1859:
1636:
3992:
is of spherical type. Such groups act cocompactly on a CAT(0) cubical complex, and, as a consequence, one can find a rational normal form for their elements and deduce a solution to the word problem (Joe
Altobelli and Charney ). An alternative normal form is provided by multifraction reduction, which
2946:
In the case of a spherical Artin–Tits group, the group is a group of fractions for the monoid, making the study much easier. Every above-mentioned problem is solved in the positive for spherical Artin–Tits groups: the word and conjugacy problems are decidable, their torsion is trivial, the center is
2663:
is finite — the alternative terminology "Artin–Tits group of finite type" is to be avoided, because of its ambiguity: a "finite type group" is just one that admits a finite generating set. Recall that a complete classification is known, the 'irreducible types' being labeled as the infinite series
1030:
1854:{\displaystyle G=\langle \sigma _{1},\ldots ,\sigma _{n-1}\mid \sigma _{i}\sigma _{j}\sigma _{i}=\sigma _{j}\sigma _{i}\sigma _{j}{\text{ for }}\vert i-j\vert =1,\sigma _{i}\sigma _{j}=\sigma _{j}\sigma _{i}{\text{ for }}\vert i-j\vert \geqslant 2\rangle }
2617:
Every (finitely generated) Artin–Tits monoid admits a finite
Garside family (Matthew Dyer and Christophe Hohlweg). As a consequence, the existence of common right-multiples in Artin–Tits monoids is decidable, and reduction of multifractions is
4410:: the associated Coxeter group acts geometrically on a Euclidean space. As a consequence, their center is trivial, and their word problem is decidable (Jon McCammond and Robert Sulway ). In 2019, a proof of the
3990:
1544:
869:
1383:
2023:
1932:
861:
810:
3794:
2612:
2580:
2548:
2355:
1109:
682:
643:
514:
212:
122:
2080:
Artin–Tits monoids are cancellative, and they admit greatest common divisors and conditional least common multiples (a least common multiple exists whenever a common multiple does).
2461:
4404:
4368:
4332:
4296:
4260:
4198:
4136:
4100:
4038:
2061:
3882:
3706:
3637:
3540:
The word and conjugacy problems of a right-angled Artin–Tits group are decidable, the former in linear time, the group is torsion-free, and there is an explicit cellular finite
400:
1970:
3001:
3470:
1442:
1337:
593:
3312:
4224:
4162:
4064:
464:
432:
5305:
4443:
3573:
3371:
2308:
1603:
2779:
3933:
3823:
3732:
3663:
3431:
3254:
3176:
2242:
2148:
1279:
1182:
1066:
3208:
3082:
2941:
2914:
2887:
2860:
2833:
2806:
2743:
2716:
2689:
2493:
2222:
2195:
2108:
5084:
4952:
3908:
3519:
2381:
1629:
1468:
341:
284:
238:
3843:
3411:
3391:
3332:
3274:
3156:
3122:
3102:
3055:
3031:
2661:
2513:
2401:
2168:
2128:
1879:
1564:
1403:
1299:
1246:
1222:
1202:
1149:
1129:
762:
742:
722:
702:
554:
534:
361:
315:
258:
162:
142:
3582:
cube complex, its "Salvetti complex". As an application, one can use right-angled Artin groups and their
Salvetti complexes to construct groups with given
3993:
gives a unique expression by an irreducible multifraction directly extending the expression by an irreducible fraction in the spherical case (Dehornoy).
4580:
Crisp, John; Paris, Luis (2001), "The solution to a conjecture of Tits on the subgroup generated by the squares of the generators of an Artin group",
3533:. A right-angled Artin group is a special case of this product, with every vertex/operand of the graph-product being a free group of rank one (the
2270:– determining the center — which is conjectured to be trivial or monogenic in the case when the group is not a direct product ("irreducible case"),
3737:
Artin–Tits groups of extra-large type are eligible for small cancellation theory. As an application, Artin–Tits groups of extra-large type are
5401:
5044:
2248:
Very few results are known for general Artin–Tits groups. In particular, the following basic questions remain open in the general case:
1025:{\displaystyle \langle s,t\rangle ^{m_{s,t}}=\langle t,s\rangle ^{m_{t,s}},{\text{ where }}m_{s,t}=m_{t,s}\in \{2,3,\ldots ,\infty \}.}
4636:
4455:
1475:
3583:
3938:
2318:
Partial results involving particular subfamilies are gathered below. Among the few known general results, one can mention:
4466:
3234:
1350:
1281:
are removed, the extension thus obtained is an Artin–Tits group. For instance, the
Coxeter group associated with the
4867:
Crisp, John; Godelle, Eddy; Wiest, Bert (2009), "The conjugacy problem in subgroups of right-angled Artin groups",
1975:
1884:
260:, where both sides have equal lengths, and there exists at most one relation for each pair of distinct generators
815:
767:
5218:
5000:
4908:
4786:
4735:
4582:
4471:
3526:
4682:
Dyer, Matthew; Hohlweg, Christophe (2016), "Small roots, low elements, and the weak order in
Coxeter groups",
3767:
3760:
Many other families of Artin–Tits groups have been identified and investigated. Here we mention two of them.
