1933:(2009), (2011), and others. In particular, Bowditch's 2013 paper introduced the notion of a "stack" of Gromov-hyperbolic metric spaces and developed an alternative framework to that of Mj for proving various results about Cannon–Thurston maps.
993:
311:
2275:
2681:
3542:
1912:
1364:
1139:
4387:
4143:
5743:
5497:
2595:
4667:
4629:
614:
567:
427:
4061:
2000:
1632:
4937:
as subgroups of other relatively hyperbolic groups in many instances also induce equivariant continuous maps between their
Bowditch boundaries; such maps are also referred to as Cannon–Thurston maps.
2997:
3592:
1706:
1284:
679:
516:
4748:
3852:
4300:
1521:
4451:
3451:
3356:
3292:
3243:
3182:
3133:
3061:
2134:
5010:
Pal obtained a generalization of Mitra's earlier result, about the existence of the Cannon–Thurston map for short exact sequences of word-hyperbolic groups, to relatively hyperbolic contex.
1408:
4925:
4332:
1467:
1082:
898:
5055:
4844:
4790:
4513:
4252:
3976:
2415:
2739:
3721:
3627:
4569:
2468:
2373:
5128:(2007), 1315–1355; 'This influential paper dates from the mid-1980's. Indeed, preprint versions are referenced in more than 30 published articles, going back as early as 1990.'
2316:
4880:
2206:
2066:
1849:
1800:
1230:
1201:
1168:
724:
376:
175:
139:
3778:
1575:
1038:
850:
808:
6125:
2949:
1742:
907:
201:
107:
4088:
3882:
463:
347:
237:
4474:
3410:
3387:
2157:
741:
After the paper of Cannon and
Thurston generated a large amount of follow-up work, with other researchers analyzing the existence or non-existence of analogs of the map
3666:
5594:
5392:
Brian H. Bowditch (2013). "Stacks of hyperbolic spaces and ends of 3-manifolds". In Craig D. Hodgson; William H. Jaco; Martin G. Scharlemann; Stephan
Tillmann (eds.).
4687:
2037:
1771:
4209:
4189:
4169:
4003:
3936:
58:
called "Group-invariant Peano curves". The preprint remained unpublished until 2007, but in the meantime had generated numerous follow-up works by other researchers.
5015:
Mj and Rafi used the Cannon–Thurston map to study which subgroups are quasiconvex in extensions of free groups and surface groups by convex cocompact subgroups of
2910:
2086:
1652:
1820:
5209:
242:
4853:
Leininger, Long and Reid used Cannon–Thurston maps to show that any finitely generated torsion-free nonfree
Kleinian group with limit set equal to
4800:
with one puncture, such that this map, in a precise sense, encodes all the Cannon–Thurston maps corresponding to arbitrary ending laminations on
2215:
6486:
5977:
5401:
6256:
2600:
4580:
As an application of the result about the existence of Cannon–Thurston maps for
Kleinian surface group representations, Mj proved that if
3471:
1854:
1296:
3135:
does not exist. Later
Matsuda and Oguni generalized the Baker–Riley approach and showed that every non-elementary word-hyperbolic group
1090:
5201:
4337:
4093:
6172:
Christopher J. Leininger; Mahan Mj; Saul
Schleimer (2011). "The universal Cannon–Thurston map and the boundary of the curve complex".
2541:
6174:
5910:
6415:
6207:
5640:
4634:
3789:
5586:
4583:
572:
525:
388:
6520:
4008:
1948:
1580:
4968:(if such a map exists) is also referred to as a Cannon–Thurston map. Of particular interest in this setting is the case where
4212:
2954:
3547:
1661:
1239:
634:
471:
4699:
3806:
6129:
5995:
5689:
5258:
4260:
1472:
4415:
3415:
3320:
3256:
3207:
3146:
3097:
3025:
6515:
5993:
Spencer
Dowdall; Ilya Kapovich; Samuel J. Taylor (2016). "Cannon–Thurston maps for hyperbolic free group extensions".
5004:
4934:
2095:
5861:
3002:
By combining and iterating these constructions, Mitra produced examples of hyperbolic subgroups of hyperbolic groups
1369:
4885:
6510:
4305:
3468:
It the context of the original Cannon–Thurston paper, and for many generalizations for the
Kleinin representations
1428:
1043:
859:
379:
5018:
4807:
4753:
5352:
4479:
4218:
3942:
2430:
5301:
2815:.) The original Cannon–Thurston theorem about fibered hyperbolic 3-manifolds is a special case of this result.
2378:
2705:
5797:
21:
5792:
3682:
3597:
6475:
Proceedings of the
International Congress of Mathematicians—Rio de Janeiro 2018. Vol. II. Invited lectures
5966:
Proceedings of the International Congress of Mathematicians—Rio de Janeiro 2018. Vol. II. Invited lectures
5451:
5084:
4526:
901:
6505:
5687:
Owen Baker; Timothy R. Riley (2020). "Cannon–Thurston maps, subgroup distortion, and hyperbolic hydra".
5147:
3676:
2769:
2491:
2475:
2436:
2321:
2007:
66:
2280:
4856:
2182:
2042:
1825:
1776:
1206:
1177:
1144:
700:
352:
151:
115:
5751:
5505:
5310:
5121:
4090:
is word-hyperbolic. In this setting Dowdall, Kapovich and Taylor proved that the Cannon–Thurston map
988:{\displaystyle \rho :H\to \mathbb {P} SL(2,\mathbb {C} )=\operatorname {Isom} _{+}(\mathbb {H} ^{3})}
39:
1413:
For Kleinian representations of surface groups, the most general result in this direction is due to
35:"Group-invariant Peano curves" (eventually published in 2007) about fibered hyperbolic 3-manifolds.
6314:
3741:
3068:
3011:
1538:
1001:
813:
771:
625:
6312:
Mahan Mj; Abhijit Pal (2011). "Relative hyperbolicity, trees of spaces and Cannon–Thurston maps".
6108:
2915:
1715:
180:
83:
6450:
6424:
6395:
6349:
6323:
6291:
6265:
6216:
6154:
6087:
6061:
6030:
6004:
5945:
5919:
5870:
5832:
5806:
5724:
5698:
5649:
5635:
5548:
5226:
5182:
5156:
4389:
is uniformly finite-to-one, with a bound on the cardinality of point preimages depending only on
4066:
3860:
432:
316:
206:
4930:
Jeon and Ohshika used Cannon–Thurston maps to establish measurable rigidity for Kleinian groups.
4456:
3392:
3369:
2139:
6105:
Pritam Ghosh (2020). "Limits of conjugacy classes under iterates of hyperbolic elements of Out(
3632:
1425:
is a surface without boundary, with a finite (possibly empty) set of cusps. Then one still has
624:, so that it provides a continuous onto function from the circle onto the 2-sphere, that is, a
6482:
5973:
5397:
4945:
4672:
4520:
3359:
3306:
2772:. However, it turns out that the Cannon–Thurston map exists in many other situations as well.
2013:
1747:
682:
4194:
4174:
4148:
3982:
3921:
1941:
In a 2017 paper Mj proved the existence of the Cannon–Thurston map in the following setting:
1851:. One of important applications of this result is that in the above situation the limit set
6434:
6379:
6370:
Abhijitn Pal (2010). "Relatively hyperbolic extensions of groups and Cannon–Thurston maps".
6333:
6275:
6226:
6138:
6071:
6014:
5929:
5880:
5816:
5760:
5708:
5659:
5603:
5557:
5514:
5361:
5318:
5267:
5218:
5166:
5093:
146:
110:
55:
32:
6446:
6391:
6345:
6287:
6240:
6187:
6150:
6083:
6048:
Ilya Kapovich and Martin Lustig (2015). "Cannon–Thurston fibers for iwip automorphisms of F
6026:
5941:
5894:
5828:
5774:
5720:
5673:
5617:
5571:
5528:
5464:
5375:
5332:
5281:
5238:
5178:
5107:
5003:
Mj and Pal obtained a generalization of Mitra's earlier result for graphs of groups to the
4882:, which is not a lattice and contains no parabolic elements, has discrete commensurator in
2888:
2071:
1637:
1040:
also acts by isometries, properly discontinuously and co-compactly, on the universal cover
6442:
6387:
6341:
6283:
6236:
6183:
6146:
6079:
6022:
5937:
5890:
5824:
5770:
5716:
5669:
5613:
5567:
5524:
5460:
5371:
5328:
5277:
5234:
5174:
5103:
3078:
In a 2013 paper, Baker and Riley constructed the first example of a word-hyperbolic group
2878:
2788:
2160:
1922:
78:
51:
28:
5124:, Review of: J. W. Cannon and W. P. Thurston, Group invariant Peano curves, Geom. Topol.
5314:
3888:) is convex-cocompact. In this case, by Mitra's general result, the Cannon–Thurston map
1917:
This result of Mj was preceded by numerous other results in the same direction, such as
5478:
Mahan Mj (2011). "Cannon–Thurston maps, i-bounded geometry and a theorem of McMullen".
4847:
2882:
2881:, where all vertex and edge groups are word-hyperbolic, and the edge-monomorphisms are
2753:
1926:
1805:
762:
750:
5562:
5543:
5272:
5253:
4171:. This result was first proved by Kapovich and Lustig under the extra assumption that
3723:
is a short exact sequence of three infinite torsion-free word-hyperbolic groups, then
20:
is any of a number of continuous group-equivariant maps between the boundaries of two
6499:
6295:
6158:
6034:
5949:
5836:
5728:
5431:
5296:
5186:
4793:
3728:
766:
6454:
6399:
6091:
2846:
is also word-hyperbolic. In this setting Mitra also described the fibers of the map
306:{\displaystyle {\tilde {S}}=\mathbb {H} ^{2}\subseteq \mathbb {H} ^{3}={\tilde {M}}}
6353:
3453:
exists and is non-injective, then there always exists a non-conical limit point of
5608:
3245:
exists and is injective. Moreover, it is known that the converse is also true: If
6254:
Woojin Jeon; Ken'ichi Ohshika (2016). "Measurable rigidity for Kleinian groups".
