israel Journal of Mathematics | issn=0021-2172 | volume=87 | issue=1 | pages=403–428| s2cid=122353183 }} *{{Citation | last1=Kapovich | first1=Michael | title=Hyperbolic manifolds and discrete groups | orig-year=2001 | publisher=Birkhäuser Boston | location=Boston, MA | series=Modern Birkhäuser Classics | isbn=978-0-8176-4912-8 | doi=10.1007/978-0-8176-4913-5 | mr=1792613 | year=2009}} *{{Citation | last1=Shalen | first1=Peter B. | editor1-last=Gersten | editor1-first=S. M. | title=Essays in group theory | publisher=Springer-Verlag | location=Berlin, New York | series=Math. Sci. Res. Inst. Publ. | isbn=978-0-387-96618-2 | mr=919830 | year=1987 | volume=8 | chapter=Dendrology of groups: an introduction | pages=265–319}}
{{DEFAULTSORT:Rips Machine}} Category:Hyperbolic geometry Category:Geometric group theory Category:Trees (topology)