2585:
2553:
2521:
2328:
1082:
648:
598:
469:
167:
95:
5452:; Koberda, Thomas (2019). "Algorithmic problems in right-angled Artin groups: complexity and applications".
4684:
2253:
90:
51:
2627:
Several important classes of Artin groups can be defined in terms of the properties of the
Coxeter matrix.
2406:
4373:
4337:
4301:
4265:
4229:
4167:
4105:
4069:
4007:
2971:
2028:
3848:
5502:
4833:
4496:
3672:
3603:
3534:
366:
1937:
2977:
5237:
5009:
4917:
4795:
4744:
4601:
4001:
3751:
Artin groups of large type are shortlex automatic with regular geodesics (Derek Holt and Sarah Rees).
3436:
1408:
1304:
559:
5135:
4869:
3279:
47:
4203:
4141:
4043:
5463:
5454:
5379:
5345:
5310:
5227:
5181:
5093:
4961:
4693:
4645:
4591:
4544:
4521:
4513:
3480:
437:
405:
63:
4445:
conjecture was announced for all affine Artin–Tits groups (Mario
Salvetti and Giovanni Paolini).
17:
5275:
4413:
3543:
3337:
2278:
1569:
5449:
5397:
3738:
3007:
2967:
2748:
2311:
2257:
3711:
3642:
3416:
3239:
3161:
2227:
2133:
1251:
1154:
1038:
5473:
5427:
5389:
5355:
5245:
5191:
5169:
5144:
5103:
5053:
5017:
4971:
4925:
4878:
4842:
4803:
4777:
4752:
4703:
4655:
4609:
4553:
4505:
2956:
2948:
1069:
5485:
5441:
5411:
5367:
5257:
5203:
5156:
5115:
5065:
5029:
4985:
4937:
4890:
4854:
4815:
4764:
4717:
4669:
4621:
4567:
3181:
3060:
2919:
2892:
2865:
2838:
2811:
2784:
2721:
2694:
2667:
2466:
2200:
2173:
2086:
5481:
5437:
5407:
5363:
5253:
5199:
5152:
5111:
5061:
5025:
4981:
4933:
4903:
4886:
4850:
4811:
4760:
4713:
4665:
4617:
4563:
3530:
3034:
2073:
3887:
3498:
2360:
1608:
1447:
320:
263:
217:
5241:
5013:
4921:
4799:
4748:
4605:
3913:
3803:
3578:
Every right-angled Artin–Tits group acts freely and cocompactly on a finite-dimensional
4998:
Appel, Kenneth I.; Schupp, Paul E. (1983), "Artin Groups and
Infinite Coxeter Groups",
4730:
4460:
3828:
3484:
3396:
3376:
3317:
3259:
3233:
For this class of Artin–Tits groups, a different labeling scheme is commonly used. Any
3141:
3107:
3087:
3040:
3016:
2952:
2646:
2498:
2386:
2153:
2113:
1864:
1549:
1388:
1284:
1231:
1207:
1187:
1134:
1114:
1073:
747:
727:
707:
687:
539:
519:
346:
300:
243:
147:
127:
5172:(2017), "Multifraction reduction I: The 3-Ore case and Artin–Tits groups of type FC",
286:. An Artin–Tits group is a group that admits an Artin–Tits presentation. Likewise, an
5496:
5057:
4558:
3742:
3492:
2641:
2076:
based on the investigation of their divisibility relations, and are well understood:
1225:
55:
5082:(2012). "Artin groups of large type are shortlex automatic with regular geodesics".
