5654:
gave a proof based on the ideas of edge-cuts. Later
Dunwoody also gave a proof of Stallings' theorem for finitely presented groups using the method of "tracks" on finite 2-complexes. Niblo obtained a proof of Stallings' theorem as a consequence of Sageev's CAT(0)-cubing relative version, where the
5588:
stabilizes an essential "hyperplane" (a simplicial tree is an example of a CAT(0)-cubing where the hyperplanes are the midpoints of edges). In certain situations such a semi-splitting can be promoted to an actual algebraic splitting, typically over a subgroup commensurable with
5092:
that are not accessible. Linnell showed that if one bounds the size of finite subgroups over which the splittings are taken then every finitely generated group is accessible in this sense as well. These results in turn gave rise to other versions of accessibility such as
5735:
outlined a proof (see pp. 228–230 in ) where the minimal surfaces argument is replaced by an easier harmonic analysis argument and this approach was pushed further by
Kapovich to cover the original case of finitely generated
4958:
Among the immediate applications of
Stallings' theorem was a proof by Stallings of a long-standing conjecture that every finitely generated group of cohomological dimension one is free and that every torsion-free
5477:. Early work on semi-splittings, inspired by Stallings' theorem, was done in the 1970s and 1980s by Scott, Swarup, and others. The work of Sageev and Gerasimov in the 1990s showed that for a subgroup
4617:
4153:
4977:
since the number of ends of a finitely generated group is easily seen to be a quasi-isometry invariant. For this reason
Stallings' theorem is considered to be one of the first results in
2075:
898:
1853:
2746:
718:
657:
547:
480:
320:
3539:
2398:
1506:
1929:
1774:
1085:
753:
512:
1809:
1973:
1550:
6226:(in Russian) Algebra, geometry, analysis and mathematical physics (Novosibirsk, 1996), pp. 91–109, 190, Izdat. Ross. Akad. Nauk Sib. Otd. Inst. Mat., Novosibirsk, 1997
5501:
5292:
4851:
2954:
2590:
2468:
2333:
1737:
1380:
1321:
1217:
1151:
1004:
683:
582:
445:
265:
5883:. Geometric group theory, Vol. 1 (Sussex, 1991), pp. 75–78, London Mathematical Society Lecture Note Series, vol. 181, Cambridge University Press, Cambridge, 1993;
5542:
5435:
5097:-Feighn accessibility of finitely presented groups (where the so-called "small" splittings are considered), acylindrical accessibility, strong accessibility, and others.
837:
782:
4761:
4431:
4368:
4294:
4068:
3917:
3848:
3813:
3478:
3366:
3222:
3183:
3070:
2874:
2788:
1243:
1030:
386:
2835:
2620:
4507:
4481:
3993:
3967:
2812:
2660:
1191:
1125:
978:
938:
857:
805:
360:
285:
228:
200:
176:
5350:
2984:
2136:
1689:
4907:
4671:
4644:
4211:
4184:
3754:
3702:
3655:
3263:
3117:
2925:
2707:
2495:
5807:
A James K. Whittemore
Lecture in Mathematics given at Yale University, 1969. Yale Mathematical Monographs, 4. Yale University Press, New Haven, Conn.-London, 1971.
1887:
1622:
2521:
2270:
3031:
5725:
5693:
5673:
5627:
5607:
5586:
5562:
5475:
5455:
5394:
5374:
5312:
5259:
5235:
5215:
5192:
5172:
5145:
5118:
5078:
5054:
5030:
5006:
4947:
4927:
4871:
4812:
4781:
4726:
4695:
4531:
4455:
4392:
4326:
4255:
4235:
4033:
4013:
3941:
3874:
3774:
3722:
3675:
3623:
3603:
3583:
3563:
3498:
3446:
3426:
3406:
3386:
3331:
3303:
3283:
3137:
3090:
3008:
2898:
2680:
2640:
2561:
2541:
2442:
2422:
2357:
2303:
2242:
2222:
2183:
2156:
2101:
2016:
1709:
1654:
1593:
1573:
1447:
1424:
1404:
1345:
1291:
1263:
1171:
1105:
1050:
958:
918:
622:
602:
406:
340:
123:
103:
66:
45:
5655:
CAT(0)-cubing is eventually promoted to being a tree. Niblo's paper also defines an abstract group-theoretic obstruction (which is a union of double cosets of
5629:
is finite (Stallings' theorem). Another situation where an actual splitting can be obtained (modulo a few exceptions) is for semi-splittings over virtually
5844:
5982:
6208:
6241:
5834:
Lemma 4.1 in C. T. C. Wall, Poincaré Complexes: I. Annals of
Mathematics, Second Series, Vol. 86, No. 2 (Sep., 1967), pp. 213-245
4540:
4076:
409:
6366:
5888:
6361:
5645:
with respect to virtually polycyclic subgroups is dealt with by the algebraic torus theorem of
Dunwoody-Swenson.
6058:
5825:
H. Hopf. Enden offener Räume und unendliche diskontinuierliche
Gruppen. Comment. Math. Helv. 16, (1944). 81-100
5746:
5732:
5695:) for obtaining an actual splitting from a semi-splitting. It is also possible to prove Stallings' theorem for
4969:
Stallings' theorem also implies that the property of having a nontrivial splitting over a finite subgroup is a
4434:
3920:
2021:
70:
6072:
M. Gromov, Hyperbolic Groups, in "Essays in Group Theory" (G. M. Gersten, ed.), MSRI Publ. 8, 1987, pp. 75-263
862:
1814:
5650:
A number of new proofs of
Stallings' theorem have been obtained by others after Stallings' original proof.
2712:
688:
627:
517:
450:
290:
179:
3503:
2362:
1452:
1892:
1742:
1055:
723:
4784:
485:
126:
1785:
2336:
1934:
1553:
1511:
1324:
6040:
5154:
Using
Dunwoody's accessibility result, Stallings' theorem about ends of groups and the fact that if
6281:
5964:
5946:
5867:
5696:
5642:
5265:
5089:
5085:
4974:
4815:
4706:
4329:
4258:
3854:
3306:
2306:
2081:
1996:
1634:
1294:
25:
5977:
5816:
H. Freudenthal. Über die Enden diskreter Räume und Gruppen. Comment. Math. Helv. 17, (1945). 1-38.
5480:
5271:
4821:
2933:
2569:
2447:
2312:
1720:
1350:
1300:
1196:
1130:
983:
662:
552:
415:
244:
5989:
UniversitĂ© de Grenoble. Annales de l'Institut Fourier, vol. 49 (1999), no. 4, pp. 1215–1224
5506:
5399:
2273:
5353:
813:
758:
5766:
4978:
4731:
4397:
4338:
4264:
4038:
3883:
3818:
3779:
3451:
3336:
3188:
3153:
3036:
2840:
2754:
1222:
1009:
365:
2817:
2602:
5791:
5756:
5634:
5238:
5195:
5033:
4702:
4486:
4460:
3972:
3946:
2797:
2645:
2162:
1715:
1176:
1110:
963:
923:
842:
790:
345:
270:
213:
185:
161:
82:
6213:
Proceedings of the London Mathematical Society (3), vol. 71 (1995), no. 3, pp. 585–617
5707:, where one first realizes a finitely presented group as the fundamental group of a compact
5320:
2963:
2106:
1659:
6194:
6144:
6103:
6026:
5927:
4880:
4649:
4622:
4189:
4162:
3727:
3680:
3628:
3236:
3095:
2987:
2903:
2685:
2473:
1863:
1598:
8:
6314:
6309:
5700:
4788:
2500:
2247:
2190:
130:
5637:
over two-ended (virtually infinite cyclic) subgroups was treated by Scott-Swarup and by
3013:
6236:
5710:
5678:
5658:
5612:
5592:
5571:
5547:
5460:
5440:
5379:
5359:
5297:
5244:
5220:
5200:
5177:
5157:
5130:
5103:
5063:
5039:
5015:
4991:
4932:
4912:
4856:
4797:
4766:
4711:
4680:
4516:
4440:
4377:
4311:
4240:
4220:
4018:
3998:
3926:
3859:
3759:
3707:
3660:
3608:
3588:
3568:
3548:
3483:
3431:
3411:
3391:
3371:
3316:
3288:
3268:
3122:
3075:
2993:
2883:
2665:
2625:
2546:
2526:
2427:
2407:
2342:
2288:
2227:
2207:
2168:
2141:
2086:
2001:
1779:
1694:
1639:
1578:
1558:
1432:
1409:
1389:
1330:
1276:
1248:
1156:
1090:
1035:
943:
903:
607:
587:
391:
325:
235:
153:
141:
108:
88:
51:
30:
6017:
6000:
6186:
6135:
6118:
6095:
5918:
5901:
5884:
203:
137:
6263:
6182:
6130:
6091:
6012:
5913:
5630:
1984:
6245:
6190:
6140:
6099:
6022:
5986:
5923:
5761:
5704:
5651:
5148:
5094:
5081:
5057:
5849:
Bulletin of the American Mathematical Society, vol. 74 (1968), pp. 361–364
5727:-manifold (see, for example, a sketch of this argument in the survey article of
5638:
4970:
231:
6355:
6173:
Kropholler, P. H.; Roller, M. A. (1989). "Relative ends and duality groups".