5449:
Mahan Mj (2009). "Cannon–Thurston maps for pared manifolds of bounded geometry".
5396:. Contemporary Mathematics, 597. American Mathematical Society. pp. 65–138.
6413:
Mahan Mj; Kasra Rafi (2018). "Algebraic ending laminations and quasiconvexity".
5480:
Actes du Séminaire de Théorie Spectrale et Géometrie. Volume 28. Année 2009–2010
5170:
2791:
word-hyperbolic subgroup, then the Cannon–Thurston map exists. (In this case if
2417:). In the same paper Mj obtains a more general version of this result, allowing
1918:
1170:. The Cannon–Thurston result can be interpreted as saying that these actions of
6383:
6337:
6018:
5933:
5366:
5347:
5297:"Local connectivity, Kleinian groups and geodesics on the blowup of the torus"
3732:
3675:
In general, it is known, as a consequence of the JSJ-decomposition theory for
3083:
621:
6438:
5664:
6231:
6202:
5855:
Woojin Jeon; Ilya Kapovich; Christopher Leininger; Ken'ichi Ohshika (2016).
5098:
5079:
2700:
2172:
1085:
2270:{\displaystyle J:G\cup \partial G\to \mathbb {H} ^{3}\cup \mathbb {S} ^{2}}
61:
In their paper Cannon and Thurston considered the following situation. Let
6075:
5820:
4393:. (However, Ghosh's result does not provide an explicit bound in terms of
3022:
is an arbitrarily high tower of exponentials, and the Cannon–Thurston map
1634:
be a discrete faithful representation without accidental parabolics. Then
761:
The original example of Cannon and Thurston can be thought of in terms of
6279:
3668:
is a finite set with cardinality bounded by a constant depending only on
2873:
In another paper Mitra considered the case where a word-hyperbolic group
2538:. Here "extends" means that the map between hyperbolic compactifications
1930:
1414:
5908:
Victor Gerasimov (2012). "Floyd maps for relatively hyperbolic groups".
3594:
is known to be uniformly finite-to-one. That means that for every point
385:
Nevertheless, Cannon and Thurston proved that this distorted inclusion
5885:
5856:
5765:
5519:
5323:
5230:
5202:"On rigidity, limit sets, and end invariants of hyperbolic 3-manifolds"
4476:
is finite-to-one. However, it is known that in this setting for every
3939:
6142:
5712:
3196:
is a quasi-isometrically embedded subgroup of a word-hyperbolic group
5654:
5482:. Seminar on Spectral Theory and Geometry, vol. 28. Univ. Grenoble I.
5418:
Semiconjugacies between actions of topologically tame Kleinian groups
5161:
2676:{\displaystyle {\hat {i}}|_{H}=i,{\hat {i}}|_{\partial H}=\partial i}
24:
extending a discrete isometric actions of the group on those spaces.
5222:
4145:
is uniformly finite-to-one, with point preimages having cardinality
3537:{\displaystyle \rho :\pi _{1}(S)\to \mathbb {P} SL(2,\mathbb {C} ),}
1907:{\displaystyle \Lambda \rho (\pi _{1}(S))\subseteq \mathbb {S} ^{2}}
6429:
6009:
3788:
are described by the theory of "convex cocompact" subgroups of the
1712:
Here the "without accidental parabolics" assumption means that for
1359:{\displaystyle \rho :\pi _{1}(S)\to \mathbb {P} SL(2,\mathbb {C} )}
6328:
6270:
6221:
6066:
5924:
5875:
5811:
5703:
1527:
has some cusps). In this setting Mj proved the following theorem:
1134:{\displaystyle \Lambda H\subseteq \partial H^{2}=\mathbb {S} ^{1}}
745:
in various other set-ups motivated by the Cannon–Thurston result.
681:, via collapsing stable and unstable laminations of the monodromy
50:
The Cannon–Thurston map first appeared in a mid-1980s preprint of
6473:
5964:
4696:, constructed a 'universal' Cannon–Thurston map from a subset of
4382:{\displaystyle \partial i:\partial F_{n}\to \partial E_{\Gamma }}
4138:{\displaystyle \partial i:\partial F_{n}\to \partial E_{\Gamma }}
2421:
to contain parabolics, under some extra technical assumptions on
1535:
be a complete connected finite volume hyperbolic surface and let
697:
is uniformly finite-to-one, with the pre-image of every point of
3800:) determines, via the Birman short exact sequence, an extension
38:
Cannon–Thurston maps provide many natural geometric examples of
6201:
Christopher J. Leininger; Darren D. Long; Alan W. Reid (2011).
4692:
Leininger, Mj and Schleimer, given a closed hyperbolic surface
4631:
is a finitely generated Kleinian group such that the limit set
3063:
exists. Later Barker and Riley showed that one can arrange for
2590:{\displaystyle {\hat {i}}:H\cup \partial H\to G\cup \partial G}
1421:
be a complete connected finite volume hyperbolic surface. Thus
6203:"Commensurators of finitely generated nonfree Kleinian groups"
5254:"The boundary of the Gieseking tree in hyperbolic three-space"
5636:"Cannon–Thurston maps for trees of hyperbolic metric spaces"
5145:
Mahan Mj (2014). "Cannon–Thurston maps for surface groups".
4816:
4762:
4662:{\displaystyle \Lambda \subseteq \partial \mathbb {H} ^{3}}
2885:. In this setting Mitra proved that for every vertex group
313:. This inclusion is highly distorted because the action of
5793:"On Cannon–Thurston maps for relatively hyperbolic groups"
4624:{\displaystyle \Gamma \leq \mathbb {P} SL(2,\mathbb {C} )}
2756:(i.e. quasiconvex) subgroup, then the Cannon–Thurston map
609:{\displaystyle \mathbb {S} ^{2}=\partial \mathbb {H} ^{3}}
562:{\displaystyle \mathbb {S} ^{1}=\partial \mathbb {H} ^{2}}
422:{\displaystyle \mathbb {H} ^{2}\subseteq \mathbb {H} ^{3}}
4408:
is a word-hyperbolic subgroup of a word-hyperbolic group
4056:{\displaystyle 1\to F_{n}\to E_{\Gamma }\to \Gamma \to 1}
3313:
is a word-hyperbolic subgroup of a word-hyperbolic group
3249:
is a word-hyperbolic subgroup of a word-hyperbolic group
1995:{\displaystyle \rho :G\to \mathbb {P} SL(2,\mathbb {C} )}
1627:{\displaystyle \rho :H\to \mathbb {P} SL(2,\mathbb {C} )}
3911:
determined by Γ. This description implies that map
3780:
is the fundamental group of a closed hyperbolic surface
3907:
are described by a collection of ending laminations on
2992:{\displaystyle \partial i:\partial A_{v}\to \partial G}
693:. In particular, this description implies that the map
27:
The notion originated from a seminal 1980s preprint of
6481:. World Sci. Publ., Hackensack, NJ. pp. 885–917.
5972:. World Sci. Publ., Hackensack, NJ. pp. 885–917.
5544:"Cannon–Thurston maps for hyperbolic group extensions"
5348:"The Cannon–Thurston map for punctured-surface groups"
3587:{\displaystyle j:\mathbb {S} ^{1}\to \mathbb {S} ^{2}}
1701:{\displaystyle j:\mathbb {S} ^{1}\to \mathbb {S} ^{2}}
1279:{\displaystyle j:\mathbb {S} ^{1}\to \mathbb {S} ^{2}}
674:{\displaystyle j:\mathbb {S} ^{1}\to \mathbb {S} ^{2}}
631:
Cannon and Thurston also explicitly described the map
511:{\displaystyle j:\mathbb {S} ^{1}\to \mathbb {S} ^{2}}
6111:
5021:
4888:
4859:
4810:
4756:
4743:{\displaystyle \partial \pi _{1}(S)=\mathbb {S} ^{1}}
4702:
4675:
4637:
4586:
4529:
4482:
4459:
4418:
4340:
4308:
4263:
4221:
4197:
4177:
4151:
4096:
4069:
4011:
3985:
3945:
3924:
3863:
3847:{\displaystyle 1\to H\to E_{\Gamma }\to \Gamma \to 1}
3809:
3744:
3685:
3635:
3600:
3550:
3474:
3418:
3395:
3372:
3323:
3259:
3210:
3149:
3100:
3028:
2957:
2918:
2891:
2708:
2603:
2544:
2439:
2381:
2324:
2283:
2218:
2185:
2142:
2098:
2074:
2045:
2016:
1951:
1857:
1828:
1808:
1779:
1750:
1718:
1664:
1640:
1583:
1541:
1475:
1431:
1372:
1299:
1242:
1209:
1180:
1147:
1093:
1046:
1004:
910:
862:
816:
774:
703:
637:
575:
528:
474:
435:
391:
355:
319:
245:
209:
183:
154:
118:
86:
4295:{\displaystyle \phi \in \operatorname {Out} (F_{n})}
3884:
is word-hyperbolic if and only if Γ ≤ Mod(
1516:{\displaystyle \Lambda \pi _{1}(S)=\mathbb {S} ^{1}}
5252:Roger C. Alperin; Warren Dicks; Joan Porti (1999).