4525:
5477:
5333:
5130:
4828:
4539:
3522:
3479:
of finite rank, corresponding to a graph with no edges, and the finitely-generated
3104:
a unique normal form that consists of a finite sequence of (copies of) elements of
78:
31:
5393:
4781:
4004:. They correspond to the extended Dynkin diagrams of the four infinite families
2960:
67:
2224:
admits a distinguished decomposition as a sequence of elements in the image of
5359:
5249:
5148:
5079:
4708:
4660:
4491:
3476:
74:
59:
297:
Alternatively, an Artin–Tits group can be specified by the set of generators
4882:
5107:
5042:
Peifer, David (1996), "Artin groups of extra-large type are biautomatic",
4929:
4906:; Brady, Noel (1997), "Morse theory and finiteness properties of groups",
4613:
5216:
McCammond, Jon; Sulway, Robert (2017), "Artin groups of
Euclidean type",
81:
who developed the theory of a more general class of groups in the 1960s.
5195:
3529:
as the extreme cases. A generalization of this construction is called a
5021:
4976:
4846:
4807:
4756:
4517:
5350:
4650:
4596:
3748:
Artin–Tits groups of extra-large type are biautomatic (David Peifer).
3579:
291:
5432:
4509:
2130:
is the associated
Coxeter group, there is a (set-theoretic) section
5468:
5315:
5186:
4966:
4698:
77:, due to his early work on braid groups in the 1920s to 1940s, and
5418:
McCammond, Jon (2017), "The mysterious geometry of Artin groups",
5384:
5232:
5098:
4542:(1966), "Normalisateurs de tores. I. Groupes de Coxeter étendus",
1301:-strand braid group is the symmetric group of all permutations of
5133:(2000), "A geometric rational form for Artin groups of FC type",
2966:
A pure Artin–Tits group of spherical type can be realized as the
5378:, CRM Series, vol. 14, Ed. Norm., Pisa, pp. 299–311,
164:
is a set of Artin–Tits relations, namely relations of the form
1539:{\displaystyle G=\langle S\mid \{st=ts\mid s,t\in S\}\rangle }
1248:
is a Coxeter group presented by reflections and the relations
4733:(1972), "Les immeubles des groupes de tresses généralisés",
4634:
Paris, Luis (2002), "Artin monoids inject in their groups",
3596:
An Artin–Tits group (and a Coxeter group) is said to be of
2357:, the only relation connecting the squares of the elements
2265:– determining torsion — which is conjectured to be trivial,
4950:
Leary, Ian (2018), "Uncountably many groups of type FP",
3475:
The class of right-angled Artin–Tits groups includes the
2275:– determining the cohomology — in particular solving the
5336:(2007), "An introduction to right-angled Artin groups",
3586:(Mladen Bestvina and Noel Brady ) see also (Ian Leary ).
4831:(1992), "Artin groups of finite type are biautomatic",
3138:
if all coefficients of the Coxeter matrix are either
294:
that, as a monoid, admits an Artin–Tits presentation.
5278:
4416:
4376:
4340:
4304:
4268:
4232:
4206:
4170:
4144:
4108:
4072:
4046:
4010:
3985:{\displaystyle \langle S'\mid R\cap S'{}^{2}\rangle }
3941:
3916:
3890:
3851:
3831:
3806:
3770:
3714:
3675:
3645:
3606:
3546:
3501:
3439:
3419:
3399:
3379:
3340:
3320:
3282:
3262:
3242:
3184:
3164:
3144:
3110:
3090:
3063:
3043:
3019:
2980:
2922:
2895:
2868:
2841:
2814:
2787:
2751:
2724:
2697:
2670:
2649:
2588:
2556:
2524:
2501:
2469:
2409:
2389:
2363:
2331:
2281:
2230:
2203:
2176:
2156:
2136:
2116:
2089:
2031:
1978:
1940:
1887:
1867:
1639:
1611:
1572:
1552:
1478:
1450:
1411:
1391:
1353:
1307:
1287:
1254:
1234:
1210:
1190:
1157:
1137:
1117:
1111:
is an Artin–Tits presentation of an Artin–Tits group
1085:
1041:
872:
818:
770:
750:
730:
710:
690:
651:
601:
562:
542:
522:
472:
440:
408:
369:
349:
323:
303:
266:
246:
220:
170:
150:
130:
98:
70:, and right-angled Artin–Tits groups, among others.