5751:
4535:
4156:
2197:
74:
6059:
http://www.ams.org/journals/proc/2008-136-12/S0002-9939-08-08973-9/home.html
6045:
Journal of Pure and Applied Algebra, vol. 89 (1993), no. 1-2, pp. 3–47
5100:
Stallings' theorem is a key tool in proving that a finitely generated group
2224:
is virtually infinite cyclic if and only if it has a finite normal subgroup
5565:
4874:
3542:
3310:
2401:
1383:
207:
17:
6276:
5959:
6311:
A geometric proof of Stallings' theorem on groups with more than one end.
6248:
Pacific Journal of Mathematics, vol. 196 (2000), no. 2, pp. 461–506
5032:
over finite subgroups always terminates in a finite number of steps. In
6294:
6258:
6158:
5941:
5862:
5728:
5124:
4963:
1988:
1858:
234:
of this topological space. A more explicit definition of the number of
6342:
Energy of harmonic functions and Gromov's proof of Stallings' theorem
5121:
4960:
2159:
6330:
Revista Matemática Complutense vol. 16(2003), no. 1, pp. 5–101
6163:
Mathematische Zeitschrift, vol. 176 (1981), no. 2, pp. 223–246
6057:
Gentimis Thanos, Asymptotic dimension of finitely presented groups,
5786:
5268:
with respect to subgroups have also been considered. For a subgroup
5979:
Sur l'accessibilité acylindrique des groupes de présentation finie.
4674:
4214:
2186:
78:
6340:
5060:
with finite edge groups is bounded by some constant depending on
5151:
where all vertex and edge groups are finite (see, for example,).
5174:
is a finitely presented group with asymptotic dimension 1 then
6001:"Accessibilité hiérarchique des groupes de présentation finie"
5194:
is virtually free one can show that for a finitely presented
6210:
Ends of group pairs and non-positively curved cube complexes.
5943:
Bounding the complexity of simplicial group actions on trees.
5264:
Relative versions of Stallings' theorem and relative ends of
4783:
admits a nontrivial (that is, without a global fixed vertex)
3119:
are almost invariant (equivalently, if and only if the set
125:
admits a nontrivial (that is, without a global fixed point)
4791:
with finite edge-stabilizers and without edge-inversions.
133:
with finite edge-stabilizers and without edge-inversions.
6260:
Cut points and canonical splittings of hyperbolic groups.
5036:
terms that the number of edges in a reduced splitting of
4303:
4300:
Stallings' theorem shows that the converse is also true.
3853:
A more precise version of this argument shows that for a
3228:
1508:. A basic fact in the theory of ends of groups says that
182:
where the degree of every vertex is finite. One can view
5352:
as the number of ends of the relative Cayley graph (the
5147:
can be represented as the fundamental group of a finite
4612:{\displaystyle G=\langle H,t|t^{-1}C_{1}t=C_{2}\rangle }
4148:{\displaystyle G=\langle H,t|t^{-1}C_{1}t=C_{2}\rangle }
206:
by giving it the natural structure of a one-dimensional
4984:
Stallings' theorem was a starting point for Dunwoody's
3605:
consist of the trivial element and all the elements of
3185:
if and only if there exists at least one essential cut
839:
is the number of "connected components at infinity" of
6328:
The geometry of abstract groups and their splittings.
6082:
Scott, Peter (1977–1978). "Ends of pairs of groups".
5713:
5681:
5661:
5615:
5595:
5574:
5550:
5509:
5483:
5463:
5443:
5402:
5382:
5362:
5323:
5300:
5274:
5247:
5223:
5203:
5180:
5160:
5133:
5106:
5066:
5042:
5018:
4994:
4935:
4915:
4883:
4859:
4824:
4800:
4769:
4734:
4714:
4683:
4652:
4625:
4543:
4519:
4489:
4463:
4443:
4400:
4380:
4341:
4314:
4267:
4243:
4223:
4192:
4165:
4079:
4041:
4021:
4001:
3975:
3949:
3929:
3886:
3862:
3821:
3782:
3762:
3730:
3710:
3683:
3663:
3631:
3611:
3591:
3571:
3551:
3506:
3486:
3454:
3434:
3414:
3394:
3374:
3339:
3319:
3291:
3271:
3239:
3191:
3156:
3125:
3098:
3078:
3039:
3016:
2996:
2966:
2936:
2906:
2886:
2843:
2820:
2800:
2757:
2715:
2688:
2668:
2648:
2628:
2605:
2572:
2549:
2529:
2503:
2476:
2450:
2430:
2410:
2365:
2345:
2315:
2291:
2250:
2230:
2210:
2171:
2144:
2109:
2089:
2024:
2004:
1937:
1895:
1866:
1817:
1788:
1745:
1723:
1697:
1662:
1642:
1601:
1581:
1561:
1514:
1455:
1435:
1412:
1392:
1353:
1333:
1303:
1279:
1251:
1225:
1199:
1179:
1159:
1133:
1113:
1093:
1058:
1038:
1012:
986:
966:
946:
926:
906:
865:
845:
816:
793:
761:
726:
691:
665:
630:
610:
590:
555:
520:
488:
453:
418:
394:
368:
348:
328:
293:
273:
247:
216:
188:
164:
111:
91:
54:
33:
6299:
Combinatorica, vol. 2 (1982), no. 1, pp. 15–23
5012:if the process of iterated nontrivial splitting of
4952:
1991:established in the 1940s the following two facts:
5719:
5687:
5667:
5621:
5601:
5580:
5556:
5536:
5495:
5469:
5449:
5429:
5388:
5368:
5344:
5306:
5286:
5253:
5229:
5209:
5186:
5166:
5139:
5112:
5072:
5048:
5024:
5000:
4941:
4921:
4901:
4865:
4845:
4806:
4775:
4755:
4720:
4689:
4665:
4638:
4611:
4525:
4501:
4475:
4449:
4425:
4386:
4362:
4320:
4288:
4249:
4229:
4205:
4178:
4147:
4062:
4027:
4007:
3987:
3961:
3935:
3911:
3868:
3842:
3807:
3768:
3748:
3716:
3696:
3669:
3649:
3617:
3597:
3577:
3557:
3533:
3492:
3472:
3440:
3420:
3400:
3380:
3360:
3325:
3297:
3277:
3257:
3216:
3177:
3131:
3111:
3084:
3064:
3025:
3002:
2978:
2948:
2919:
2892:
2868:
2829:
2806:
2782:
2740:
2701:
2674:
2654:
2634:
2614:
2584:
2555:
2535:
2515:
2489:
2462:
2436:
2416:
2392:
2351:
2327:
2297:
2264:
2236:
2216:
2177:
2150:
2130:
2095:
2069:
2010:
1967:
1923:
1881:
1847:
1803:
1768:
1731:
1703:
1683:
1648:
1616:
1587:
1567:
1544:
1500:
1441:
1418:
1398:
1374:
1339:
1315:
1285:
1257:
1237:
1211:
1185:
1165:
1145:
1119:
1099:
1079:
1044:
1024:
998:
972:
952:
932:
912:
892:
851:
831:
799:
776:
747:
712:
677:
651:
616:
596:
576:
541:
506:
474:
439:
400:
380:
354:
334:
314:
279:
259:
222:
194:
170:
117:
97:
60:
39:
6224:Semi-splittings of groups and actions on cubings.
5998:
5788:On torsion-free groups with infinitely many ends.
2280:
2200:proved in 1967 the following complementary fact:
144:case (1968) and then in the general case (1971).
85:the theorem says that a finitely generated group
6353:
6172:
5864:The accessibility of finitely presented groups.
5633:subgroups. Here the case of semi-splittings of
47:has more than one end if and only if the group
4705:this result can be restated as follows: For a
5805:Group theory and three-dimensional manifolds.
5564:admitting an essential isometric action on a
6119:"Relative version of a theorem of Stallings"
5999:Delzant, Thomas; Potyagailo, Leonid (2001).
4606:
4550:
4142:
4086:
2064:
2040:
1979:
6284:, vol. 140 (2000), no. 3, pp. 605–637
6266:, vol. 180 (1998), no. 2, pp. 145–186
6116:
5967:, vol. 129 (1997), no. 3, pp. 527–565
5949:, vol. 103 (1991), no. 3, pp. 449–469
4370:if and only if one of the following holds:
3147:A simple but important observation states:
1627:
6081:
1552:does not depend on the choice of a finite
6134:
6016:
5917:
3333:has at least one essential cut and hence
2070:{\displaystyle e(G)\in \{0,1,2,\infty \}}
1826:
1791:
1753:
1725:
69:admits a nontrivial decomposition as an
6160:Decomposition theorems for group pairs.
5899:
5846:Groups of dimension 1 are locally free.
2642:consists of all (topological) edges of
893:{\displaystyle e(\Gamma )=m<\infty }
6354:
6053:
6051:
5961:Acylindrical accessibility for groups.
5794:(2), vol. 88 (1968), pp. 312–334
5088:is accessible but that there do exist
4304:Formal statement of Stallings' theorem
3229:Cuts and splittings over finite groups
1848:{\displaystyle e(\mathbb {Z} ^{2})=1.}
22:Stallings theorem about ends of groups
6068:
6066:
6042:Covering theory for graphs of groups.