4446:{\displaystyle \partial i:\partial H\to \partial G}
3938:is a convex-cocompact purely atoroidal subgroup of
3446:{\displaystyle \partial i:\partial H\to \partial G}
3351:{\displaystyle \partial i:\partial H\to \partial G}
3287:{\displaystyle \partial i:\partial H\to \partial G}
3238:{\displaystyle \partial i:\partial H\to \partial G}
3177:{\displaystyle \partial i:\partial H\to \partial G}
3128:{\displaystyle \partial i:\partial H\to \partial G}
3056:{\displaystyle \partial i:\partial H\to \partial G}
6119:
5857:"Conical limit points and the Cannon–Thurston map"
5049:
4919:
4874:
4838:
4784:
4742:
4681:
4661:
4623:
4563:
4507:
4468:
4445:
4381:
4326:
4294:
4246:
4203:
4183:
4163:
4137:
4082:
4055:
3997:
3970:
3930:
3876:
3846:
3772:
3715:
3660:
3621:
3586:
3536:
3445:
3404:
3381:
3350:
3286:
3237:
3176:
3127:
3055:
2991:
2943:
2904:
2733:
2675:
2589:
2462:
2425:. He also provided a description of the fibers of
2409:
2367:
2310:
2269:
2200:
2151:
2128:
2080:
2060:
2031:
1994:
1906:
1843:
1814:
1794:
1765:
1736:
1700:
1646:
1626:
1569:
1515:
1461:
1402:
1358:
1278:
1224:
1195:
1162:
1133:
1076:
1032:
987:
892:
844:
802:
718:
673:
608:
561:
510:
457:
421:
370:
341:
305:
231:
195:
169:
133:
101:
4575:Generalizations, applications and related results
1929:(2007) and (2013), Miyachi (2002), Souto (2006),
6472:Mahan Mj, Mahan (2018). "Cannon–Thurston maps".
5963:Mahan Mj, Mahan (2018). "Cannon–Thurston maps".
5595:Proceedings of the American Mathematical Society
4334:to be convex cocompact) the Cannon–Thurston map
3204:is word-hyperbolic, and the Cannon–Thurston map
2834:then, by a result of Mosher, the quotient group
2129:{\displaystyle j:\partial G\to \mathbb {S} ^{2}}
3727:is isomorphic to a free product of some closed
1403:{\displaystyle j:\Lambda H\to \mathbb {S} ^{2}}
77:itself is a closed hyperbolic surface, and its
4920:{\displaystyle \mathbb {P} SL(2,\mathbb {C} )}
3139:can be embedded in some word-hyperbolic group
2858:in terms of "algebraic ending laminations" on
2530:between their hyperbolic boundaries, the map
5587:"A hyperbolic-by-hyperbolic hyperbolic group"
5078:James W. Cannon; William P. Thurston (2007).
4327:{\displaystyle \Gamma =\langle \phi \rangle }
4257:Ghosh proved that for an arbitrary atoroidal
2683:, is continuous. In this setting, if the map
1462:{\displaystyle \mathbb {H} ^{2}={\tilde {S}}}
1077:{\displaystyle \mathbb {H} ^{2}={\tilde {S}}}
893:{\displaystyle \mathbb {H} ^{3}={\tilde {M}}}
8:
5432:"Cannon–Thurston maps for thick free groups"
5210:Journal of the American Mathematical Society
5120:Darryl McCullough, MR2326947 (2008i:57016),
5050:{\displaystyle \operatorname {Out} t(F_{n})}
4839:{\displaystyle \partial {\mathcal {C}}(S,z)}
4785:{\displaystyle \partial {\mathcal {C}}(S,z)}
4321:
4315:
2002:be a discrete faithful representation where
1366:, if there exists an induced continuous map
5791:Yoshifumi Matsuda; Shin-ichi Oguni (2014).
4508:{\displaystyle p\in \Lambda _{\partial G}H}
4247:{\displaystyle \operatorname {Out} (F_{n})}
3971:{\displaystyle \operatorname {Out} (F_{n})}
3143:in such a way that the Cannon–Thurston map
6054:Journal of the London Mathematical Society
5744:"Cannon–Thurston maps do not always exist"
5629:
5627:
5498:"Cannon–Thurston maps for Kleinian groups"
2506:is also word-hyperbolic. If the inclusion
904:. Thus one gets a discrete representation
757:Kleinian representations of surface groups
6428:
6327:
6269:
6230:
6220:
6113:
6112:
6110:
6065:
6008:
5923:
5884:
5874:
5810:
5764:
5702:
5663:
5653:
5607:
5561:
5518:
5365:
5322:
5271:
5160:
5097:
5038:
5020:
4910:
4909:
4890:
4889:
4887:
4866:
4862:
4861:
4858:
4815:
4814:
4809:
4761:
4760:
4755:
4734:
4730:
4729:
4710:
4701:
4674:
4653:
4649:
4648:
4636:
4614:
4613:
4594:
4593:
4585:
4543:
4528:
4493:
4481:
4458:
4417:
4373:
4357:
4339:
4307:
4283:
4262:
4235:
4220:
4196:
4176:
4150:
4129:
4113:
4095:
4074:
4068:
4035:
4022:
4010:
3984:
3959:
3944:
3923:
3868:
3862:
3826:
3808:
3755:
3743:
3684:
3640:
3634:
3613:
3609:
3608:
3599:
3578:
3574:
3573:
3563:
3559:
3558:
3549:
3524:
3523:
3504:
3503:
3485:
3473:
3417:
3394:
3371:
3322:
3258:
3209:
3148:
3099:
3027:
2974:
2956:
2929:
2917:
2896:
2890:
2713:
2707:
2655:
2650:
2638:
2637:
2622:
2617:
2605:
2604:
2602:
2546:
2545:
2543:
2452:
2446:
2442:
2441:
2438:
2410:{\displaystyle x_{0}\in \mathbb {H} ^{3}}
2401:
2397:
2396:
2386:
2380:
2347:
2323:
2293:
2288:
2282:
2261:
2257:
2256:
2246:
2242:
2241:
2217:
2192:
2188:
2187:
2184:
2141:
2120:
2116:
2115:
2097:
2073:
2052:
2048:
2047:
2044:
2015:
1985:
1984:
1965:
1964:
1950:
1921:(1994), Alperin, Dicks and Porti (1999),
1898:
1894:
1893:
1871:
1856:
1835:
1831:
1830:
1827:
1807:
1786:
1782:
1781:
1778:
1749:
1717:
1692:
1688:
1687:
1677:
1673:
1672:
1663:
1639:
1617:
1616:
1597:
1596:
1582:
1552:
1540:
1507:
1503:
1502:
1483:
1474:
1448:
1447:
1438:
1434:
1433:
1430:
1394:
1390:
1389:
1371:
1349:
1348:
1329:
1328:
1310:
1298:
1270:
1266:
1265:
1255:
1251:
1250:
1241:
1216:
1212:
1211:
1208:
1187:
1183:
1182:
1179:
1154:
1150:
1149:
1146:
1125:
1121:
1120:
1110:
1092:
1063:
1062:
1053:
1049:
1048:
1045:
1015:
1003:
976:
972:
971:
958:
944:
943:
924:
923:
909:
879:
878:
869:
865:
864:
861:
827:
815:
785:
773:
710:
706:
705:
702:
665:
661:
660:
650:
646:
645:
636:
600:
596:
595:
582:
578:
577:
574:
553:
549:
548:
535:
531:
530:
527:
502:
498:
497:
487:
483:
482:
473:
440:
434:
413:
409:
408:
398:
394:
393:
390:
362:
358:
357:
354:
324:
318:
292:
291:
282:
278:
277:
267:
263:
262:
247:
246:
244:
214:
208:
182:
161:
157:
156:
153:
125:
121:
120:
117:
88:
87:
85:
6307:
6305:
2734:{\displaystyle \Lambda _{\partial G}(H)}
1289:One can ask, given a hyperbolic surface
5850:
5848:
5846:
5491:
5489:
5387:
5385:
5067:
4005:) then for the corresponding extension
3188:Multiplicity of the Cannon–Thurston map
2862:, parameterized by the boundary points
69:that fibers over the circle with fiber
6365:
6363:
5140:
5138:
5136:
5134:
5073:
5071:
2879:fundamental group of a graph of groups
5786:
5784:
5742:Owen Baker; Timothy R. Riley (2013).
4804:. As an application, they prove that
3716:{\displaystyle 1\to H\to G\to Q\to 1}
3622:{\displaystyle p\in \mathbb {S} ^{2}}
7:
6257:Ergodic Theory and Dynamical Systems
4564:{\displaystyle (\partial i)^{-1}(p)}
3796:). Every subgroup Γ ≤ Mod(
2039:contains no parabolic isometries of
141:. Similarly, the universal cover of
3412:. However, the converse fails: If
2826:are two word-hyperbolic groups and
2482:Existence and non-existence results
4811:
4757:
4703:
4676:
4644:
4638:
4587:
4533:
4494:
4490:
4460:
4437:
4428:
4419:
4412:such that the Cannon–Thurston map
4401:bound always holds in this case.)