3124:
and their inverses ("symmetric greedy normal form")
2310:conjecture, i.e., finding an acyclic complex whose
5299:
4437:
4398:
4362:
4326:
4290:
4254:
4218:
4192:
4156:
4130:
4094:
4058:
4032:
3984:
3927:
3902:
3876:
3837:
3817:
3788:
3726:
3700:
3657:
3631:
3567:
3513:
3464:
3425:
3405:
3385:
3365:
3326:
3306:
3268:
3248:
3202:
3170:
3150:
3116:
3096:
3076:
3057:is a group of fractions for the associated monoid
3049:
3025:
2995:
2935:
2908:
2881:
2854:
2827:
2800:
2773:
2737:
2710:
2683:
2655:
2606:
2574:
2542:
2507:
2487:
2455:
2395:
2375:
2349:
2302:
2236:
2216:
2189:
2162:
2142:
2122:
2102:
2055:
2017:
1964:
1926:
1873:
1853:
1623:
1597:
1558:
1538:
1462:
1436:
1397:
1378:{\displaystyle G=\langle S\mid \emptyset \rangle }
1377:
1331:
1293:
1273:
1240:
1216:
1196:
1176:
1143:
1123:
1103:
1060:
1024:
855:
804:
756:
736:
716:
696:
676:
637:
587:
548:
528:
508:
458:
426:
394:
355:
335:
309:
278:
252:
232:
206:
156:
136:
116:
3178:, i.e., all relations are commutation relations
863:, etc. — the Artin–Tits relations take the form
5085:Proceedings of the London Mathematical Society
4953:Proceedings of the London Mathematical Society
4784:(1972), "Artin-Gruppen und Coxeter-Gruppen",
2018:{\displaystyle m_{\sigma _{i},\sigma _{j}}=2}
1927:{\displaystyle m_{\sigma _{i},\sigma _{j}}=3}
8:
3979:
3942:
3783:
3771:
2601:
2589:
2582:embeds in the Artin–Tits group presented by
2569:
2557:
2537:
2525:
2344:
2332:
2044:
2032:
1953:
1941:
1848:
1839:
1827:
1767:
1755:
1646:
1533:
1530:
1494:
1485:
1372:
1360:
1326:
1308:
1098:
1086:
1016:
992:
924:
911:
886:
873:
832:
819:
784:
771:
665:
652:
144:is a (usually finite) set of generators and
111:
99:
5270:Paolini, Giovanni; Salvetti, Mario (2019),
3741:-free and have solvable conjugacy problem (
3575:(John Crisp, Eddy Godelle, and Bert Wiest).
3013:In modern terminology, an Artin–Tits group
2947:monogenic in the irreducible case, and the
856:{\displaystyle \langle s,t\rangle ^{3}=sts}
805:{\displaystyle \langle s,t\rangle ^{2}=st}
5467:
5431:
5383:
5349:
5314:
5277:
5231:
5185:
5097:
4975:
4965:
4707:
4697:
4659:
4649:
4595:
4557:
4415:
4390:
4379:
4378:
4375:
4354:
4343:
4342:
4339:
4318:
4307:
4306:
4303:
4282:
4271:
4270:
4267:
4246:
4235:
4234:
4231:
4205:
4184:
4173:
4172:
4169:
4143:
4122:
4111:
4110:
4107:
4086:
4075:
4074:
4071:
4045:
4024:
4013:
4012:
4009:
3973:
3971:
3940:
3915:
3889:
3856:
3850:
3830:
3805:
3769:
3713:
3680:
3674:
3644:
3611:
3605:
3545:
3500:
3487:. Every right-angled Artin group of rank
3444:
3438:
3418:
3398:
3378:
3345:
3339:
3319:
3281:
3261:
3241:
3183:
3163:
3143:
3109:
3089:
3068:
3062:
3042:
3018:
2987:
2983:
2982:
2979:
2927:
2921:
2900:
2894:
2873:
2867:
2846:
2840:
2819:
2813:
2792:
2786:
2756:
2750:
2729:
2723:
2702:
2696:
2675:
2669:
2648:
2587:
2555:
2523:
2500:
2468:
2447:
2437:
2424:
2414:
2408:
2388:
2362:
2330:
2322:Artin–Tits groups are infinite countable.