2741:{\displaystyle \delta A=\delta A^{*}}
713:{\displaystyle e(\Gamma )\leqslant n}
659:. If there does not exist an integer
652:{\displaystyle e(\Gamma )\leqslant n}
542:{\displaystyle e(\Gamma )\leqslant n}
475:{\displaystyle e(\Gamma )\leqslant m}
315:{\displaystyle e(\Gamma )\leqslant n}
267:be a non-negative integer. The graph
238:is presented below for completeness.
105:has more than one end if and only if
5870:, vol. 81 (1985), no. 3, pp. 449-457
5568:where a subgroup commensurable with
3756:starts with a nontrivial element of
3657:starts with a nontrivial element of
3534:{\displaystyle \Gamma =\Gamma (G,S)}
2393:{\displaystyle \Gamma =\Gamma (G,S)}
1501:{\displaystyle e(G)=e(\Gamma (G,S))}
604:is the smallest nonnegative integer
6275:M. J. Dunwoody, and E. L. Swenson.
6175:Journal of Pure and Applied Algebra
6123:Journal of Pure and Applied Algebra
6084:Journal of Pure and Applied Algebra
6048:
5906:Journal of Pure and Applied Algebra
5857:
5855:
3033:is finite. It is easy to see that
1924:{\displaystyle 1<|X|<\infty }
1052:infinite connected components. If
13:
6317:, vol. 105 (2004), pp. 61–76
6063:
4840:
4818:, Stallings' theorem implies that
3724:whose normal form expressions for
3625:whose normal form expressions for
3513:
3507:
2801:
2649:
2372:
2366:
2061:
1962:
1918:
1769:{\displaystyle e(\mathbb {Z} )=2.}
1521:
1477:
1354:
1226:
1180:
1114:
1080:{\displaystyle e(\Gamma )=\infty }
1074:
1065:
1013:
967:
927:
887:
872:
846:
823:
794:
768:
748:{\displaystyle e(\Gamma )=\infty }
742:
733:
698:
637:
562:
527:
460:
425:
369:
349:
300:
274:
217:
189:
165:
14:
6378:
6345:, preprint, 2007, arXiv:0707.4231
5641:. The case of semi-splittings of
3815:is an essential cut in Γ so that
3072:is a cut if and only if the sets
1268:
507:{\displaystyle 0\leqslant n<m}
147:
5852:
4953:Applications and generalizations
3142:
1804:{\displaystyle \mathbb {Z} ^{2}}
6333:
6320:
6302:
6287:
6269:
6251:
6235:G. P. Scott, and G. A. Swarup.
6229:
6216:
6201:
6166:
6151:
6117:Swarup, G. Ananda (1977–1978).
6110:
6075:
6033:
5992:
5970:
5952:
5934:
3480:is a finite generating set for
1968:{\displaystyle e(F(X))=\infty }
1545:{\displaystyle e(\Gamma (G,S))}
1265:infinite connected components.
322:if for every finite collection
16:In the mathematical subject of
5893:
5873:
5837:
5828:
5819:
5810:
5797:
5779:
5747:Free product with amalgamation
5609:, such as for the case where
5525:
5513:
5437:is called a semi-splitting of
5418:
5406:
5339:
5327:
5294:of a finitely generated group
5056:as the fundamental group of a
4834:
4828:
4744:
4738:
4563:
4435:free product with amalgamation
4351:
4345:
4277:
4271:
4099:
4051:
4045:
3921:free product with amalgamation
3831:
3825:
3802:
3783:
3528:
3516:
3408:be finite generating sets for
3349:
3343:
3211:
3192:
3166:
3160:
3059:
3040:
2863:
2844:
2777:
2758:
2387:
2375:
2281:Cuts and almost invariant sets
2119:
2113:
2034:
2028:
1956:
1953:
1947:
1941:
1911:
1903:
1876:
1870:
1836:
1821:
1757:
1749:
1672:
1666:
1611:
1605:
1539:
1536:
1524:
1518:
1495:
1492:
1480:
1474:
1465:
1459:
1369:
1357:
1068:
1062:
875:
869:
826:
820:
771:
765:
736:
730:
701:
695:
640:
634:
565:
559:
530:
524:
463:
457:
428:
422:
303:
297:
1:
6238:An algebraic annulus theorem.
6018:10.1016/S0040-9383(99)00078-6
4988:. A finitely generated group
3776:. It is not hard to see that
2272:is either infinite cyclic or
6278:The algebraic torus theorem.
6187:10.1016/0022-4049(89)90014-5
6136:10.1016/0022-4049(77)90042-1
6096:10.1016/0022-4049(77)90051-2
5919:10.1016/0022-4049(83)90037-3
5902:"On accessibility of groups"
5496:{\displaystyle H\leqslant G}
5287:{\displaystyle H\leqslant G}
4846:{\displaystyle e(G)=\infty }
4457:is a finite group such that
3943:is a finite group such that
3704:consists of all elements of
2949:{\displaystyle A\subseteq G}
2585:{\displaystyle A\subseteq G}
2463:{\displaystyle A\subseteq G}
2328:{\displaystyle S\subseteq G}
2185:contains an infinite cyclic
1732:{\displaystyle \mathbb {Z} }
1375:{\displaystyle \Gamma (G,S)}
1316:{\displaystyle S\subseteq G}
1212:{\displaystyle F\subseteq K}
1146:{\displaystyle n\geqslant 0}
999:{\displaystyle F\subseteq K}
678:{\displaystyle n\geqslant 0}
577:{\displaystyle e(\Gamma )=m}
440:{\displaystyle e(\Gamma )=m}
260:{\displaystyle n\geqslant 0}
81:. In the modern language of
7:
5940:M. Bestvina and M. Feighn.
5740:
5537:{\displaystyle e(G,H)>1}
5430:{\displaystyle e(G,H)>1}
5316:the number of relative ends
5217:the hyperbolic boundary of
4035:are finitely generated and
2709:. Note that by definition
10:
6383:
1153:there exists a finite set
1087:, then for any finite set
940:there exists a finite set
900:, then for any finite set
832:{\displaystyle e(\Gamma )}
777:{\displaystyle e(\Gamma )}
151:
136:The theorem was proved by
5697:finitely presented groups
5643:finitely generated groups
5544:corresponds to the group
5266:finitely generated groups
5090:finitely generated groups
4756:{\displaystyle e(G)>1}
4426:{\displaystyle G=H*_{C}K}
4363:{\displaystyle e(G)>1}
4289:{\displaystyle e(G)>1}
4063:{\displaystyle e(G)>1}
3912:{\displaystyle G=H*_{C}K}
3843:{\displaystyle e(G)>1}
3808:{\displaystyle (A,A^{*})}
3473:{\displaystyle S=X\cup Y}
3361:{\displaystyle e(G)>1}
3307:finitely generated groups
3217:{\displaystyle (A,A^{*})}
3178:{\displaystyle e(G)>1}
3065:{\displaystyle (A,A^{*})}
2869:{\displaystyle (A,A^{*})}
2783:{\displaystyle (A,A^{*})}
2662:connecting a vertex from
1980:Freudenthal-Hopf theorems
1238:{\displaystyle \Gamma -K}
1025:{\displaystyle \Gamma -F}
381:{\displaystyle \Gamma -F}
6367:Theorems in group theory
6282:Inventiones Mathematicae
5965:Inventiones Mathematicae
5947:Inventiones Mathematicae
5868:Inventiones Mathematicae
5772:
5086:finitely presented group
4975:finitely generated group
4816:finitely generated group
4707:finitely generated group
4330:finitely generated group
4259:finitely generated group
3855:finitely generated group
2830:{\displaystyle \delta A}
2615:{\displaystyle \delta A}
2307:finitely generated group
2082:finitely generated group
1997:finitely generated group
1635:finitely generated group
1628:Basic facts and examples
1295:finitely generated group
71:amalgamated free product
26:finitely generated group
5900:Linnell, P. A. (1983).
4502:{\displaystyle C\neq K}
4476:{\displaystyle C\neq H}
3988:{\displaystyle C\neq K}
3962:{\displaystyle C\neq H}
2807:{\displaystyle \Gamma }
2655:{\displaystyle \Gamma }
1186:{\displaystyle \Gamma }
1120:{\displaystyle \Gamma }
973:{\displaystyle \Gamma }
933:{\displaystyle \Gamma }
852:{\displaystyle \Gamma }
800:{\displaystyle \Gamma }
355:{\displaystyle \Gamma }
280:{\displaystyle \Gamma }
223:{\displaystyle \Gamma }
195:{\displaystyle \Gamma }
171:{\displaystyle \Gamma }
6362:Geometric group theory
5767:Geometric group theory
5721:
5689:
5669:
5635:word-hyperbolic groups
5623:
5603:
5582:
5558:
5538:
5497:
5471:
5451:
5431:
5390:
5370:
5346:
5345:{\displaystyle e(G,H)}
5308:
5288:
5255:
5231:
5211:
5188:
5168:
5141:
5114:
5074:
5050:
5026:
5002:
4979:geometric group theory
4943:
4923:
4903:
4867:
4847:
4808:
4777:
4757:
4722:
4691:
4673:are isomorphic finite
4667:
4640:
4613:
4527:
4503:
4477:
4451:
4427:
4388:
4364:
4322:
4290:
4251:
4231:
4213:are isomorphic finite
4207:
4180:
4149:
4064:
4029:
4009:
3989:
3963:
3937:
3913:
3870:
3844:
3809:
3770:
3750:
3718:
3698:
3671:
3651:
3619:
3599:
3579:
3559:
3535:
3494:
3474:
3442:
3422:
3402:
3382:
3362:
3327:
3299:
3279:
3259:
3218:
3179:
3139:is almost invariant).