4397:, and it is still unknown if the 2
4374:
4366:
4350:
4341:
4309:
4198:
4191:is infinite cyclic, that is, that
4178:
4130:
4122:
4106:
4097:
4075:
4044:
4036:
3925:
3903:does exist. The fibers of the map
3869:
3835:
3827:
3437:
3428:
3419:
3396:
3373:
3342:
3333:
3324:
3317:such that the Cannon–Thurston map
3298:is uasi-isometrically embedded in
3278:
3269:
3260:
3253:such that the Cannon–Thurston map
3229:
3220:
3211:
3168:
3159:
3150:
3119:
3110:
3101:
3094:such that the Cannon–Thurston map
3047:
3038:
3029:
2983:
2967:
2958:
2714:
2710:
2667:
2656:
2581:
2566:
2463:{\displaystyle \mathbb {H} ^{3}/G}
2368:{\displaystyle J(g)=gx_{0},g\in G}
2294:
2231:
2212:Here "induces" means that the map
2143:
2105:
1858:
1476:
1379:
1103:
1094:
900:by isometries, and this action is
591:
544:
14:
6372:Proc. Indian Acad. Sci. Math. Sci
6175:Commentarii Mathematici Helvetici
5911:Geometric and Functional Analysis
2311:{\displaystyle J|_{\partial G}=j}
616:. Moreover, in this case the map
6416:Algebraic and Geometric Topology
6208:Algebraic and Geometric Topology
5641:Journal of Differential Geometry
5394:Geometry and topology down under
4944:is a group acting as a discrete
4875:{\displaystyle \mathbb {S} ^{2}}
3784:, such hyperbolic extensions of
3461:with exactly one preimage under
3389:has exactly one pre-image under
2201:{\displaystyle \mathbb {S} ^{2}}
2061:{\displaystyle \mathbb {H} ^{3}}
1844:{\displaystyle \mathbb {H} ^{2}}
1795:{\displaystyle \mathbb {H} ^{3}}
1225:{\displaystyle \mathbb {H} ^{3}}
1196:{\displaystyle \mathbb {H} ^{2}}
1163:{\displaystyle \mathbb {S} ^{1}}
719:{\displaystyle \mathbb {S} ^{2}}
371:{\displaystyle \mathbb {H} ^{3}}
170:{\displaystyle \mathbb {H} ^{3}}
134:{\displaystyle \mathbb {H} ^{2}}
5862:Conformal Geometry and Dynamics
3082:and a word-hyperbolic (in fact
5080:"Group invariant Peano curves"
5044:
5031:
4980:is the hyperbolic boundary of
4914:
4900:
4833:
4821:
4779:
4767:
4722:
4716:
4618:
4604:
4558:
4552:
4540:
4530:
4434:
4363:
4289:
4276:
4241:
4228:
4119:
4047:
4041:
4028:
4015:
3965:
3952:
3838:
3832:
3819:
3813:
3767:
3761:
3707:
3701:
3695:
3689:
3655:
3649:
3569:
3528:
3514:
3500:
3497:
3491:
3434:
3339:
3305:It is known, for more general
3294:exists and is injective, then
3275:
3226:
3165:
3116:
3044:
2980:
2935:
2728:
2722:
2651:
2643:
2618:
2610:
2572:
2551:
2334:
2328:
2289:
2237:
2111:
2026:
2020:
1989:
1975:
1961:
1886:
1883:
1877:
1864:
1760:
1754:
1683:
1621:
1607:
1593:
1564:
1558:
1495:
1489:
1453:
1385:
1353:
1339:
1325:
1322:
1316:
1293:and a discrete representation
1261:
1068:
1027:
1021:
982:
967:
948:
934:
920:
884:
839:
833:
797:
791:
656:
493:
452:
446:
336:
330:
297:
252:
226:
220:
93:
1:
6130:Groups, Geometry and Dynamics
5996:Israel Journal of Mathematics
5690:Groups, Geometry and Dynamics
5609:10.1090/S0002-9939-97-04249-4
5563:10.1016/S0040-9383(97)00036-0
5273:10.1016/S0166-8641(97)00270-8
5259:Topology and Its Applications
4988:is relatively hyperbolic and
4404:It remains unknown, whenever
4211:is generated by an atoroidal
3773:{\displaystyle H=\pi _{1}(S)}
1570:{\displaystyle H=\pi _{1}(S)}
1033:{\displaystyle H=\pi _{1}(S)}
845:{\displaystyle G=\pi _{1}(M)}
803:{\displaystyle H=\pi _{1}(S)}
6120:{\displaystyle \mathbb {F} }
5057:and of mapping class groups.
4996:is the Bowditch boundary of
4935:relatively hyperbolic groups
3915:is uniformly finite-to-one.
2944:{\displaystyle i:A_{v}\to G}
2754:quasi-isometrically embedded
2691:-equivariant, and the image
2518:extends to a continuous map
1737:{\displaystyle 1\neq h\in H}
726:having cardinality at most 2
196:{\displaystyle S\subseteq M}
102:{\displaystyle {\tilde {S}}}
5752:Forum of Mathematics, Sigma
5634:Mahan Mitra, Mahan (1998).
5295:Curtis T. McMullen (2001).
5171:10.4007/annals.2014.179.1.1
4948:on two metrizable compacta
4850:and locally path-connected.
4083:{\displaystyle E_{\Gamma }}
3877:{\displaystyle E_{\Gamma }}
1822:is a parabolic isometry of
1773:is a parabolic isometry of
683:pseudo-Anosov homeomorphism
458:{\displaystyle \pi _{1}(S)}
342:{\displaystyle \pi _{1}(S)}
232:{\displaystyle \pi _{1}(S)}
145:can be identified with the
109:can be identified with the
6537:
5346:Brian H. Bowditch (2007).
4469:{\displaystyle \partial i}
3405:{\displaystyle \partial i}
3382:{\displaystyle \partial G}
2883:quasi-isometric embeddings
2152:{\displaystyle \partial G}
6384:10.1007/s12044-010-0009-0
6338:10.1007/s10711-010-9519-2
6019:10.1007/s11856-016-1426-2
5934:10.1007/s00039-012-0175-6
5367:10.1007/s00209-006-0012-4
5353:Mathematische Zeitschrift
3661:{\displaystyle j^{-1}(p)}
3067:to have arbitrarily high
2687:exists, it is unique and
2474:Cannon–Thurston maps and
2167:, and where the image of
749:Cannon–Thurston maps and
6439:10.2140/agt.2018.18.1883
5506:Forum of Mathematics, Pi
5302:Inventiones Mathematicae
4682:{\displaystyle \Lambda }
3544:the Cannon–Thurston map
2951:the Cannon–Thurston map
2912:, for the inclusion map
2502:be a subgroup such that
2032:{\displaystyle \rho (G)}
1766:{\displaystyle \rho (h)}
429:extends to a continuous
22:hyperbolic metric spaces
6232:10.2140/agt.2011.11.605
5798:Journal of Group Theory
5452:Geometry & Topology
5099:10.2140/gt.2007.11.1315
5085:Geometry & Topology
4972:is word-hyperbolic and
4204:{\displaystyle \Gamma }
4184:{\displaystyle \Gamma }
4164:{\displaystyle \leq 2n}
3998:{\displaystyle n\geq 3}
3931:{\displaystyle \Gamma }
2779:is word-hyperbolic and
1937:General Kleinian groups
765:representations of the
6521:Geometric group theory
6121:
5665:10.4310/jdg/1214460609
5051:
4921:
4876:
4840:
4786:
4744:
4683:
4663:
4625:
4565:
4509:
4470:
4447:
4383:
4328:
4296:
4248:
4205:
4185:
4165:
4139:
4084:
4057:
3999:
3972:
3932:
3878:
3848:
3774:
3717:
3677:word-hyperbolic groups
3662:
3623:
3588:
3538:
3447:
3406:
3383:
3358:exists then for every
3352:
3288:
3239:
3178:
3129:
3057:
2993:
2945:
2906:
2811:is not quasiconvex in
2775:Mitra proved that if
2735:
2677:
2591:
2476:word-hyperbolic groups
2464:
2411:
2369:
2312:
2271:
2202:
2153:
2130:
2082:
2062:
2033:
1996:
1914:is locally connected.
1908:
1845:
1816:
1796:
1767:
1738:
1702:
1648:
1628:
1571:
1517:
1463:
1404:
1360:
1280:
1226:
1197:
1164:
1135:
1078:
1034:
989:
902:properly discontinuous
894:
846:
804:
720:
689:for this fibration of
675:
610:
563:
512:
459:
423:
372:
343:
307:
233:
197:
171:
135:
103:
6122:
5821:10.1515/jgt-2013-0024
5148:Annals of Mathematics
5052:
5005:relatively hyperbolic
4922:
4877:
4841:
4787:
4745:
4689:is locally connected.
4684:
4664:
4626:
4566:
4510:
4471:
4448:
4384:
4329:
4297:
4249:
4206:
4186:
4166:
4140:
4085:
4058:
4000:
3973:
3933:
3879:
3849:
3775:
3718:
3663:
3629:, the full pre-image
3624:
3589:
3539:
3448:
3407:
3384:
3353:
3289:
3240:
3179:
3130:
3058:
2994:
2946:
2907:
2905:{\displaystyle A_{v}}
2770:topological embedding
2736:
2678:
2592:
2492:word-hyperbolic group
2465:
2412:
2370:
2313:
2277:is continuous, where
2272:
2203:
2154:
2131:
2088:induces a continuous
2083:
2081:{\displaystyle \rho }
2063:
2034:
2008:word-hyperbolic group
1997:
1909:
1846:
1817:
1797:
1768:
1739:
1703:
1654:induces a continuous
1649:
1647:{\displaystyle \rho }
1629:
1572:
1518:
1464:
1405:
1361:
1281:
1227:
1198:
1165:
1136:
1079:
1035:
990:
895:
847:
805:
734:is the genus of
721:
676:
611:
564:
513:
460:
424:
373:
344:
308:
239:-invariant inclusion
234:
198:
172:
136:
104:
67:hyperbolic 3-manifold
6280:10.1017/etds.2015.15
6109:
5542:Mahan Mitra (1998).