2280:
2229:
2208:
2202:
2181:
2175:
2155:
2135:
2115:
2094:
2088:
2030:
2001:
1988:
1983:
1977:
1939:
1910:
1897:
1892:
1886:
1866:
1822:
1816:
1806:
1793:
1783:
1750:
1744:
1734:
1724:
1711:
1701:
1691:
1672:
1653:
1638:
1610:
1577:
1571:
1551:
1477:
1449:
1416:
1410:
1390:
1352:
1306:
1286:
1259:
1253:
1233:
1209:
1189:
1162:
1156:
1136:
1116:
1084:
1046:
1040:
977:
958:
949:
932:
927:
894:
889:
871:
835:
817:
787:
769:
749:
729:
709:
689:
668:
650:
600:
567:
561:
541:
521:
471:
439:
407:
374:
368:
348:
322:
302:
265:
245:
219:
169:
149:
129:
97:
5426:(Winter Braids VII (Caen, 2017)): 1–30,
3006:Artin–Tits groups of spherical type are
2260:— which are conjectured to be decidable,
4483:
3800:("flag complex") if, for every subset
3789:{\displaystyle \langle S\mid R\rangle }
2623:Particular classes of Artin–Tits groups
2607:{\displaystyle \langle S\mid R\rangle }
2575:{\displaystyle \langle S\mid R\rangle }
2543:{\displaystyle \langle S\mid R\rangle }
2350:{\displaystyle \langle S\mid R\rangle }
1104:{\displaystyle \langle S\mid R\rangle }
677:{\displaystyle \langle s,t\rangle ^{m}}
638:{\displaystyle stst\ldots =tsts\ldots }
509:{\displaystyle stst\ldots =tsts\ldots }
207:{\displaystyle stst\ldots =tsts\ldots }
117:{\displaystyle \langle S\mid R\rangle }
3495:of a right-angled Artin group of rank
89:An Artin–Tits presentation is a group
3084:and there exists for each element of
2636:An Artin–Tits group is said to be of
2550:, the Artin–Tits monoid presented by
2456:{\displaystyle s^{2}t^{2}=t^{2}s^{2}}
7:
5376:Basic questions on Artin–Tits groups
4399:{\displaystyle {\widetilde {G}}_{2}}
4363:{\displaystyle {\widetilde {F}}_{4}}
4327:{\displaystyle {\widetilde {E}}_{8}}
4291:{\displaystyle {\widetilde {E}}_{7}}
4255:{\displaystyle {\widetilde {E}}_{6}}
4193:{\displaystyle {\widetilde {D}}_{n}}
4131:{\displaystyle {\widetilde {C}}_{n}}
4095:{\displaystyle {\widetilde {B}}_{n}}
4033:{\displaystyle {\widetilde {A}}_{n}}
2072:Artin–Tits monoids are eligible for
2056:{\displaystyle \vert i-j\vert >1}
684:to denote an alternating product of
46:, are a family of infinite discrete
5374:Godelle, Eddy; Paris, Luis (2012),
5045:Journal of Pure and Applied Algebra
4000:if the associated Coxeter group is
3877:{\displaystyle m_{s,t}\neq \infty }
2631:Artin–Tits groups of spherical type
1546:is the free abelian group based on
5307:conjecture for affine Artin groups
3996:An Artin–Tits group is said to be
3871:
3701:{\displaystyle m_{s,t}\geqslant 4}
3632:{\displaystyle m_{s,t}\geqslant 3}
3459:
3420:
3243:
3212:(free) partially commutative group
3165:
3134:An Artin–Tits group is said to be
2518:For every Artin–Tits presentation
1431:
1369:
1013:
582:
556:, if any. By convention, one puts
395:{\displaystyle m_{s,t}\geqslant 2}
27:Family of infinite discrete groups
25:
4637:Commentarii Mathematici Helvetici
4456:Free partially commutative monoid
4226:, and of the five sporadic types
1965:{\displaystyle \vert i-j\vert =1}
5174:Journal of Combinatorial Algebra
2996:{\displaystyle \mathbb {C} ^{n}}
2110:is an Artin–Tits monoid, and if
1151:obtained by adding the relation
402:that is the length of the words
54:. They are closely related with
18:Free partially commutative group
4406:. Affine Artin–Tits groups are
3591:Artin–Tits groups of large type
3465:{\displaystyle m_{s,t}=\infty }
1437:{\displaystyle m_{s,t}=\infty }
1332:{\displaystyle \{1,\ldots ,n\}}
588:{\displaystyle m_{s,t}=\infty }
5478:10.1016/j.jalgebra.2018.10.023
5294:
5282:
4432:
4420:
3562:
3550:
2970:of the complement of a finite
2768:
2762:
2297:
2285:
1:
3307:{\displaystyle 1,2,\ldots ,n}
2963:, by combinatorial methods ).