3133:
3113:
3086:
3066:
3027:
3004:
2980:
2979:{\displaystyle g\in G}
2950:
2921:
2894:
2870:
2831:
2808:
2784:
2742:
2703:
2676:
2656:
2636:
2616:
2586:
2557:
2537:
2517:
2491:
2464:
2438:
2418:
2394:
2353:
2329:
2299:
2266:
2238:
2218:
2179:
2152:
2132:
2131:{\displaystyle e(G)=2}
2097:
2071:
2012:
1969:
1925:
1883:
1849:
1805:
1770:
1733:
1705:
1685:
1684:{\displaystyle e(G)=0}
1650:
1618:
1589:
1569:
1546:
1502:
1443:
1420:
1400:
1376:
1341:
1317:
1287:
1259:
1239:
1213:
1187:
1167:
1147:
1121:
1101:
1081:
1046:
1026:
1000:
974:
954:
934:
914:
894:
853:
833:
801:
778:
749:
714:
679:
653:
618:
598:
578:
543:
508:
476:
441:
402:
382:
356:
336:
316:
281:
261:
224:
196:
172:
119:
99:
62:
41:
5881:An inaccessible group
5792:Annals of Mathematics
5722:
5690:
5670:
5624:
5604:
5583:
5559:
5539:
5498:
5472:
5452:
5432:
5391:
5371:
5347:
5309:
5289:
5256:
5239:topological dimension
5232:
5212:
5196:word-hyperbolic group
5189:
5169:
5142:
5115:
5075:
5051:
5027:
5003:
4944:
4924:
4904:
4902:{\displaystyle G=A*B}
4868:
4848:
4809:
4778:
4758:
4723:
4692:
4668:
4666:{\displaystyle C_{2}}
4641:
4639:{\displaystyle C_{1}}
4614:
4528:
4504:
4478:
4452:
4428:
4389:
4365:
4323:
4291:
4252:
4232:
4208:
4206:{\displaystyle C_{2}}
4181:
4179:{\displaystyle C_{1}}
4150:
4065:
4030:
4010:
3990:
3964:
3938:
3914:
3871:
3845:
3810:
3771:
3751:
3749:{\displaystyle G=H*K}
3719:
3699:
3697:{\displaystyle A^{*}}
3672:
3652:
3650:{\displaystyle G=H*K}
3620:
3600:
3580:
3560:
3536:
3495:
3475:
3443:
3423:
3403:
3383:
3363:
3328:
3300:
3280:
3260:
3258:{\displaystyle G=H*K}
3219:
3180:
3134:
3114:
3112:{\displaystyle A^{*}}
3087:
3067:
3028:
3005:
2981:
2951:
2922:
2920:{\displaystyle A^{*}}
2895:
2871:
2832:
2809:
2785:
2743:
2704:
2702:{\displaystyle A^{*}}
2677:
2657:
2637:
2617:
2587:
2558:
2538:
2518:
2492:
2490:{\displaystyle A^{*}}
2465:
2439:
2419:
2395:
2354:
2330:
2300:
2267:
2239:
2219:
2180:
2153:
2133:
2098:
2072:
2013:
1970:
1926:
1884:
1850:
1806:
1771:
1734:
1716:infinite cyclic group
1706:
1686:
1651:
1619:
1590:
1570:
1547:
1503:
1444:
1421:
1401:
1377:
1342:
1318:
1288:
1260:
1240:
1214:
1188:
1168:
1148:
1122:
1102:
1082:
1047:
1027:
1001:
975:
955:
935:
915:
895:
854:
834:
802:
786:the number of ends of
779:
750:
715:
680:
654:
619:
599:
579:
544:
509:
477:
442:
403:
383:
357:
337:
317:
282:
262:
225:
197:
173:
120:
100:
63:
42:
5711:
5679:
5659:
5613:
5593:
5572:
5548:
5507:
5481:
5461:
5441:
5400:
5380:
5360:
5354:Schreier coset graph
5321:
5298:
5272:
5245:
5241:zero if and only if
5221:
5201:
5178:
5158:
5131:
5104:
5064:
5040:
5016:
4992:
4986:accessibility theory
4933:
4913:
4881:
4857:
4822:
4798:
4767:
4732:
4712:
4681:
4650:
4623:
4541:
4517:
4487:
4461:
4441:
4398:
4378:
4339:
4312:
4265:
4241:
4221:
4190:
4163:
4077:
4039:
4019:
3999:
3973:
3947:
3927:
3884:
3860:
3819:
3780:
3760:
3728:
3708:
3681:
3661:
3629:
3609:
3589:
3569:
3549:
3504:
3484:
3452:
3448:accordingly so that
3432:
3412:
3392:
3372:
3337:
3317:
3289:
3269:
3237:
3189:
3154:
3123:
3096:
3076:
3037:
3014:
2994:
2988:symmetric difference
2964:
2934:
2904:
2884:
2841:
2818:
2798:
2755:
2713:
2686:
2666:
2646:
2626:
2603:
2570:
2547:
2527:
2501:
2474:
2448:
2428:
2408:
2363:
2343:
2313:
2289:
2248:
2228:
2208:
2169:
2142:
2107:
2087:
2022:
2002:
1935:
1893:
1882:{\displaystyle F(X)}
1864:
1815:
1786:
1743:
1721:
1695:
1660:
1640:
1617:{\displaystyle e(G)}
1599:
1579:
1559:
1512:
1453:
1433:
1410:
1390:
1351:
1331:
1301:
1277:
1249:
1223:
1197:
1177:
1157:
1131:
1127:and for any integer
1111:
1091:
1056:
1036:
1010:
984:
964:
944:
924:
904:
863:
843:
814:
791:
759:
724:
689:
663:
628:
608:
588:
553:
518:
486:
451:
416:
410:connected components
392:
366:
346:
326:
291:
271:
245:
214:
210:. Then the ends of
186:
162:
109:
89:
52:
31:
6315:Geometriae Dedicata
5843:John R. Stallings.
5785:John R. Stallings.
5701:Riemannian geometry
4794:For the case where
4701:In the language of
4394:admits a splitting
2682:with a vertex from
2516:{\displaystyle G-A}
2265:{\displaystyle G/W}
287:is said to satisfy
6296:Cutting up graphs.
6244:2007-07-15 at the
5985:2011-06-05 at the
5717:
5685:
5665:
5619:
5599:
5578:
5554:
5534:
5493:
5467:
5447:
5427:
5396:. The case where
5386:
5366:
5342:
5304:
5284:
5261:is virtually free.
5251:
5227:
5207:
5184:
5164:
5137:
5110:
5084:proved that every
5070:
5046:
5022:
4998:
4939:
4919:
4899:
4863:
4843:
4814:is a torsion-free
4804:
4773:
4753:
4718:
4687:
4663:
4636:
4609:
4523:
4499:
4473:
4447:
4423:
4384:
4360:
4318:
4286:
4247:
4227:
4203:
4176:
4145:
4060:
4025:
4005:
3985:
3959:
3933:
3909:
3866:
3840:
3805:
3766:
3746:
3714:
3694:
3667:
3647:
3615:
3595:
3575:
3555:
3531:
3490:
3470:
3438:
3418:
3398:
3378:
3358:
3323:
3295:
3275:
3255:
3214:
3175:
3129:
3109:
3082:
3062:
3026:{\displaystyle Ag}
3023:
3000:
2976:
2946:
2917:
2890:
2866:
2827:
2804:
2780:
2738:
2699:
2672:
2652:
2632:
2612:
2582:
2553:
2533:
2513:
2487:
2460:
2434:
2414:
2390:
2349:
2325:
2295:
2262:
2234:
2214:
2198:Charles T. C. Wall
2175:
2148:
2128:
2093:
2067:
2008:
1987:and independently
1965:
1921:
1879:
1845:
1801:
1780:free abelian group
1766:
1729:
1701:
1681:
1646:
1614:
1585:
1565:
1542:
1498:
1439:
1416:
1396:
1372:
1337:
1313:
1283:
1255:
1235:
1209:
1183:
1163:
1143:
1117:
1097:
1077:
1042:
1022:
996:
970:
950:
930:
910:
890:
849:
829:
797:
774:
745:
710:
675:
649:
614:
594:
574:
539:
504:
472:
437:
398:
378:
352:
332:
312:
277:
257:
220:
192:
168:
154:End (graph theory)
115:
95:
58:
37:
6308:Graham A. Niblo.
6222:V. N. Gerasimov.