5200:Yair Minsky (1994).
5122:Mathematical Reviews
5019:
4886:
4857:
4808:
4754:
4700:
4673:
4635:
4584:
4527:
4480:
4457:
4453:exists, if the map
4416:
4338:
4306:
4261:
4219:
4195:
4175:
4149:
4094:
4067:
4009:
3983:
3943:
3922:
3861:
3857:Moreover, the group
3807:
3742:
3683:
3633:
3598:
3548:
3472:
3416:
3393:
3370:
3321:
3257:
3208:
3147:
3098:
3026:
2955:
2916:
2889:
2706:
2601:
2542:
2437:
2379:
2375:(for some basepoint
2322:
2281:
2216:
2183:
2140:
2096:
2072:
2043:
2014:
1949:
1855:
1826:
1806:
1777:
1748:
1716:
1662:
1638:
1581:
1539:
1473:
1429:
1370:
1297:
1240:
1232:induce a continuous
1207:
1178:
1145:
1091:
1044:
1002:
908:
860:
814:
772:
701:
635:
573:
526:
472:
433:
389:
380:geometrically finite
353:
317:
243:
207:
181:
152:
116:
84:
40:space-filling curves
6315:Geometriae Dedicata
6076:10.1112/jlms/jdu069
5585:Lee Mosher (1997).
5430:Juan Souto (2006).
5315:2001InMat.146...35M
4940:More generally, if
4669:is connected, then
4571:has cardinality 1.
4521:conical limit point
4302:(without requiring
3790:mapping class group
3360:conical limit point
3192:As noted above, if
3069:primitive recursive
3012:subgroup distortion
2536:Cannon–Thurston map
810:. As a subgroup of
626:space-filling curve
18:Cannon–Thurston map
6516:Geometric topology
6117:
5766:10.1017/fms.2013.4
5520:10.1017/fmp.2017.2
5324:10.1007/PL00005809
5047:
4917:
4872:
4836:
4782:
4740:
4679:
4659:
4621:
4561:
4505:
4466:
4443:
4379:
4324:
4292:
4244:
4201:
4181:
4161:
4135:
4080:
4053:
3995:
3968:
3928:
3874:
3844:
3770:
3713:
3658:
3619:
3584:
3534:
3443:
3402:
3379:
3348:
3307:convergence groups
3284:
3235:
3174:
3125:
3053:
2989:
2941:
2902:
2807:are infinite then
2731:
2699:) is equal to the
2673:
2587:
2460:
2431:ending laminations
2407:
2365:
2308:
2267:
2198:
2149:
2126:
2078:
2058:
2029:
1992:
1904:
1841:
1812:
1792:
1763:
1734:
1698:
1644:
1624:
1567:
1513:
1459:
1400:
1356:
1276:
1222:
1193:
1160:
1131:
1074:
1030:
985:
890:
842:
800:
716:
671:
606:
559:
508:
465:-equivariant map
455:
419:
368:
339:
303:
229:
193:
167:
147:hyperbolic 3-space
131:
99:
16:In mathematics, a
6511:Dynamical systems
6488:978-981-3272-91-0
5979:978-981-3272-91-0
5602:(12): 3447–3455.
5496:Mahan Mj (2017).
5403:978-0-8218-8480-5
4960:-equivariant map
4946:convergence group
4213:fully irreducible
3309:reasons, that if
2646:
2613:
2554:
2092:-equivariant map
1815:{\displaystyle h}
1658:-equivariant map
1456:
1236:-equivariant map
1071:
887:
300:
255:
96:
6528:
6492:
6480:
6459:
6458:
6432:
6423:(4): 1883–1916.
6410:
6404:
6403:
6367:
6358:
6357:
6331:
6309:
6300:
6299:
6273:
6264:(8): 2498–2511.
6251:
6245:
6244:
6234:
6224:
6198:
6192:
6191:
6169:
6163:
6162:
6126:
6124:
6123:
6118:
6116:
6102:
6096:
6095:
6069:
6045:
6039:
6038:
6012:
5990:
5984:
5983:
5971:
5960:
5954:
5953:
5927:
5918:(5): 1361–1399.
5905:
5899:
5898:
5888:
5886:10.1090/ecgd/294
5878:
5852:
5841:
5840:
5814:
5788:
5779:
5778:
5768:
5748:
5739:
5733:
5732:
5706:
5684:
5678:
5677:
5667:
5657:
5631:
5622:
5621:
5611:
5591:
5582:
5576:
5575:
5565:
5539:
5533:
5532:
5522:
5502:
5493:
5484:
5483:
5475:
5469:
5468:
5446:
5440:
5439:
5427:
5421:
5420:, 2002, preprint
5416:Hideki Miyachi,
5414:
5408:
5407:
5389:
5380:
5379:
5369:
5343:
5337:
5336:
5326:
5292:
5286:
5285:
5275:
5249:
5243:
5242:
5206:
5197:
5191:
5190:
5164:
5142:
5129:
5118:
5112:
5111:
5101:
5092:(3): 1315–1356.
5075:
5056:
5054:
5053:
5048:
5043:
5042:
4926:
4924:
4923:
4918:
4913:
4893:
4881:
4879:
4878:
4873:
4871:
4870:
4865:
4845:
4843:
4842:
4837:
4820:
4819:
4791:
4789:
4788:
4783:
4766:
4765:
4750:to the boundary
4749:
4747:
4746:
4741:
4739:
4738:
4733:
4715:
4714:
4688:
4686:
4685:
4680:
4668:
4666:
4665:
4660:
4658:
4657:
4652:
4630:
4628:
4627:
4622:
4617:
4597:
4570:
4568:
4567:
4562:
4551:
4550:
4514:
4512:
4511:
4506:
4501:
4500:
4475:
4473:
4472:
4467:
4452:
4450:
4449:
4444:
4388:
4386:
4385:
4380:
4378:
4377:
4362:
4361:
4333:
4331:
4330:
4325:
4301:
4299:
4298:
4293:
4288:
4287:
4253:
4251:
4250:
4245:
4240:
4239:
4210:
4208:
4207:
4202:
4190:
4188:
4187:
4182:
4170:
4168:
4167:
4162:
4144:
4142:
4141:
4136:
4134:
4133:
4118:
4117:
4089:
4087:
4086:
4081:
4079:
4078:
4062:
4060:
4059:
4054:
4040:
4039:
4027:
4026:
4004:
4002:
4001:
3996:
3977:
3975:
3974:
3969:
3964:
3963:
3937:
3935:
3934:
3929:
3883:
3881:
3880:
3875:
3873:
3872:
3853:
3851:
3850:
3845:
3831:
3830:
3779:
3777:
3776:
3771:
3760:
3759:
3722:
3720:
3719:
3714:
3667:
3665:
3664:
3659:
3648:
3647:
3628:
3626:
3625:
3620:
3618:
3617:
3612:
3593:
3591:
3590:
3585:
3583:
3582:
3577:
3568:
3567:
3562:
3543:
3541:
3540:
3535:
3527:
3507:
3490:
3489:
3452:
3450:
3449:
3444:
3411:
3409:
3408:
3403:
3388:
3386:
3385:
3380:
3357:
3355:
3354:
3349:
3293:
3291:
3290:
3285:
3244:
3242:
3241:
3236:
3184:does not exist.
