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| In [[geometric group theory]], the '''Rips machine''' is a method of studying the [[group action|action]] of [[group (mathematics)|groups]] on [[Real tree|'''R'''-trees]]. It was introduced in unpublished work of [[Eliyahu Rips]] in about 1991.
| | Hi, everybody! My name is Frieda. <br>It is a little about myself: I live in Canada, my city of New Westminster. <br>It's called often Eastern or cultural capital of BC. I've married 4 years ago.<br>I have 2 children - a son (Karolyn) and the daughter (Rickie). We all like Mineral collecting.<br>My site is [http://www.ohrtorah.net/wholesale http://www.ohrtorah.net/wholesale] |
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| 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 {{harvtxt|Morgan|Shalen|1991}} that any [[finitely generated group]] acting freely on an '''R'''-tree is a [[free product]] of free abelian and surface groups {{harv|Bestvina|Feighn|1995}}.
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| ==Actions of surface groups on R-trees==
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| By [[Bass–Serre theory]], a group acting freely on a simplicial tree is free. This is no longer true for '''R'''-trees, as {{harvtxt|Morgan|Shalen|1991}} showed that the fundamental groups of surfaces of [[Euler characteristic]] less than −1 also act freely on a '''R'''-trees.
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| 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.
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| ==Applications==
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| 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 tree]]s arise naturally in several contexts in [[geometric topology]]: for example as boundary points of the [[Teichmüller space]]<ref>Richard Skora. ''Splittings of surfaces.'' Bulletin of the American Mathematical Societ (N.S.), vol. 23 (1990), no. 1, pp. 85–90</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 convergence|Gromov-Hausdorff limits]] of, appropriately rescaled, [[Kleinian group]] actions,<ref>Mladen Bestvina. ''Degenerations of the hyperbolic space.'' Duke Mathematical Journal. vol. 56 (1988), no. 1, pp. 143–161</ref><ref name="k"/> and so on. The use of <math>\mathbb R</math>-trees machinery provides substantial shortcuts in modern proofs of [[Geometrization conjecture|Thurston's Hyperbolization Theorem]] for [[Haken manifold|Haken 3-manifolds]].<ref name="k">M. Kapovich. ''Hyperbolic manifolds and discrete groups.''
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| Progress in Mathematics, 183. Birkhäuser. Boston, MA, 2001. ISBN 0-8176-3904-7</ref><ref>J.-P. Otal. ''The hyperbolization theorem for fibered 3-manifolds.''
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| 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</ref> Similarly, <math>\mathbb R</math>-trees play a key role in the study of [[Marc Culler|Culler]]-[[Karen Vogtmann|Vogtmann]]'s Outer space<ref>Marshall Cohen, and Martin Lustig. ''Very small group actions on <math>\mathbb R</math>-trees and Dehn twist automorphisms.'' Topology, vol. 34 (1995), no. 3, pp. 575–617</ref><ref>Gilbert Levitt and Martin Lustig. ''Irreducible automorphisms of F<sub>n</sub> have north-south dynamics on compactified outer space.'' Journal de l'Institut de Mathématiques de Jussieu, vol. 2 (2003), no. 1, pp. 59–72</ref> as well as in other areas of [[geometric group theory]]; for example, [[Ultralimit#Asymptotic cones|asymptotic cones]] of groups often have a tree-like structure and give rise to group actions on [[real tree]]s.<ref>[[Cornelia Druţu]] and Mark Sapir. ''Tree-graded spaces and asymptotic cones of groups.'' (With an appendix by Denis Osin and Sapir.) Topology, vol. 44 (2005), no. 5, pp. 959–1058</ref><ref>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</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 group]]s, 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 group]]s.<ref>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</ref><ref>Zlil Sela. ''Diophantine geometry over groups. I. Makanin-Razborov diagrams.'' Publications Mathématiques. Institut de Hautes Études Scientifiques, No. 93 (2001), pp. 31–105</ref>
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| ==References==
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| <references />
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| *{{Citation | last1=Bestvina | first1=Mladen | last2=Feighn | first2=Mark | title=Stable actions of groups on real trees | doi=10.1007/BF01884300 | mr=1346208 | year=1995 | journal=[[Inventiones Mathematicae]] | issn=0020-9910 | volume=121 | issue=2 | pages=287–321}}
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| *{{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 | mr=1286836 | year=1994 | journal=Israel Journal of Mathematics | issn=0021-2172 | volume=87 | issue=1 | pages=403–428}}
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| *{{Citation | last1=Kapovich | first1=Michael | title=Hyperbolic manifolds and discrete groups | origyear=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}}
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| *{{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 (journal)|Topology. An International Journal of Mathematics]] | issn=0040-9383 | volume=30 | issue=2 | pages=143–154}}
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| *{{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}}
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| ==External links==
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| *{{citation|url=http://www.homepages.ucl.ac.uk/~ucahhjr/Notes/rips.pdf|title=Rips theory|first=Henry|last=Wilton|year=2003}}
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| {{DEFAULTSORT:Rips Machine}}
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| [[Category:Hyperbolic geometry]]
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| [[Category:Geometric group theory]]
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| [[Category:Trees (topology)]]
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Hi, everybody! My name is Frieda.
It is a little about myself: I live in Canada, my city of New Westminster.
It's called often Eastern or cultural capital of BC. I've married 4 years ago.
I have 2 children - a son (Karolyn) and the daughter (Rickie). We all like Mineral collecting.
My site is http://www.ohrtorah.net/wholesale