5757:Bass–Serre theory
5720:{\displaystyle 4}
5688:{\displaystyle G}
5668:{\displaystyle H}
5622:{\displaystyle H}
5602:{\displaystyle H}
5581:{\displaystyle H}
5557:{\displaystyle G}
5470:{\displaystyle H}
5450:{\displaystyle G}
5389:{\displaystyle H}
5369:{\displaystyle G}
5307:{\displaystyle G}
5254:{\displaystyle G}
5230:{\displaystyle G}
5210:{\displaystyle G}
5187:{\displaystyle G}
5167:{\displaystyle G}
5140:{\displaystyle G}
5113:{\displaystyle G}
5073:{\displaystyle G}
5049:{\displaystyle G}
5034:Bass–Serre theory
5025:{\displaystyle G}
5001:{\displaystyle G}
4942:{\displaystyle B}
4922:{\displaystyle A}
4866:{\displaystyle G}
4807:{\displaystyle G}
4776:{\displaystyle G}
4721:{\displaystyle G}
4703:Bass–Serre theory
4690:{\displaystyle H}
4526:{\displaystyle G}
4450:{\displaystyle C}
4387:{\displaystyle G}
4321:{\displaystyle G}
4250:{\displaystyle G}
4230:{\displaystyle H}
4028:{\displaystyle K}
4008:{\displaystyle H}
3936:{\displaystyle C}
3869:{\displaystyle G}
3769:{\displaystyle K}
3717:{\displaystyle G}
3670:{\displaystyle H}
3618:{\displaystyle G}
3598:{\displaystyle A}
3578:{\displaystyle S}
3558:{\displaystyle G}
3493:{\displaystyle G}
3441:{\displaystyle K}
3421:{\displaystyle H}
3401:{\displaystyle Y}
3381:{\displaystyle X}
3326:{\displaystyle G}
3298:{\displaystyle K}
3278:{\displaystyle H}
3132:{\displaystyle A}
3085:{\displaystyle A}
3003:{\displaystyle A}
2893:{\displaystyle A}
2880:if both the sets
2837:is finite. A cut
2675:{\displaystyle A}
2635:{\displaystyle A}
2556:{\displaystyle G}
2536:{\displaystyle A}
2437:{\displaystyle S}
2417:{\displaystyle G}
2352:{\displaystyle G}
2298:{\displaystyle G}
2274:infinite dihedral
2237:{\displaystyle W}
2217:{\displaystyle G}
2178:{\displaystyle G}
2151:{\displaystyle G}
2096:{\displaystyle G}
2011:{\displaystyle G}
1704:{\displaystyle G}
1649:{\displaystyle G}
1624:is well-defined.
1588:{\displaystyle G}
1568:{\displaystyle S}
1442:{\displaystyle G}
1428:number of ends of
1419:{\displaystyle S}
1399:{\displaystyle G}
1340:{\displaystyle G}
1286:{\displaystyle G}
1258:{\displaystyle n}
1166:{\displaystyle K}
1100:{\displaystyle F}
1045:{\displaystyle m}
953:{\displaystyle K}
913:{\displaystyle F}
617:{\displaystyle n}
597:{\displaystyle m}
482:and if for every
412:. By definition,
401:{\displaystyle n}
335:{\displaystyle F}
204:topological space
138:John R. Stallings
118:{\displaystyle G}
98:{\displaystyle G}
83:Bass–Serre theory
61:{\displaystyle G}
40:{\displaystyle G}
6374:
6346:
6337:
6331:
6324:
6318:
6306:
6300:
6293:M. J. Dunwoody.
6291:
6285:
6273:
6267:
6264:Acta Mathematica
6257:B. H. Bowditch.
6255:
6249:
6233:
6227:
6220:
6214:
6205:
6199:
6198:
6170:
6164:
6155:
6149:
6148:
6138:
6114:
6108:
6107:
6090:(1–3): 179–198.
6079:
6073:
6070:
6061:
6055:
6046:
6037:
6031:
6030:
6020:
5996:
5990:
5974:
5968:
5956:
5950:
5938:
5932:
5931:
5921:
5897:
5891:
5879:M. J. Dunwoody.
5877:
5871:
5861:M. J. Dunwoody.
5859:
5850:
5841:
5835:
5832:
5826:
5823:
5817:
5814:
5808:
5803:John Stallings.
5801:
5795:
5783:
5726:
5724:
5723:
5718:
5705:minimal surfaces
5694:
5692:
5691:
5686:
5674:
5672:
5671:
5666:
5628:
5626:
5625:
5620:
5608:
5606:
5605:
5600:
5587:
5585:
5584:
5579:
5563:
5561:
5560:
5555:
5543:
5541:
5540:
5535:
5502:
5500:
5499:
5494:
5476:
5474:
5473:
5468:
5456:
5454:
5453:
5448:
5436:
5434:
5433:
5428:
5395:
5393:
5392:
5387:
5376:with respect to
5375:
5373:
5372:
5367:
5351:
5349:
5348:
5343:
5313:
5311:
5310:
5305:
5293:
5291:
5290:
5285:
5260:
5258:
5257:
5252:
5236:
5234:
5233:
5228:
5216:
5214:
5213:
5208:
5193:
5191:
5190:
5185:
5173:
5171:
5170:
5165:
5146:
5144:
5143:
5138:
5119:
5117:
5116:
5111:
5079:
5077:
5076:
5071:
5055:
5053:
5052:
5047:
5031:
5029:
5028:
5023:
5007:
5005:
5004:
4999:
4948:
4946:
4945:
4940:
4928:
4926:
4925:
4920:
4908:
4906:
4905:
4900:
4873:admits a proper
4872:
4870:
4869:
4864:
4852:
4850:
4849:
4844:
4813:
4811:
4810:
4805:
4787:on a simplicial
4782:
4780:
4779:
4774:
4762:
4760:
4759:
4754:
4727:
4725:
4724:
4719:
4696:
4694:
4693:
4688:
4672:
4670:
4669:
4664:
4662:
4661:
4645:
4643:
4642:
4637:
4635:
4634:
4618:
4616:
4615:
4610:
4605:
4604:
4589:
4588:
4579:
4578:
4566:
4532:
4530:
4529:
4524:
4508:
4506:
4505:
4500:
4482:
4480:
4479:
4474:
4456:
4454:
4453:
4448:
4432:
4430:
4429:
4424:
4419:
4418:
4393:
4391:
4390:
4385:
4369:
4367:
4366:
4361:
4327:
4325:
4324:
4319:
4295:
4293:
4292:
4287:
4256:
4254:
4253:
4248:
4236:
4234:
4233:
4228:
4212:
4210:
4209:
4204:
4202:
4201:
4185:
4183:
4182:
4177:
4175:
4174:
4154:
4152:
4151:
4146:
4141:
4140:
4125:
4124:
4115:
4114:
4102:
4069:
4067:
4066:
4061:
4034:
4032:
4031:
4026:
4014:
4012:
4011:
4006:
3994:
3992:
3991:
3986:
3968:
3966:
3965:
3960:
3942:
3940:
3939:
3934:
3918:
3916:
3915:
3910:
3905:
3904:
3875:
3873:
3872:
3867:
3849:
3847:
3846:
3841:
3814:
3812:
3811:
3806:
3801:
3800:
3775:
3773:
3772:
3767:
3755:
3753:
3752:
3747:
3723:
3721:
3720:
3715:
3703:
3701:
3700:
3695:
3693:
3692:
3676:
3674:
3673:
3668:
3656:
3654:
3653:
3648:
3624:
3622:
3621:
3616:
3604:
3602:
3601:
3596:
3584:
3582:
3581:
3576:
3565:with respect to
3564:
3562:
3561:
3556:
3540:
3538:
3537:
3532:
3499:
3497:
3496:
3491:
3479:
3477:
3476:
3471:
3447:
3445:
3444:
3439:
3427:
3425:
3424:
3419:
3407:
3405:
3404:
3399:
3387:
3385:
3384:
3379:
3367:
3365:
3364:
3359:
3332:
3330:
3329:
3324:
3304:
3302:
3301:
3296:
3284:
3282:
3281:
3276:
3264:
3262:
3261:
3256:
3223:
3221:
3220:
3215:
3210:
3209:
3184:
3182:
3181:
3176:
3138:
3136:
3135:
3130:
3118:
3116:
3115:
3110:
3108:
3107:
3091:
3089:
3088:
3083:
3071:
3069:
3068:
3063:
3058:
3057:
3032:
3030:
3029:
3024:
3009:
3007:
3006:
3001:
2985:
2983:
2982:
2977:
2958:almost invariant
2955:
2953:
2952:
2947:
2926:
2924:
2923:
2918:
2916:
2915:
2899:
2897:
2896:
2891:
2875:
2873:
2872:
2867:
2862:
2861:
2836:
2834:
2833:
2828:
2813:
2811:
2810:
2805:
2789:
2787:
2786:
2781:
2776:
2775:
2751:An ordered pair
2747:
2745:
2744:
2739:
2737:
2736:
2708:
2706:
2705:
2700:
2698:
2697:
2681:
2679:
2678:
2673:
2661:
2659:
2658:
2653:
2641:
2639:
2638:
2633:
2621:
2619:
2618:
2613:
2591:
2589:
2588:
2583:
2562:
2560:
2559:
2554:
2542:
2540:
2539:
2534:
2522:
2520:
2519:
2514:
2496:
2494:
2493:
2488:
2486:
2485:
2469:
2467:
2466:
2461:
2443:
2441:
2440:
2435:
2424:with respect to
2423:
2421:
2420:
2415:
2399:
2397:
2396:
2391:
2358:
2356:
2355:
2350:
2334:
2332:
2331:
2326:
2304:
2302:
2301:
2296:
2271:
2269:
2268:
2263:
2258:
2243:
2241:
2240:
2235:
2223:
2221:
2220:
2215:
2184:
2182:
2181:
2176:
2157:
2155:
2154:
2149:
2137:
2135:
2134:
2129:
2102:
2100:
2099:
2094:
2076:
2074:
2073:
2068:
2017:
2015:
2014:
2009:
1985:Hans Freudenthal
1974:
1972:
1971:
1966:
1930:
1928:
1927:
1922:
1914:
1906:
1888:
1886:
1885:
1880:
1854:
1852:
1851:
1846:
1835:
1834:
1829:
1810:
1808:
1807:
1802:
1800:
1799:
1794:
1775:
1773:
1772:
1767:
1756:
1738:
1736:
1735:
1730:
1728:
1710:
1708:
1707:
1702:
1690:
1688:
1687:
1682:
1655:
1653:
1652:
1647:
1623:
1621:
1620:
1615:
1594:
1592:
1591:
1586:
1574:
1572:
1571:
1566:
1551:
1549:
1548:
1543:
1507:
1505:
1504:
1499:
1448:
1446:
1445:
1440:
1425:
1423:
1422:
1417:
1406:with respect to
1405:
1403:
1402:
1397:
1381:
1379:
1378:
1373:
1346:
1344:
1343:
1338:
1322:
1320:
1319:
1314:
1292:
1290:
1289:
1284:
1264:
1262:
1261:
1256:
1244:
1242:
1241:
1236:
1218:
1216:
1215:
1210:
1192:
1190:
1189:
1184:
1172:
1170:
1169:
1164:
1152:
1150:
1149:
1144:
1126:
1124:
1123:
1118:
1106:
1104:
1103:
1098:
1086:
1084:
1083:
1078:
1051:
1049:
1048:
1043:
1031:
1029:
1028:
1023:
1005:
1003:
1002:
997:
979:
977:
976:
971:
959:
957:
956:
951:
939:
937:
936:
931:
919:
917:
916:
911:
899:
897:
896:
891:
858:
856:
855:
850:
838:
836:
835:
830:
806:
804:
803:
798:
783:
781:
780:
775:
754:
752:
751:
746:
719:
717:
716:
711:
684:
682:
681:
676:
658:
656:
655:
650:
623:
621:
620:
615:
603:
601:
600:
595:
583:
581:
580:
575:
548:
546:
545:
540:
513:
511:
510:
505:
481:
479:
478:
473:
446:
444:
443:
438:
407:
405:
404:
399:
387:
385:
384:
379:
361:
359:
358:
353:
341:
339:
338:
333:
321:
319:
318:
313:
286:
284:
283:
278:
266:
264:
263:
258:
229:
227:
226:
221:
201:
199:
198:
193:
177:
175:
174:
169:
129:on a simplicial
124:
122:
121:
116:
104:
102:
101:
96:
67:
65:
64:
59:
46:
44:
43:
38:
6382:
6381:
6377:
6376:
6375:
6373:
6372:
6371:
6352:
6351:
6350:
6349:
6338:
6334:
6326:C. T. C. Wall.
6325:
6321:
6307:
6303:
6292:
6288:
6274:
6270:
6256:
6252:
6246:Wayback Machine
6234:
6230:
6221:
6217:
6207:Michah Sageev.
6206:
6202:
6171:
6167:
6156:
6152:
6115:
6111:
6080:
6076:
6071:
6064:
6056:
6049:
6038:
6034:
5997:
5993:
5987:Wayback Machine
5975:
5971:
5957:
5953:
5939:
5935:
5898:
5894:
5878:
5874:
5860:
5853:
5842:
5838:
5833:
5829:
5824:
5820:
5815:
5811:
5802:
5798:
5784:
5780:
5775:
5762:Graph of groups
5743:
5712:
5709:
5708:
5680:
5677:
5676:
5660:
5657:
5656:
5614:
5611:
5610:
5594:
5591:
5590:
5573:
5570:
5569:
5549:
5546:
5545:
5508:
5505:
5504:
5482:
5479:
5478:
5462:
5459:
5458:
5442:
5439:
5438:
5401:
5398:
5397:
5381:
5378:
5377:
5361:
5358:
5357:
5322:
5319:
5318:
5299:
5296:
5295:
5273:
5270:
5269:
5246:
5243:
5242:
5222:
5219:
5218:
5202:
5199:
5198:
5179:
5176:
5175:
5159:
5156:
5155:
5149:graph of groups
5132:
5129:
5128:
5127:if and only if
5105:
5102:
5101:
5065:
5062:
5061:
5058:graph of groups
5041:
5038:
5037:
5017:
5014:
5013:
4993:
4990:
4989:
4973:invariant of a
4955:
4934:
4931:
4930:
4914:
4911:
4910:
4882:
4879:
4878:
4858:
4855:
4854:
4853:if and only if
4823:
4820:
4819:
4799:
4796:
4795:
4768:
4765:
4764:
4763:if and only if
4733:
4730:
4729:
4713:
4710:
4709:
4682:
4679:
4678:
4657:
4653:
4651:
4648:
4647:
4630:
4626:
4624:
4621:
4620:
4600:
4596:
4584:
4580:
4571:
4567:
4562:
4542:
4539:
4538:
4518:
4515:
4514:
4488:
4485:
4484:
4462:
4459:
4458:
4442:
4439:
4438:
4414:
4410:
4399:
4396:
4395:
4379:
4376:
4375:
4340:
4337:
4336:
4313:
4310:
4309:
4306:
4266:
4263:
4262:
4242:
4239:
4238:
4222:
4219:
4218:
4197:
4193:
4191:
4188:
4187:
4170:
4166:
4164:
4161:
4160:
4136:
4132:
4120:
4116:
4107:
4103:
4098:
4078:
4075:
4074:
4040:
4037:
4036:
4020:
4017:
4016:
4000:
3997:
3996:
3974:
3971:
3970:
3948:
3945:
3944:
3928:
3925:
3924:
3900:
3896:
3885:
3882:
3881:
3861:
3858:
3857:
3820:
3817:
3816:
3796:
3792:
3781:
3778:
3777:
3761:
3758:
3757:
3729:
3726:
3725:
3709:
3706:
3705:
3688:
3684:
3682:
3679:
3678:
3662:
3659:
3658:
3630:
3627:
3626:
3610:
3607:
3606:
3590:
3587:
3586:
3570:
3567:
3566:
3550:
3547:
3546:
3505:
3502:
3501:
3485:
3482:
3481:
3453:
3450:
3449:
3433:
3430:
3429:
3413:
3410:
3409:
3393:
3390:
3389:
3373:
3370:
3369:
3338:
3335:
3334:
3318:
3315:
3314:
3305:are nontrivial
3290:
3287:
3286:
3270:
3267:
3266:
3238:
3235:
3234:
3231:
3205:
3201:
3190:
3187:
3186:
3155:
3152:
3151:
3145:
3124:
3121:
3120:
3103:
3099:
3097:
3094:
3093:
3077:
3074:
3073:
3053:
3049:
3038:
3035:
3034:
3015:
3012:
3011:
2995:
2992:
2991:
2965:
2962:
2961:
2935:
2932:
2931:
2911:
2907:
2905:
2902:
2901:
2885:
2882:
2881:
2857:
2853:
2842:
2839:
2838:
2819:
2816:
2815:
2799:
2796:
2795:
2771:
2767:
2756:
2753:
2752:
2732:
2728:
2714:
2711:
2710:
2693:
2689:
2687:
2684:
2683:
2667:
2664:
2663:
2647:
2644:
2643:
2627:
2624:
2623:
2604:
2601:
2600:
2571:
2568:
2567:
2548:
2545:
2544:
2528:
2525:
2524:
2502:
2499:
2498:
2497:the complement
2481:
2477:
2475:
2472:
2471:
2449:
2446:
2445:
2444:. For a subset
2429:
2426:
2425:
2409:
2406:
2405:
2364:
2361:
2360:
2344:
2341:
2340:
2314:
2311:
2310:
2290:
2287:
2286:
2283:
2254:
2249:
2246:
2245:
2229:
2226:
2225:
2209:
2206:
2205:
2170:
2167:
2166:
2163:infinite cyclic
2143:
2140:
2139:
2138:if and only if
2108:
2105:
2104:
2088:
2085:
2084:
2023:
2020:
2019:
2003:
2000:
1999:
1982:
1936:
1933:
1932:
1910:
1902:
1894:
1891:
1890:
1865:
1862:
1861:
1830:
1825:
1824:
1816:
1813:
1812:
1795:
1790:
1789:
1787:
1784:
1783:
1752:
1744:
1741:
1740:
1724:
1722:
1719:
1718:
1696:
1693:
1692:
1691:if and only if
1661:
1658:
1657:
1641:
1638:
1637:
1630:
1600:
1597:
1596:
1580:
1577:
1576:
1560:
1557:
1556:
1513:
1510:
1509:
1454:
1451:
1450:
1434:
1431:
1430:
1411:
1408:
1407:
1391:
1388:
1387:
1352:
1349:
1348:
1332:
1329:
1328:
1302:
1299:
1298:
1278:
1275:
1274:
1271:
1250:
1247:
1246:
1224:
1221:
1220:
1198:
1195:
1194:
1178:
1175:
1174:
1158:
1155:
1154:
1132:
1129:
1128:
1112:
1109:
1108:
1092:
1089:
1088:
1057:
1054:
1053:
1037:
1034:
1033:
1011:
1008:
1007:
985:
982:
981:
965:
962:
961:
945:
942:
941:
925:
922:
921:
905:
902:
901:
864:
861:
860:
844:
841:
840:
815:
812:
811:
792:
789:
788:
760:
757:
756:
725:
722:
721:
690:
687:
686:
664:
661:
660:
629:
626:
625:
609:
606:
605:
589:
586:
585:
554:
551:
550:
549:is false. Thus
519:
516:
515:
487:
484:
483:
452:
449:
448:
417:
414:
413:
393:
390:
389:
367:
364:
363:
347:
344:
343:
327:
324:
323:
292:
289:
288:
272:
269:
268:
246:
243:
242:
236:ends of a graph
215:
212:
211:
187:
184:
183:
178:be a connected
163:
160:
159:
156:
150:
140:, first in the
110:
107:
106:
90:
87:
86:
53:
50:
49:
32:
29:
28:
12:
11:
5:
6380:
6370:
6369:
6364:
6348:
6347:
6332:
6319:
6301:
6286:
6268:
6250:
6228:
6215:
6200:
6181:(2): 197–210.