3183:
3181:
3180:
3175:
3134:
3132:
3131:
3126:
3062:
3060:
3059:
3054:
2998:
2996:
2995:
2990:
2979:
2978:
2950:
2948:
2947:
2942:
2934:
2933:
2911:
2909:
2908:
2903:
2901:
2900:
2768:exists and is a
2740:
2738:
2737:
2732:
2721:
2720:
2682:
2680:
2679:
2674:
2663:
2662:
2654:
2648:
2647:
2639:
2627:
2626:
2621:
2615:
2614:
2606:
2596:
2594:
2593:
2588:
2556:
2555:
2547:
2469:
2467:
2466:
2461:
2456:
2451:
2450:
2445:
2416:
2414:
2413:
2408:
2406:
2405:
2400:
2391:
2390:
2374:
2372:
2371:
2366:
2352:
2351:
2317:
2315:
2314:
2309:
2301:
2300:
2292:
2276:
2274:
2273:
2268:
2266:
2265:
2260:
2251:
2250:
2245:
2207:
2205:
2204:
2199:
2197:
2196:
2191:
2158:
2156:
2155:
2150:
2135:
2133:
2132:
2127:
2125:
2124:
2119:
2087:
2085:
2084:
2079:
2067:
2065:
2064:
2059:
2057:
2056:
2051:
2038:
2036:
2035:
2030:
2001:
1999:
1998:
1993:
1988:
1968:
1913:
1911:
1910:
1905:
1903:
1902:
1897:
1876:
1875:
1850:
1848:
1847:
1842:
1840:
1839:
1834:
1821:
1819:
1818:
1813:
1801:
1799:
1798:
1793:
1791:
1790:
1785:
1772:
1770:
1769:
1764:
1743:
1741:
1740:
1735:
1707:
1705:
1704:
1699:
1697:
1696:
1691:
1682:
1681:
1676:
1653:
1651:
1650:
1645:
1633:
1631:
1630:
1625:
1620:
1600:
1576:
1574:
1573:
1568:
1557:
1556:
1522:
1520:
1519:
1514:
1512:
1511:
1506:
1488:
1487:
1468:
1466:
1465:
1460:
1458:
1457:
1449:
1443:
1442:
1437:
1409:
1407:
1406:
1401:
1399:
1398:
1393:
1365:
1363:
1362:
1357:
1352:
1332:
1315:
1314:
1285:
1283:
1282:
1277:
1275:
1274:
1269:
1260:
1259:
1254:
1231:
1229:
1228:
1223:
1221:
1220:
1215:
1202:
1200:
1199:
1194:
1192:
1191:
1186:
1169:
1167:
1166:
1161:
1159:
1158:
1153:
1140:
1138:
1137:
1132:
1130:
1129:
1124:
1115:
1114:
1083:
1081:
1080:
1075:
1073:
1072:
1064:
1058:
1057:
1052:
1039:
1037:
1036:
1031:
1020:
1019:
994:
992:
991:
986:
981:
980:
975:
963:
962:
947:
927:
899:
897:
896:
891:
889:
888:
880:
874:
873:
868:
851:
849:
848:
843:
832:
831:
809:
807:
806:
801:
790:
789:
725:
723:
722:
717:
715:
714:
709:
680:
678:
677:
672:
670:
669:
664:
655:
654:
649:
615:
613:
612:
607:
605:
604:
599:
587:
586:
581:
568:
566:
565:
560:
558:
557:
552:
540:
539:
534:
517:
515:
514:
509:
507:
506:
501:
492:
491:
486:
464:
462:
461:
456:
445:
444:
428:
426:
425:
420:
418:
417:
412:
403:
402:
397:
377:
375:
374:
369:
367:
366:
361:
348:
346:
345:
340:
329:
328:
312:
310:
309:
304:
302:
301:
293:
287:
286:
281:
272:
271:
266:
257:
256:
248:
238:
236:
235:
230:
219:
218:
202:
200:
199:
194:
177:. The inclusion
176:
174:
173:
168:
166:
165:
160:
140:
138:
137:
132:
130:
129:
124:
111:hyperbolic plane
108:
106:
105:
100:
98:
97:
89:
56:William Thurston
33:William Thurston
6536:
6535:
6531:
6530:
6529:
6527:
6526:
6525:
6496:
6495:
6489:
6478:
6471:
6468:
6466:Further reading
6463:
6462:
6412:
6411:
6407:
6369:
6368:
6361:
6311:
6310:
6303:
6253:
6252:
6248:
6200:
6199:
6195:
6171:
6170:
6166:
6143:10.4171/GGD/540
6107:
6106:
6104:
6103:
6099:
6051:
6047:
6046:
6042:
5992:
5991:
5987:
5980:
5969:
5962:
5961:
5957:
5907:
5906:
5902:
5854:
5853:
5844:
5790:
5789:
5782:
5746:
5741:
5740:
5736:
5713:10.4171/ggd/543
5686:
5685:
5681:
5633:
5632:
5625:
5589:
5584:
5583:
5579:
5541:
5540:
5536:
5500:
5495:
5494:
5487:
5477:
5476:
5472:
5448:
5447:
5443:
5429:
5428:
5424:
5415:
5411:
5404:
5391:
5390:
5383:
5345:
5344:
5340:
5294:
5293:
5289:
5251:
5250:
5246:
5223:10.2307/2152785
5204:
5199:
5198:
5194:
5144:
5143:
5132:
5119:
5115:
5077:
5076:
5069:
5064:
5034:
5017:
5016:
4956:, a continuous
4884:
4883:
4860:
4855:
4854:
4806:
4805:
4752:
4751:
4728:
4706:
4698:
4697:
4671:
4670:
4647:
4633:
4632:
4582:
4581:
4577:
4539:
4525:
4524:
4489:
4478:
4477:
4455:
4454:
4414:
4413:
4369:
4353:
4336:
4335:
4304:
4303:
4279:
4259:
4258:
4231:
4217:
4216:
4193:
4192:
4173:
4172:
4147:
4146:
4125:
4109:
4092:
4091:
4070:
4065:
4064:
4031:
4018:
4007:
4006:
3981:
3980:
3955:
3941:
3940:
3920:
3919:
3902:
3864:
3859:
3858:
3822:
3805:
3804:
3751:
3740:
3739:
3681:
3680:
3636:
3631:
3630:
3607:
3596:
3595:
3572:
3557:
3546:
3545:
3481:
3470:
3469:
3414:
3413:
3391:
3390:
3368:
3367:
3319:
3318:
3255:
3254:
3206:
3205:
3190:
3145:
3144:
3096:
3095:
3024:
3023:
2970:
2953:
2952:
2925:
2914:
2913:
2892:
2887:
2886:
2709:
2704:
2703:
2649:
2616:
2599:
2598:
2540:
2539:
2484:
2479:
2440:
2435:
2434:
2395:
2382:
2377:
2376:
2343:
2320:
2319:
2287:
2279:
2278:
2255:
2240:
2214:
2213:
2186:
2181:
2180:
2161:Gromov boundary
2138:
2137:
2114:
2094:
2093:
2070:
2069:
2046:
2041:
2040:
2012:
2011:
1947:
1946:
1939:
1892:
1867:
1853:
1852:
1829:
1824:
1823:
1804:
1803:
1802:if and only if
1780:
1775:
1774:
1746:
1745:
1714:
1713:
1686:
1671:
1660:
1659:
1636:
1635:
1579:
1578:
1548:
1537:
1536:
1501:
1479:
1471:
1470:
1432:
1427:
1426:
1388:
1368:
1367:
1306:
1295:
1294:
1264:
1249:
1238:
1237:
1210:
1205:
1204:
1181:
1176:
1175:
1148:
1143:
1142:
1141:being equal to
1119:
1106:
1089:
1088:
1047:
1042:
1041:
1011:
1000:
999:
970:
954:
906:
905:
863:
858:
857:
823:
812:
811:
781:
770:
769:
759:
754:
751:Kleinian groups
704:
699:
698:
659:
644:
633:
632:
594:
576:
571:
570:
547:
529:
524:
523:
496:
481:
470:
469:
436:
431:
430:
407:
392:
387:
386:
356:
351:
350:
320:
315:
314:
276:
261:
241:
240:
210:
205:
204:
179:
178:
155:
150:
149:
119:
114:
113:
82:
81:
79:universal cover
52:James W. Cannon
48:
12:
11:
5:
6534:
6532:
6524:
6523:
6518:
6513:
6508:
6498:
6497:
6494:
6493:
6487:
6467:
6464:
6461:
6460:
6405:
6359:
6301:
6246:
6215:(1): 605–624.
6193:
6182:(4): 769–816.
6164:
6137:(1): 177–211.
6115:
6097:
6060:(1): 203–224.
6049:
6040:
6003:(2): 753–797.
5985:
5978:
5955:
5900:
5842:
5780:
5734:
5697:(1): 255–282.
5679:
5648:(1): 135–164.
5623:
5577:
5556:(3): 527–538.
5534:
5485:
5470:
5441:
5422:
5409:
5402:
5381:
5338:
5287:
5266:(3): 219–259.
5244:
5217:(3): 539–588.