2515:(John Crisp and Luis Paris ).
5394:10.1007/978-88-7642-431-1_13
5058:10.1016/0022-4049(95)00094-1
4559:10.1016/0021-8693(66)90053-6
4494:(1947). "Theory of Braids".
4467:Non-commutative cryptography
4219:{\displaystyle n\geqslant 3}
4157:{\displaystyle n\geqslant 2}
4059:{\displaystyle n\geqslant 1}
3413:are connected by an edge in
30:In the mathematical area of
5420:Winter Braids Lecture Notes
2781:and six exceptional groups
1385:is the free group based on
516:is the relation connecting
459:{\displaystyle tsts\ldots }
427:{\displaystyle stst\ldots }
73:The groups are named after
5519:
2955:, by geometrical methods,
595:when there is no relation
5360:10.1007/s10711-007-9148-6
5300:{\displaystyle K(\pi ,1)}
5250:10.1007/s00222-017-0728-2
4709:10.1016/j.aim.2016.06.022
4661:10.1007/s00014-002-8353-z
4438:{\displaystyle K(\pi ,1)}
3568:{\displaystyle K(\pi ,1)}
3366:{\displaystyle m_{s,t}=2}
3129:Right-angled Artin groups
2303:{\displaystyle K(\pi ,1)}
1598:{\displaystyle m_{s,t}=2}
645:. Formally, if we define
5219:Inventiones Mathematicae
5001:Inventiones Mathematicae
4909:Inventiones Mathematicae
4787:Inventiones Mathematicae
4736:Inventiones Mathematicae
4583:Inventiones Mathematicae
4472:Elementary abelian group
2774:{\displaystyle I_{2}(n)}
2314:is the considered group.
1068:can be organized into a
44:generalized braid groups
5149:10.1023/A:1005216814166
4685:Advances in Mathematics
3727:{\displaystyle s\neq t}
3658:{\displaystyle s\neq t}
3531:graph product of groups
3426:{\displaystyle \Gamma }
3249:{\displaystyle \Gamma }
3171:{\displaystyle \infty }
2325:In an Artin–Tits group
2244:("greedy normal form").
2237:{\displaystyle \sigma }
2197:, and every element of
2143:{\displaystyle \sigma }
1274:{\displaystyle s^{2}=1}
1177:{\displaystyle s^{2}=1}
1061:{\displaystyle m_{s,t}}
5301:
4439:
4400:
4364:
4328:
4292:
4256:
4220:
4194:
4158:
4132:
4096:
4060:
4034:
3986:
3929:
3904:
3878:
3839:
3819:
3790:
3728:
3702:
3665:; it is said to be of
3659:
3633:
3569:
3515:
3491:can be constructed as
3466:
3427:
3407:
3387:
3367:
3328:
3308:
3270:
3250:
3204:
3172:
3152:
3118:
3098:
3078:
3051:
3027:
2997:
2972:hyperplane arrangement
2937:
2910:
2883:
2856:
2829:
2802:
2775:
2739:
2712:
2685:
2657:
2608:
2576:
2544:
2509:
2489:
2457:
2397:
2377:
2351:
2304:
2238:
2218:
2191:
2164:
2144:
2124:
2104:
2057:
2019:
1966:
1928:
1875:
1861:is the braid group on
1855:
1625:
1599:
1560:
1540:
1464:
1438:
1399:
1379:
1333:
1295:
1275:
1242:
1218:
1198:
1178:
1145:
1125:
1105:
1062:
1026:
857:
806:
758:
738:
718:
698:
678:
639:
589:
550:
530:
510:
460:
428:
396:
357:
337:
311:
280:
254:
234:
208:
158:
138:
118:
5302:
4930:10.1007/s002220050168
4883:10.1112/jtopol/jtp018
4834:Mathematische Annalen
4614:10.1007/s002220100138
4497:Annals of Mathematics
4463:(an unrelated notion)
4440:
4401:
4365:
4329:
4293:
4257:
4221:
4195:
4159:
4133:
4097:
4061:
4035:
3987:
3930:
3905:
3879:
3840:
3820:
3791:
3729:
3703:
3660:
3634:
3584:finiteness properties
3570:
3535:infinite cyclic group
3516:
3483:, corresponding to a
3467:
3428:
3408:
3388:
3368:
3329:
3309:
3271:
3251:
3205:
3203:{\displaystyle st=ts}
3173:
3153:
3119:
3099:
3079:
3077:{\displaystyle A^{+}}
3052:
3028:
2998:
2938:
2936:{\displaystyle H_{4}}
2911:
2909:{\displaystyle H_{3}}
2884:
2882:{\displaystyle F_{4}}
2857:
2855:{\displaystyle E_{8}}
2830:
2828:{\displaystyle E_{7}}
2803:
2801:{\displaystyle E_{6}}
2776:
2740:
2738:{\displaystyle D_{n}}
2713:
2711:{\displaystyle B_{n}}
2686:
2684:{\displaystyle A_{n}}
2658:
2609:
2577:
2545:
2510:
2490:
2488:{\displaystyle st=ts}
2458:
2398:
2378:
2352:
2305:
2239:
2219:
2217:{\displaystyle A^{+}}
2192:
2190:{\displaystyle A^{+}}
2165:
2145:
2125:
2105:
2103:{\displaystyle A^{+}}
2058:
2020:
1967:
1929:
1876:
1856:
1626:
1600:
1561:
1541:
1465:
1439:
1400:
1380:
1334:
1296:
1276:
1243:
1219:
1199:
1179:
1146:
1126:
1106:
1063:
1027:
858:
807:
759:
739:
719:
699:
679:
640:
590:
551:
531:
511:
461:
429:
397:
363:, the natural number
358:
338:
312:
281:
255:
235:
209:
159:
139:
119:
5276:
4414:
4374:
4338:
4302:
4266:
4230:
4204:
4168:
4142:
4106:
4070:
4044:
4008:
3939:
3914:
3888:
3849:
3829:
3804:
3768:
3764:An Artin–Tits group
3712:
3673:
3643:
3604:
3544:
3499:
3437:
3417:
3397:
3377:
3338:
3318:
3280:
3260:
3240:
3182:
3162:
3142:
3108:
3088:
3061:
3041:
3017:
2978:
2920:
2893:
2866:
2839:
2812:
2785:
2749:
2722:
2695:
2668:
2647:
2586:
2554:
2522:
2499:
2467:
2407:
2387:
2361:
2329:
2279:
2228:
2201:
2174:
2154:
2134:
2114:
2087:
2029:
1976:
1938:
1885:
1865:
1637:
1609:
1570:
1550:
1476:
1448:
1409:
1389:
1351:
1305:
1285:
1252:
1232:
1208:
1188:
1155:
1135:
1115:
1083:
1039:
870:
816:
768:
748:
728:
708:
688:
649:
599:
560:
540:
520:
470:
438:
406:
367:
347:
321:
301:
264:
244:
218:
168:
148:
128:
96:
5338:Geometriae Dedicata
5242:2017InMat.210..231M
5136:Geometriae Dedicata
5108:10.1112/plms/pdr035
5014:1983InMat..72..201A
4922:1997InMat.129..445B
4870:Journal of Topology
4800:1972InMat..17..245B
4749:1972InMat..17..273D
4606:2001InMat.145...19C
3903:{\displaystyle s,t}
3708:for all generators
3639:for all generators
3514:{\displaystyle r-1}
3481:free abelian groups
2376:{\displaystyle s,t}
1624:{\displaystyle s,t}
1463:{\displaystyle s,t}
336:{\displaystyle s,t}
279:{\displaystyle s,t}
233:{\displaystyle s,t}
64:free abelian groups
5455:Journal of Algebra
5450:Kahrobaei, Delaram
5297:
5022:10.1007/BF01389320
4977:10.1112/plms.12135
4847:10.1007/BF01444642
4808:10.1007/BF01406235
4757:10.1007/BF01406236
4545:Journal of Algebra
4435:
4396:
4360:
4324:
4288:
4252:
4216:
4190:
4154:
4128:
4092:
4056:
4030:
3982:
3928:{\displaystyle S'}
3925:
3900:
3874:
3835:
3818:{\displaystyle S'}
3815:
3786:
3745:and Paul Schupp).