6165:
6150:
6129:(1–3): 75–82.
6109:
6074:
6062:
6047:
6032:
6011:(3): 617–629.
5991:
5969:
5951:
5933:
5892:
5872:
5851:
5836:
5827:
5818:
5809:
5796:
5777:
5776:
5774:
5771:
5770:
5769:
5764:
5759:
5754:
5749:
5742:
5739:
5738:
5737:
5716:
5703:techniques of
5684:
5664:
5647:
5646:
5618:
5598:
5577:
5553:
5533:
5530:
5527:
5524:
5521:
5518:
5515:
5512:
5503:the condition
5492:
5489:
5486:
5466:
5446:
5426:
5423:
5420:
5417:
5414:
5411:
5408:
5405:
5385:
5365:
5341:
5338:
5335:
5332:
5329:
5326:
5303:
5283:
5280:
5277:
5262:
5250:
5226:
5206:
5183:
5163:
5152:
5136:
5109:
5098:
5069:
5045:
5021:
5008:is said to be
4997:
4982:
4971:quasi-isometry
4967:
4954:
4951:
4938:
4918:
4898:
4895:
4892:
4889:
4886:
4877:decomposition
4862:
4842:
4839:
4836:
4833:
4830:
4827:
4803:
4772:
4752:
4749:
4746:
4743:
4740:
4737:
4717:
4699:
4698:
4686:
4660:
4656:
4633:
4629:
4608:
4603:
4599:
4595:
4592:
4587:
4583:
4577:
4574:
4570:
4565:
4561:
4558:
4555:
4552:
4549:
4546:
4522:
4510:
4498:
4495:
4492:
4472:
4469:
4466:
4446:
4422:
4417:
4413:
4409:
4406:
4403:
4383:
4359:
4356:
4353:
4350:
4347:
4344:
4317:
4305:
4302:
4298:
4297:
4285:
4282:
4279:
4276:
4273:
4270:
4246:
4226:
4200:
4196:
4173:
4169:
4144:
4139:
4135:
4131:
4128:
4123:
4119:
4113:
4110:
4106:
4101:
4097:
4094:
4091:
4088:
4085:
4082:
4071:
4059:
4056:
4053:
4050:
4047:
4044:
4024:
4004:
3984:
3981:
3978:
3958:
3955:
3952:
3932:
3908:
3903:
3899:
3895:
3892:
3889:
3865:
3839:
3836:
3833:
3830:
3827:
3824:
3804:
3799:
3795:
3791:
3788:
3785:
3765:
3745:
3742:
3739:
3736:
3733:
3713:
3691:
3687:
3666:
3646:
3643:
3640:
3637:
3634:
3614:
3594:
3574:
3554:
3530:
3527:
3524:
3521:
3518:
3515:
3512:
3509:
3489:
3469:
3466:
3463:
3460:
3457:
3437:
3417:
3397:
3377:
3368:. Indeed, let
3357:
3354:
3351:
3348:
3345:
3342:
3322:
3294:
3274:
3254:
3251:
3248:
3245:
3242:
3230:
3227:
3226:
3225:
3213:
3208:
3204:
3200:
3197:
3194:
3174:
3171:
3168:
3165:
3162:
3159:
3144:
3141:
3128:
3106:
3102:
3081:
3061:
3056:
3052:
3048:
3045:
3042:
3022:
3019:
2999:
2975:
2972:
2969:
2945:
2942:
2939:
2927:are infinite.
2914:
2910:
2889:
2865:
2860:
2856:
2852:
2849:
2846:
2826:
2823:
2803:
2779:
2774:
2770:
2766:
2763:
2760:
2735:
2731:
2727:
2724:
2721:
2718:
2696:
2692:
2671:
2651:
2631:
2611:
2608:
2581:
2578:
2575:
2552:
2532:
2512:
2509:
2506:
2484:
2480:
2459:
2456:
2453:
2433:
2413:
2389:
2386:
2383:
2380:
2377:
2374:
2371:
2368:
2348:
2337:generating set
2324:
2321:
2318:
2294:
2282:
2279:
2278:
2277:
2261:
2257:
2253:
2233:
2213:
2195:
2194:
2174:
2147:
2127:
2124:
2121:
2118:
2115:
2112:
2092:
2078:
2066:
2063:
2060:
2057:
2054:
2051:
2048:
2045:
2042:
2039:
2036:
2033:
2030:
2027:
2007:
1981:
1978:
1977:
1976:
1964:
1961:
1958:
1955:
1952:
1949:
1946:
1943:
1940:
1920:
1917:
1913:
1909:
1905:
1901:
1898:
1878:
1875:
1872:
1869:
1855:
1844:
1841:
1838:
1833:
1828:
1823:
1820:
1798:
1793:
1776:
1765:
1762:
1759:
1755:
1751:
1748:
1727:
1712:
1700:
1680:
1677:
1674:
1671:
1668:
1665:
1645:
1629:
1626:
1613:
1610:
1607:
1604:
1584:
1564:
1554:generating set
1541:
1538:
1535:
1532:
1529:
1526:
1523:
1520:
1517:
1497:
1494:
1491:
1488:
1485:
1482:
1479:
1476:
1473:
1470:
1467:
1464:
1461:
1458:
1449:is defined as
1438:
1415:
1395:
1371:
1368:
1365:
1362:
1359:
1356:
1336:
1325:generating set
1312:
1309:
1306:
1282:
1270:
1269:Ends of groups
1267:
1254:
1234:
1231:
1228:
1208:
1205:
1202:
1182:
1162:
1142:
1139:
1136:
1116:
1096:
1076:
1073:
1070:
1067:
1064:
1061:
1041:
1021:
1018:
1015:
995:
992:
989:
969:
949:
929:
909:
889:
886:
883:
880:
877:
874:
871:
868:
848:
828:
825:
822:
819:
796:
773:
770:
767:
764:
744:
741:
738:
735:
732:
729:
709:
706:
703:
700:
697:
694:
674:
671:
668:
648:
645:
642:
639:
636:
633:
613:
593:
573:
570:
567:
564:
561:
558:
538:
535:
532:
529:
526:
523:
514:the statement
503:
500:
497:
494:
491:
471:
468:
465:
462:
459:
456:
436:
433:
430:
427:
424:
421:
397:
377:
374:
371:
351:
331:
311:
308:
305:
302:
299:
296:
276:
256:
253:
250:
219:
191:
167:
152:Main article:
149:
148:Ends of graphs
146:
114:
94:
77:over a finite
57:
36:
24:states that a
9:
6:
4:
3:
2:
6379:
6368:
6365:
6363:
6360:
6359:
6357:
6344:
6343:
6339:M. Kapovich.