5192:
5130:
5113:
5066:
5065:
5063:
5060:
5059:
5058:
5046:
5041:
5037:
5033:
5030:
5027:
5024:
5012:
5011:
5008:
5001:
4938:
4933:Inclusions of
4931:
4928:
4916:
4912:
4908:
4905:
4902:
4899:
4896:
4892:
4869:
4864:
4851:
4848:path-connected
4835:
4832:
4829:
4826:
4823:
4818:
4813:
4781:
4778:
4775:
4772:
4769:
4764:
4759:
4737:
4732:
4727:
4724:
4721:
4718:
4713:
4709:
4705:
4690:
4678:
4656:
4651:
4646:
4643:
4640:
4620:
4616:
4612:
4609:
4606:
4603:
4600:
4596:
4592:
4589:
4576:
4573:
4560:
4557:
4554:
4549:
4546:
4542:
4538:
4535:
4532:
4504:
4499:
4496:
4492:
4488:
4485:
4465:
4462:
4442:
4439:
4436:
4433:
4430:
4427:
4424:
4421:
4376:
4372:
4368:
4365:
4360:
4356:
4352:
4349:
4346:
4343:
4323:
4320:
4317:
4314:
4311:
4291:
4286:
4282:
4278:
4275:
4272:
4269:
4266:
4243:
4238:
4234:
4230:
4227:
4224:
4200:
4180:
4160:
4157:
4154:
4132:
4128:
4124:
4121:
4116:
4112:
4108:
4105:
4102:
4099:
4077:
4073:
4052:
4049:
4046:
4043:
4038:
4034:
4030:
4025:
4021:
4017:
4014:
3994:
3991:
3988:
3967:
3962:
3958:
3954:
3951:
3948:
3927:
3900:
3871:
3867:
3855:
3854:
3843:
3840:
3837:
3834:
3829:
3825:
3821:
3818:
3815:
3812:
3769:
3766:
3763:
3758:
3754:
3750:
3747:
3729:surface groups
3712:
3709:
3706:
3703:
3700:
3697:
3694:
3691:
3688:
3657:
3654:
3651:
3646:
3643:
3639:
3616:
3611:
3606:
3603:
3581:
3576:
3571:
3566:
3561:
3556:
3553:
3533:
3530:
3526:
3522:
3519:
3516:
3513:
3510:
3506:
3502:
3499:
3496:
3493:
3488:
3484:
3480:
3477:
3442:
3439:
3436:
3433:
3430:
3427:
3424:
3421:
3401:
3398:
3378:
3375:
3347:
3344:
3341:
3338:
3335:
3332:
3329:
3326:
3283:
3280:
3277:
3274:
3271:
3268:
3265:
3262:
3234:
3231:
3228:
3225:
3222:
3219:
3216:
3213:
3189:
3186:
3173:
3170:
3167:
3164:
3161:
3158:
3155:
3152:
3124:
3121:
3118:
3115:
3112:
3109:
3106:
3103:
3071:distortion in
3052:
3049:
3046:
3043:
3040:
3037:
3034:
3031:
2988:
2985:
2982:
2977:
2973:
2969:
2966:
2963:
2960:
2940:
2937:
2932:
2928:
2924:
2921:
2899:
2895:
2877:splits as the
2730:
2727:
2724:
2719:
2716:
2712:
2672:
2669:
2666:
2661:
2658:
2653:
2645:
2642:
2636:
2633:
2630:
2625:
2620:
2612:
2609:
2586:
2583:
2580:
2577:
2574:
2571:
2568:
2565:
2562:
2559:
2553:
2550:
2483:
2480:
2478:
2472:
2459:
2455:
2449:
2444:
2404:
2399:
2394:
2389:
2385:
2364:
2361:
2358:
2355:
2350:
2346:
2342:
2339:
2336:
2333:
2330:
2327:
2307:
2304:
2299:
2296:
2291:
2286:
2264:
2259:
2254:
2249:
2244:
2239:
2236:
2233:
2230:
2227:
2224:
2221:
2210:
2209:
2195:
2190:
2148:
2145:
2123:
2118:
2113:
2110:
2107:
2104:
2101:
2077:
2055:
2050:
2028:
2025:
2022:
2019:
1991:
1987:
1983:
1980:
1977:
1974:
1971:
1967:
1963:
1960:
1957:
1954:
1938:
1935:
1901:
1896:
1891:
1888:
1885:
1882:
1879:
1874:
1870:
1866:
1863:
1860:
1838:
1833:
1811:
1789:
1784:
1762:
1759:
1756:
1753:
1744:, the element
1733:
1730:
1727:
1724:
1721:
1710:
1709:
1695:
1690:
1685:
1680:
1675:
1670:
1667:
1643:
1623:
1619:
1615:
1612:
1609:
1606:
1603:
1599:
1595:
1592:
1589:
1586:
1566:
1563:
1560:
1555:
1551:
1547:
1544:
1510:
1505:
1500:
1497:
1494:
1491:
1486:
1482:
1478:
1455:
1452:
1446:
1441:
1436:
1397:
1392:
1387:
1384:
1381:
1378:
1375:
1355:
1351:
1347:
1344:
1341:
1338:
1335:
1331:
1327:
1324:
1321:
1318:
1313:
1309:
1305:
1302:
1273:
1268:
1263:
1258:
1253:
1248:
1245:
1219:
1214:
1190:
1185:
1157:
1152:
1128:
1123:
1118:
1113:
1109:
1105:
1102:
1099:
1096:
1070:
1067:
1061:
1056:
1051:
1029:
1026:
1023:
1018:
1014:
1010:
1007:
984:
979:
974:
969:
966:
961:
957:
953:
950:
946:
942:
939:
936:
933:
930:
926:
922:
919:
916:
913:
886:
883:
877:
872:
867:
841:
838:
835:
830:
826:
822:
819:
799:
796:
793:
788:
784:
780:
777:
758:
755:
753:
747:
713:
708:
668:
663:
658:
653:
648:
643:
640:
603:
598:
593:
590:
585:
580:
556:
551:
546:
543:
538:
533:
520:
519:
505:
500:
495:
490:
485:
480:
477:
454:
451:
448:
443:
439:
416:
411:
406:
401:
396:
365:
360:
338:
335:
332:
327:
323:
299:
296:
290:
285:
280:
275:
270:
265:
260:
254:
251:
228:
225:
222:
217:
213:
192:
189:
186:
164:
159:
128:
123:
95:
92:
47:
44:
13:
10:
9:
6:
4:
3:
2:
6533:
6522:
6519:
6517:
6514:
6512:
6509:
6507:
6504:
6503:
6501:
6490:
6484:
6477:
6476:
6470:
6469:
6465:
6456:
6452:
6448:
6444:
6440:
6436:
6431:
6426:
6422:
6418:
6417:
6409:
6406:
6401:
6397:
6393:
6389:
6385:
6381:
6377:
6373:
6366:
6364:
6360:
6355:
6351:
6347:
6343:
6339:
6335:
6330:
6325:
6321:
6317:
6316:
6308:
6306:
6302:
6297:
6293:
6289:
6285:
6281:
6277:
6272:
6267:
6263:
6259:
6258:
6250:
6247:
6242:
6238:
6233:
6228:
6223:
6218:
6214:
6210:
6209:
6204:
6197:
6194:
6189:
6185:
6181:
6177:
6176:
6168:
6165:
6160:
6156:
6152:
6148:
6144:
6140:
6136:
6132:
6131:
6101:
6098:
6093:
6089:
6085:
6081:
6077:
6073:
6068:
6063:
6059:
6055:
6044:
6041:
6036:
6032:
6028:
6024:
6020:
6016:
6011:
6006:
6002:
5998:
5997:
5989:
5986:
5981:
5975:
5968:
5967:
5959:
5956:
5951:
5947:
5943:
5939:
5935:
5931:
5926:
5921:
5917:
5913:
5912:
5904:
5901:
5896:
5892:
5887:
5882:
5877:
5872:
5868:
5864:
5863:
5858:
5851:
5849:
5847:
5843:
5838:
5834:
5830:
5826:
5822:
5818:
5813:
5808:
5804:
5800:
5799:
5794:
5787:
5785:
5781:
5776:
5772:
5767:
5762:
5758:
5754:
5753:
5745:
5738:
5735:
5730:
5726:
5722:
5718:
5714:
5710:
5705:
5700:
5696:
5692:
5691:
5683:
5680:
5675:
5671:
5666:
5661:
5656:
5651:
5647:
5643:
5642:
5637:
5630:
5628:
5624:
5619:
5615:
5610:
5605:
5601:
5597:
5596:
5588:
5581:
5578:
5573:
5569:
5564:
5559:
5555:
5551:
5550:
5545:
5538:
5535:
5530:
5526:
5521:
5516:
5512:
5508:
5507:
5499:
5492:
5490:
5486:
5481:
5474:
5471:
5466:
5462:
5458:
5454:
5453:
5445:
5442:
5437:
5433:
5426:
5423:
5419:
5413:
5410:
5405:
5399:
5395:
5388:
5386:
5382:
5377:
5373:
5368:
5363:
5359:
5355:
5354:
5349:
5342:
5339:
5334:
5330:
5325:
5320:
5316:
5312:
5308:
5304:
5303:
5298:
5291:
5288:
5283:
5279:
5274:
5269:
5265:
5261:
5260:
5255:
5248:
5245:
5240:
5236:
5232:
5228:
5224:
5220:
5216:
5212:
5211:
5203:
5196:
5193:
5188:
5184:
5180:
5176:
5172:
5168:
5163:
5158:
5154:
5150:
5149:
5141:
5139:
5137:
5135:
5131:
5127:
5123:
5117:
5114:
5109:
5105:
5100:
5095:
5091:
5087:
5086:
5081:
5074:
5072:
5068:
5061:
5039:
5035:
5028:
5025:
5022:
5014:
5013:
5009:
5006:
5002:
4999:
4995:
4992: =
4991:
4987:
4983:
4979:
4976: =
4975:
4971:
4967:
4964: →
4963:
4959:
4955:
4951:
4947:
4943:
4939:
4936:
4932:
4929:
4906:
4903:
4897:
4894:
4867:
4852:
4849:
4830:
4827:
4824:
4803:
4799:
4795:
4794:curve complex
4776:
4773:
4770:
4735:
4725:
4719:
4711:
4707:
4695:
4691:
4654:
4641:
4610:
4607:
4601:
4598:
4590:
4579:
4578:
4574:
4572:
4555:
4547:
4544:
4536:
4522:
4518:
4502:
4497:
4486:
4483:
4463:
4440:
4431:
4425:
4422:
4411:
4407:
4402:
4400:
4396:
4392:
4370:
4358:
4354:
4347:
4344:
4318:
4312:
4284:
4280:
4273:
4270:
4267:
4264:
4255:
4236:
4232:
4225:
4222:
4214:
4158:
4155:
4152:
4126:
4114:
4110:
4103:
4100:
4071:
4050:
4032:
4023:
4019:
4012:
3992:
3989:
3986:
3978:
3960:
3956:
3949:
3946:
3916:
3914:
3910:
3906:
3899:
3896: →
3895:
3891:
3887:
3865:
3841:
3823:
3816:
3810:
3803:
3802:
3801:
3799:
3795:
3791:
3787:
3783:
3764:
3756:
3752:
3748:
3745:
3736:
3734:
3730:
3726:
3710:
3704:
3698:
3692:
3686:
3678:
3673:
3671:
3652:
3644:
3641:
3637:
3614:
3604:
3601:
3579:
3564:
3554:
3551:
3531:
3520:
3517:
3511:
3508:
3494:
3486:
3482:
3478:
3475:
3466:
3464:
3460:
3456:
3440:
3431:
3425:
3422:
3399:
3376:
3365:
3361:
3345:
3336:
3330:
3327:
3316:
3312:
3308:
3303:
3301:
3297:
3281:
3272:
3266:
3263:
3252:
3248:
3232:
3223:
3217:
3214:
3203:
3199:
3195:
3187:
3185:
3171:
3162:
3156:
3153:
3142:
3138:
3122:
3113:
3107:
3104:
3093:
3090: ≤
3089:
3085:
3081:
3076:
3074:
3070:
3066:
3050:
3041:
3035:
3032:
3021:
3017:
3013:
3009:
3006: ≤
3005:
3000:
2986:
2975:
2971:
2964:
2961:
2938:
2930:
2926:
2922:
2919:
2897:
2893:
2884:
2880:
2876:
2871:
2869:
2866: ∈
2865:
2861:
2857:
2853:
2849:
2845:
2841:
2838: =
2837:
2833:
2830:is normal in
2829:
2825:
2822: ≤
2821:
2816:
2814:
2810:
2806:
2802:
2799: =
2798:
2794:
2790:
2786:
2783: ≤
2782:
2778:
2773:
2771:
2767:
2763:
2759:
2755:
2751:
2748: ≤
2747:
2742:
2725:
2717:
2702:
2698:
2694:
2690:
2686:
2670:
2664:
2659:
2640:
2634:
2631:
2628:
2623:
2607:
2584:
2578:
2575:
2569:
2563:
2560:
2557:
2548:
2537:
2533:
2529:
2525:
2521:
2517:
2514: →
2513:
2509:
2505:
2501:
2498: ≤
2497:
2493:
2489:
2481:
2477:
2473:
2471:
2457:
2453:
2447:
2432:
2428:
2424:
2420:
2402:
2392:
2387:
2383:
2362:
2359:
2356:
2353:
2348:
2344:
2340:
2337:
2331:
2325:
2305:
2302:
2297:
2284:
2262:
2252:
2247:
2234:
2228:
2225:
2222:
2219:
2193:
2178:
2174:
2170:
2166:
2162:
2146:
2121:
2108:
2102:
2099:
2091:
2075:
2053:
2023:
2017:
2009:
2005:
1981:
1978:
1972:
1969:
1958:
1955:
1952:
1944:
1943:
1942:
1936:
1934:
1932:
1928:
1924:
1920:
1915:
1899:
1889:
1880:
1872:
1868:
1861:
1836:
1809:
1787:
1757:
1751:
1731:
1728:
1725:
1722:
1719:
1693:
1678:
1668:
1665:
1657:
1641:
1613:
1610:
1604:
1601:
1590:
1587:
1584:
1561:
1553:
1549:
1545:
1542:
1534:
1530:
1529:
1528:
1526:
1508:
1498:
1492:
1484:
1480:
1450:
1444:
1439:
1424:
1420:
1416:
1411:
1395:
1382:
1376:
1373:
1345:
1342:
1336:
1333:
1319:
1311:
1307:
1303:
1300:
1292:
1287:
1271:
1256:
1246:
1243:
1235:
1217:
1188:
1173:
1155:
1126:
1116:
1111:
1107:
1100:
1097:
1087:
1065:
1059:
1054:
1024:
1016:
1012:
1008:
1005:
996:
977:
964:
959:
955:
951:
940:
937:
931:
928:
917:
914:
911:
903:
881:
875:
870:
855:
836:
828:
824:
820:
817:
794:
786:
782:
778:
775:
768:
767:surface group
764:
756:
752:
748:
746:
744:
739:
737:
733:
729:
711:
696:
692:
688:
684:
666:
651:
641:
638:
629:
627:
623:
619:
601:
588:
583:
554:
541:
536:
503:
488:
478:
475:
468:
467:
466:
449:
441:
437:
414:
404:
399:
383:
381:
363:
333:
325:
321:
294:
288:
283:
273:
268:
258:
249:
223:
215:
211:
190:
187:
184:
162:
148:
144:
126:
112:
90:
80:
76:
72:
68:
64:
59:
57:
53:
45:
43:
41:
36:
34:
30:
25:
23:
19:
6506:Group theory
6474:
6420:
6414:
6408:
6378:(1): 57–68.
6375:
6371:
6319:
6313:
6261:
6255:
6249:
6212:
6206:
6196:
6179:
6173:
6167:
6134:
6128:
6100:
6057:
6053:
6043:
6000:
5994:
5988:
5965:
5958:
5915:
5909:
5903:
5869:(4): 58–80.
5866:
5860:
5805:(1): 41–47.
5802:
5796:
5756:
5750:
5737:
5694:
5688:
5682:
5655:math/9609209
5645:
5639:
5599:
5593:
5580:
5553:
5547:
5537:
5510:
5504:
5479:
5473:
5456:
5450:
5444:
5435:
5425:
5417:
5412:
5393:
5357:
5351:
5341:
5309:(1): 35–91.
5306:
5300:
5290:
5263:
5257:
5247:
5214:
5208:
5195:
5162:math/0607509
5152:
5146:
5125:
5116:
5089:
5083:
4997:
4993:
4989:
4985:
4981:
4977:
4973:
4969:
4965:
4961:
4957:
4953:
4949:
4941:
4801:
4797:
4693:
4516:
4409:
4405:
4403:
4398:
4394:
4390:
4256:
3917:
3912:
3908:
3904:
3897:
3893:
3889:
3885:
3856:
3797:
3793:
3785:
3781:
3737:
3724:
3674:
3669:
3467:
3462:
3458:
3454:
3363:
3314:
3310:
3304:
3299:
3295:
3250:
3246:
3201:
3197:
3193:
3191:
3140:
3136:
3091:
3087:
3079:
3077:
3072:
3064:
3019:
3015:
3007:
3003:
3001:
2999:does exist.
2874:
2872:
2867:
2863:
2859:
2855:
2851:
2847:
2843:
2839:
2835:
2831:
2827:
2823:
2819:
2817:
2812:
2808:
2804:
2800:
2796:
2792:
2784:
2780:
2776:
2774:
2765:
2761:
2757:
2749:
2745:
2743:
2696:
2692:
2688:
2684:
2535:
2534:is called a
2531:
2527:
2523:
2519:
2515:
2511:
2507:
2503:
2499:
2495:
2487:
2485:
2429:in terms of
2426:
2422:
2418:
2211:
2176:
2168:
2164:
2089:
2010:, and where
2003:
1940:
1916:
1711:
1655:
1532:
1524:
1422:
1418:
1417:(2014). Let
1412:
1290:
1288:
1233:
1171:
997:
853:
852:, the group
760:
742:
740:
735:
731:
727:
694:
690:
686:
630:
617:
521:
384:
142:
74:
70:
65:be a closed
62:
60:
49:
37:
29:James Cannon
26:
17:
15:
5155:(1): 1–80.
4984:, or where
4215:element of
3086:) subgroup
2597:, given by
1084:, with the
203:lifts to a
6500:Categories
6430:1506.08036
6010:1506.06974
5459:: 89–245.
5062:References
4523:, the set
4515:such that
4063:the group
3733:free group
3679:, that if
3010:where the
998:The group
622:surjective
6329:0708.3578
6322:: 59–78.
6296:119149073
6271:1406.4594
6222:0908.2272
6159:119295501
6067:1207.3494
6035:255427886
5950:253648281
5925:1001.4482
5876:1401.2638
5837:119169019
5812:1206.5868
5729:119299936
5704:1209.0815
5360:: 35–76.
5187:119160004
5026:
4812:∂
4758:∂
4708:π
4704:∂
4677:Λ
4645:∂
4642:⊆
4639:Λ
4591:≤
4588:Γ
4545:−
4534:∂
4495:∂
4491:Λ
4487:∈
4461:∂
4438:∂
4435:→
4429:∂
4420:∂
4375:Γ
4367:∂
4364:→
4351:∂
4342:∂
4322:⟩
4319:ϕ
4316:⟨
4310:Γ
4274:
4268:∈
4265:ϕ
4226:
4199:Γ
4179:Γ
4153:≤
4131:Γ
4123:∂
4120:→
4107:∂
4098:∂
4076:Γ
4048:→
4045:Γ
4042:→
4037:Γ
4029:→
4016:→
3990:≥
3950:
3926:Γ
3870:Γ
3839:→
3836:Γ
3833:→
3828:Γ
3820:→
3814:→
3753:π
3731:and of a
3708:→
3702:→
3696:→
3690:→
3642:−
3605:∈
3570:→
3501:→
3483:π
3476:ρ
3438:∂
3435:→
3429:∂
3420:∂
3397:∂
3374:∂
3343:∂
3340:→
3334:∂
3325:∂
3279:∂
3276:→
3270:∂
3261:∂
3230:∂
3227:→
3221:∂
3212:∂
3169:∂
3166:→
3160:∂
3151:∂
3120:∂
3117:→
3111:∂
3102:∂
3048:∂
3045:→
3039:∂
3030:∂
2984:∂
2981:→
2968:∂
2959:∂
2936:→
2715:∂
2711:Λ
2701:limit set
2668:∂
2657:∂
2644:^
2611:^
2582:∂
2579:∪
2573:→
2567:∂
2564:∪
2552:^
2393:∈
2360:∈
2295:∂
2253:∪
2238:→
2232:∂
2229:∪
2173:limit set
2144:∂
2112:→
2106:∂
2076:ρ
2018:ρ
1962:→
1953:ρ
1890:⊆
1869:π
1862:ρ
1859:Λ
1752:ρ
1729:∈
1723:≠
1684:→
1642:ρ
1594:→
1585:ρ
1550:π
1523:(even if
1481:π
1477:Λ
1454:~
1386:→
1380:Λ
1326:→
1308:π
1301:ρ
1262:→
1104:∂
1101:⊆
1095:Λ
1086:limit set
1069:~
1013:π
965:
921:→
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