125:(every point in the Thurston boundary of the TeichmĂĽller 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
93:-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.
516:
247:
205:
175:
145:
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795:
476:, SMF/AMS Texts and Monographs, vol. 7, American Mathematical Society, Providence, RI and Société Mathématique de France, Paris,
675:
481:
457:
253:, Sela's version of the JSJ-decomposition theory and the work of Sela on the Tarski Conjecture for free groups and the theory of
249:-trees, together with Bass–Serre theory, is a key tool in the work of Sela on solving the isomorphism problem for (torsion-free)
856:
732:
851:
148:
25:
861:
178:
57:
in which every arc is isometric to some real interval. Rips proved the conjecture of Morgan and Shalen that any
666:
Sela, Zlil (2002), "Diophantine geometry over groups and the elementary theory of free and hyperbolic groups",
329:
58:
638:
636:; Sapir, Mark (2008), "Groups acting on tree-graded spaces and splittings of relatively hyperbolic groups",
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17:
78:
250:
122:
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29:
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595:
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364:
283:
254:
118:
821:
791:
751:
671:
633:
589:; Sapir, Mark (2005), "Tree-graded spaces and asymptotic cones of groups (With an appendix by
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51:
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443:
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Shalen, Peter B. (1987), "Dendrology of groups: an introduction", in
Gersten, S. M. (ed.),
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763:
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314:
831:
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Sela, Zlil (2001), "Diophantine geometry over groups. I. Makanin-Razborov diagrams",
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368:
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147:-tree endowed with an isometric action of the fundamental group of the surface), as
101:
The Rips machine assigns to a stable isometric action of a finitely generated group
66:
54:
40:
422:
327:
Bestvina, Mladen; Feighn, Mark (1995), "Stable actions of groups on real trees",
81:, a group acting freely on a simplicial tree is free. This is no longer true for
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85:-trees, as Morgan and Shalen showed that the fundamental groups of surfaces of
787:
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619:
590:
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448:
220:
755:
352:
306:
277:
Morgan, John W.; Shalen, Peter B. (1991), "Free actions of surface groups on
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of groups often have a tree-like structure and give rise to group actions on
722:
Gaboriau, D.; Levitt, G.; Paulin, F. (1994), "Pseudogroups of isometries of
224:
114:
33:
105:
a certain "normal form" approximation of that action by a stable action of
670:, vol. II, Beijing: Higher Education Press, Beijing, pp. 87–92,
692:
Publications Mathématiques de l'Institut des Hautes Études
Scientifiques
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548:
Levitt, Gilbert; Lustig, Martin (2003), "Irreducible automorphisms of F
343:
609:
496:
Cohen, Marshall; Lustig, Martin (1995), "Very small group actions on
177:-trees machinery provides substantial shortcuts in modern proofs of
442:, Progress in Mathematics, vol. 183, Birkhäuser, Boston, MA,
409:
Bestvina, Mladen (1988), "Degenerations of the hyperbolic space",
816:, Math. Sci. Res. Inst. Publ., vol. 8, Berlin, New York:
782:, Modern Birkhäuser Classics, Boston, MA: Birkhäuser Boston,
668:
552:
have north-south dynamics on compactified outer space",
502:
233:
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131:
113:
in the sense of Bass–Serre theory. Group actions on
474:
The hyperbolization theorem for fibered 3-manifolds
510:
241:
199:
169:
139:
554:Journal de l'Institut de Mathématiques de Jussieu
381:Skora, Richard (1990), "Splittings of surfaces",
109:on a simplicial tree and hence a splitting of
383:Bulletin of the American Mathematical Society
8:
215:'s Outer space as well as in other areas of
39:. It was introduced in unpublished work of
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780:Hyperbolic manifolds and discrete groups
440:Hyperbolic manifolds and discrete groups
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121:: for example as boundary points of the
89:less than −1 also act freely on a
266:
207:-trees play a key role in the study of
117:arise naturally in several contexts in
518:-trees and Dehn twist automorphisms",
726:and Rips' theorem on free actions on
7:
73:Actions of surface groups on R-trees
69:of free abelian and surface groups.
433:
431:
179:Thurston's Hyperbolization Theorem
14:
396:10.1090/S0273-0979-1990-15907-5
155:actions, and so on. The use of
1:
733:Israel Journal of Mathematics
423:10.1215/S0012-7094-88-05607-4
535:10.1016/0040-9383(94)00038-M
511:{\displaystyle \mathbb {R} }
298:10.1016/0040-9383(91)90002-L
242:{\displaystyle \mathbb {R} }
200:{\displaystyle \mathbb {R} }
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140:{\displaystyle \mathbb {R} }
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778:Kapovich, Michael (2009) ,
878:
472:Otal, Jean-Pierre (2001),
438:Kapovich, Michael (2001),
788:10.1007/978-0-8176-4913-5
705:10.1007/s10240-001-8188-y
653:10.1016/j.aim.2007.08.012
620:10.1016/j.top.2005.03.003
566:10.1017/S1474748003000033
449:10.1007/978-0-8176-4913-5
411:Duke Mathematical Journal
330:Inventiones Mathematicae
59:finitely generated group
639:Advances in Mathematics
149:Gromov-Hausdorff limits
857:Geometric group theory
814:Essays in group theory
512:
251:word-hyperbolic groups
243:
217:geometric group theory
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171:
141:
18:geometric group theory
513:
244:
202:
172:
142:
820:, pp. 265–319,
500:
231:
189:
159:
129:
87:Euler characteristic
61:acting freely on an
50:-tree is a uniquely
852:Hyperbolic geometry
593:and Mark Sapir.)",
747:10.1007/BF02773004
508:
344:10.1007/BF01884300
239:
197:
167:
137:
119:geometric topology
827:978-0-387-96618-2
797:978-0-8176-4912-8
183:Haken 3-manifolds
123:TeichmĂĽller space
79:Bass–Serre theory
52:arcwise-connected
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862:Trees (topology)
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808:
774:
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646:(3): 1313–1367,
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716:Further reading
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664:
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634:DruĹŁu, Cornelia
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631:
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603:(5): 959–1058,
587:DruĹŁu, Cornelia
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219:; for example,
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127:
126:
99:
75:
43:in about 1991.
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740:(1): 403–428,
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549:
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528:(3): 575–617,
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417:(1): 143–161,
401:
385:, New Series,
373:
337:(2): 287–321,
319:
291:(2): 143–154,
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237:
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165:
153:Kleinian group
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185:. Similarly,
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610:math/0405030
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255:limit groups
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97:Applications
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67:free product
62:
55:metric space
47:
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41:Eliyahu Rips
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22:Rips machine
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65:-tree is a
846:Categories
698:: 31–105,
591:Denis Osin
261:References
225:real trees
115:real trees
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596:Topology
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