6336:
6329:
6323:
6316:
6313:
6312:
6305:
6298:
6297:
6290:
6283:
6280:
6279:
6272:
6265:
6262:
6261:
6254:
6247:
6243:
6240:
6239:
6232:
6225:
6219:
6212:
6211:
6204:
6196:
6192:
6188:
6184:
6180:
6176:
6169:
6162:
6161:
6154:
6146:
6142:
6137:
6132:
6128:
6124:
6120:
6113:
6105:
6101:
6097:
6093:
6089:
6085:
6078:
6069:
6067:
6060:
6054:
6052:
6044:
6043:
6036:
6028:
6024:
6019:
6014:
6010:
6006:
6002:
5995:
5988:
5984:
5981:
5980:
5973:
5966:
5963:
5962:
5955:
5948:
5945:
5944:
5937:
5929:
5925:
5920:
5915:
5911:
5907:
5903:
5896:
5890:
5889:0-521-43529-3
5886:
5882:
5876:
5869:
5866:
5865:
5858:
5856:
5848:
5847:
5840:
5831:
5822:
5813:
5806:
5800:
5793:
5790:
5789:
5782:
5778:
5768:
5765:
5763:
5760:
5758:
5755:
5753:
5752:HNN extension
5750:
5748:
5745:
5744:
5734:
5730:
5714:
5706:
5702:
5698:
5682:
5662:
5653:
5649:
5648:
5644:
5640:
5636:
5632:
5616:
5596:
5575:
5567:
5566:CAT(0)-cubing
5551:
5531:
5528:
5522:
5519:
5516:
5510:
5490:
5487:
5484:
5464:
5444:
5424:
5421:
5415:
5412:
5409:
5403:
5383:
5363:
5355:
5336:
5333:
5330:
5324:
5317:
5301:
5281:
5278:
5275:
5267:
5263:
5248:
5240:
5224:
5204:
5197:
5181:
5161:
5153:
5150:
5134:
5126:
5123:
5107:
5099:
5096:
5091:
5087:
5083:
5067:
5059:
5043:
5035:
5019:
5011:
4995:
4987:
4983:
4980:
4976:
4972:
4968:
4965:
4962:
4957:
4956:
4950:
4936:
4916:
4896:
4893:
4890:
4887:
4884:
4876:
4860:
4837:
4831:
4825:
4817:
4801:
4792:
4790:
4786:
4770:
4750:
4747:
4741:
4735:
4715:
4708:
4704:
4684:
4676:
4658:
4654:
4631:
4627:
4601:
4597:
4593:
4590:
4585:
4581:
4575:
4572:
4568:
4559:
4556:
4553:
4547:
4544:
4537:
4536:HNN extension
4533:
4520:
4511:
4496:
4493:
4490:
4470:
4467:
4464:
4444:
4436:
4420:
4415:
4411:
4407:
4404:
4401:
4381:
4373:
4372:
4371:
4357:
4354:
4348:
4342:
4333:
4331:
4315:
4301:
4283:
4280:
4274:
4268:
4260:
4244:
4224:
4216:
4198:
4194:
4171:
4167:
4158:
4157:HNN-extension
4137:
4133:
4129:
4126:
4121:
4117:
4111:
4108:
4104:
4095:
4092:
4089:
4083:
4080:
4072:
4057:
4054:
4048:
4042:
4022:
4002:
3982:
3979:
3976:
3956:
3953:
3950:
3930:
3922:
3906:
3901:
3897:
3893:
3890:
3887:
3879:
3878:
3877:
3863:
3856:
3851:
3837:
3834:
3828:
3822:
3797:
3793:
3789:
3786:
3763:
3743:
3740:
3737:
3734:
3731:
3711:
3689:
3685:
3664:
3644:
3641:
3638:
3635:
3632:
3612:
3592:
3572:
3552:
3544:
3525:
3522:
3519:
3510:
3487:
3467:
3464:
3461:
3458:
3455:
3435:
3415:
3395:
3375:
3355:
3352:
3346:
3340:
3320:
3312:
3308:
3292:
3272:
3252:
3249:
3246:
3243:
3240:
3206:
3202:
3198:
3195:
3172:
3169:
3163:
3157:
3150:
3149:
3148:
3143:Cuts and ends
3140:
3126:
3104:
3100:
3079:
3054:
3050:
3046:
3043:
3020:
3017:
2997:
2989:
2973:
2970:
2967:
2960:if for every
2959:
2943:
2940:
2937:
2928:
2912:
2908:
2887:
2879:
2858:
2854:
2850:
2847:
2824:
2821:
2793:
2772:
2768:
2764:
2761:
2749:
2733:
2729:
2725:
2722:
2719:
2716:
2694:
2690:
2669:
2629:
2609:
2606:
2599:
2595:
2594:edge boundary
2579:
2576:
2573:
2566:For a subset
2564:
2550:
2530:
2510:
2507:
2504:
2482:
2478:
2457:
2454:
2451:
2431:
2411:
2403:
2384:
2381:
2378:
2369:
2346:
2338:
2322:
2319:
2316:
2308:
2292:
2275:
2259:
2255:
2251:
2231:
2211:
2203:
2202:
2201:
2199:
2192:
2188:
2172:
2164:
2161:
2145:
2125:
2122:
2116:
2110:
2090:
2083:
2079:
2058:
2055:
2052:
2049:
2046:
2043:
2037:
2031:
2025:
2005:
1998:
1994:
1993:
1992:
1990:
1986:
1959:
1950:
1944:
1938:
1915:
1907:
1899:
1896:
1873:
1867:
1860:
1856:
1842:
1839:
1831:
1818:
1796:
1781:
1777:
1763:
1760:
1746:
1717:
1713:
1698:
1678:
1675:
1669:
1663:
1643:
1636:
1632:
1631:
1625:
1608:
1602:
1582:
1562:
1555:
1533:
1530:
1527:
1515:
1489:
1486:
1483:
1471:
1468:
1462:
1456:
1436:
1429:
1413:
1393:
1385:
1366:
1363:
1360:
1334:
1326:
1310:
1307:
1304:
1296:
1280:
1266:
1252:
1245:has at least
1232:
1229:
1206:
1203:
1200:
1160:
1140:
1137:
1134:
1094:
1071:
1059:
1039:
1019:
1016:
993:
990:
987:
947:
907:
884:
881:
878:
866:
817:
808:
787:
762:
755:. The number
739:
727:
707:
704:
692:
672:
669:
666:
646:
643:
631:
611:
591:
571:
568:
556:
536:
533:
521:
501:
498:
495:
492:
489:
469:
466:
454:
434:
431:
419:
411:
395:
375:
372:
329:
309:
306:
294:
254:
251:
248:
239:
237:
233:
209:
205:
181:
155:
145:
143:
139:
134:
132:
128:
112:
92:
84:
80:
76:
75:HNN extension
72:
68:
55:
34:
27:
23:
19:
6341:
6335:
6327:
6322:
6310:
6304:
6295:
6289:
6277:
6271:
6259:
6253:
6237:
6231:
6223:
6218:
6209:
6203:
6178:
6174:
6168:
6159:
6153:
6126:
6122:
6112:
6087:
6083:
6077:
6041:
6035:
6008:
6004:
5994:
5978:
5976:T. Delzant.
5972:
5960:
5954:
5942:
5936:
5912:(1): 39–46.
5909:
5905:
5895:
5880:
5875:
5863:
5845:
5839:
5830:
5821:
5812:
5804:
5799:
5787:
5781:
5315:
5314:one defines
5009:
4985:
4949:nontrivial.
4875:free product
4793:
4700:
4513:
4334:
4307:
4299:
3852:
3543:Cayley graph
3311:Cayley graph
3232:
3146:
2957:
2929:
2877:
2791:
2790:is called a
2750:
2597:
2593:
2565:
2402:Cayley graph
2335:be a finite
2284:
2196:
1983:
1782:of rank two
1427:
1384:Cayley graph
1323:be a finite
1272:
1173:of edges of
1107:of edges of
1032:has exactly
960:of edges of
920:of edges of
810:Informally,
809:
785:
388:has at most
342:of edges of
240:
208:cell complex
157:
142:torsion-free
135:
48:
21:
18:group theory
15:
6157:H. MĂĽller.
2598:co-boundary
6356:Categories
5631:polycyclic
5010:accessible
4964:free group
4909:with both
4619:where and
4512:The group
4374:The group
3224:in Γ.
2956:is called
2876:is called
2470:denote by
2244:such that
2189:of finite
2165:(that is,
1989:Heinz Hopf
1859:free group
1711:is finite.
1595:, so that
1219:such that
1006:such that
784:is called
685:such that
624:such that
362:the graph
6039:H. Bass.
5958:Z. Sela.
5488:⩽
5279:⩽
5122:virtually
4961:virtually
4894:∗
4841:∞
4728:we have
4675:subgroups
4607:⟩
4573:−
4551:⟨
4494:≠
4468:≠
4412:∗
4215:subgroups
4143:⟩
4109:−
4087:⟨
3980:≠
3954:≠
3898:∗
3798:∗
3741:∗
3690:∗
3642:∗
3514:Γ
3508:Γ
3465:∪
3309:then the
3250:∗
3207:∗
3105:∗
3055:∗
2971:∈
2941:⊆
2930:A subset
2913:∗
2878:essential
2859:∗
2822:δ
2802:Γ
2773:∗
2734:∗
2726:δ
2717:δ
2695:∗
2650:Γ
2607:δ
2577:⊆
2508:−
2483:∗
2455:⊆
2373:Γ
2367:Γ
2320:⊆
2160:virtually
2062:∞
2038:∈
1963:∞
1919:∞
1522:Γ
1478:Γ
1355:Γ
1308:⊆
1230:−
1227:Γ
1204:⊆
1181:Γ
1138:⩾
1115:Γ
1075:∞
1066:Γ
1017:−
1014:Γ
991:⊆
968:Γ
928:Γ
888:∞
873:Γ
847:Γ
824:Γ
795:Γ
769:Γ
743:∞
734:Γ
705:⩽
699:Γ
670:⩾
644:⩽
638:Γ
563:Γ
534:⩽
528:Γ
493:⩽
467:⩽
461:Γ
426:Γ
408:infinite
373:−
370:Γ
350:Γ
307:⩽
301:Γ
275:Γ
252:⩾
218:Γ
190:Γ
166:Γ
6242:Archived
6005:Topology
5983:Archived
5741:See also
5652:Dunwoody
5639:Bowditch
5095:Bestvina
5082:Dunwoody
4966:is free.
3677:. Thus
3500:and let
2990:between
2359:and let
2204:A group
2187:subgroup
2103:we have
2080:For any
2018:we have
1995:For any
1931:we have
1811:we have
1778:For the
1739:we have
1714:For the
1656:we have
1347:and let
230:are the
79:subgroup
6195:1025923
6145:0466326
6104:0487104
6027:1838998
5928:0716233
5736:groups.
3541:be the
2596:or the
2400:be the
1382:be the
6193:
6143:
6102:
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