3724:
3698:
3655:
3629:
3565:
3511:
3462:
3423:
3403:
3383:
3363:
3324:
3304:
3266:
3246:
3228:locally free group
3200:
3168:
3148:
3114:
3094:
3074:
3047:
3023:
3008:biautomatic groups
2993:
2933:
2906:
2879:
2852:
2825:
2798:
2771:
2735:
2708:
2681:
2653:
2640:if the associated
2604:
2572:
2540:
2505:
2485:
2453:
2393:
2373:
2347:
2300:
2258:conjugacy problems
2234:
2214:
2187:
2160:
2140:
2120:
2100:
2068:General properties
2053:
2015:
1962:
1924:
1871:
1851:
1621:
1595:
1556:
1536:
1460:
1434:
1395:
1375:
1329:
1291:
1271:
1238:
1214:
1194:
1174:
1141:
1131:, the quotient of
1121:
1101:
1058:
1022:
853:
802:
754:
734:
714:
694:
674:
635:
585:
546:
526:
506:
456:
424:
392:
353:
333:
307:
276:
250:
230:
204:
154:
134:
114:
50:defined by simple
5403:978-88-7642-430-4
5196:10.4171/JCA/1-2-3
5170:Dehornoy, Patrick
4778:Brieskorn, Egbert
4408:of Euclidean type
4387:
4351:
4315:
4279:
4243:
4181:
4119:
4083:
4021:
3838:{\displaystyle S}
3406:{\displaystyle t}
3386:{\displaystyle s}
3327:{\displaystyle M}
3314:defines a matrix
3276:vertices labeled
3269:{\displaystyle n}
3151:{\displaystyle 2}
3117:{\displaystyle W}
3097:{\displaystyle A}
3050:{\displaystyle A}
3026:{\displaystyle A}
2968:fundamental group
2656:{\displaystyle W}
2508:{\displaystyle R}
2396:{\displaystyle S}
2312:fundamental group
2163:{\displaystyle W}
2123:{\displaystyle W}
1874:{\displaystyle n}
1825:
1753:
1559:{\displaystyle S}
1398:{\displaystyle S}
1294:{\displaystyle n}
1241:{\displaystyle W}
1228:. Conversely, if
1217:{\displaystyle R}
1197:{\displaystyle s}
1144:{\displaystyle A}
1124:{\displaystyle A}
952:
951: where
757:{\displaystyle s}
744:, beginning with
737:{\displaystyle m}
717:{\displaystyle t}
697:{\displaystyle s}
549:{\displaystyle t}
529:{\displaystyle s}
356:{\displaystyle S}
310:{\displaystyle S}
288:Artin–Tits monoid
253:{\displaystyle S}
157:{\displaystyle R}
137:{\displaystyle S}
40:Artin–Tits groups
16:(Redirected from
5510:
5489:
5471:
5444:
5435:
5414:
5387:
5370:
5353:
5320:
5319:
5318:
5306:
5304:
5303:
5298:
5267:
5261:
5260:
5235:
5213:
5207:
5206:
5189:
5166:
5160:
5159:
5129:Altobelli, Joe;
5126:
5120:
5119:
5101:
5075:
5069:
5068:
5039:
5033:
5032:
4995:
4989:
4988:
4979:
4969:
4947:
4941:
4940:
4904:Bestvina, Mladen
4900:
4894:
4893:
4864:
4858:
4857:
4825:
4819:
4818:
4774:
4768:
4767:
4727:
4721:
4720:
4711:
4701:
4679:
4673:
4672:
4663:
4653:
4631:
4625:
4624:
4599:
4577:
4571:
4570:
4561:
4536:
4530:
4529:
4488:
4444:
4442:
4441:
4436:
4405:
4403:
4402:
4397:
4395:
4394:
4389:
4388:
4380:
4369:
4367:
4366:
4361:
4359:
4358:
4353:
4352:
4344:
4333:
4331:
4330:
4325:
4323:
4322:
4317:
4316:
4308:
4297:
4295:
4294:
4289:
4287:
4286:
4281:
4280:
4272:
4261:
4259:
4258:
4253:
4251:
4250:
4245:
4244:
4236:
4225:
4223:
4222:
4217:
4199:
4197:
4196:
4191:
4189:
4188:
4183:
4182:
4174:
4163:
4161:
4160:
4155:
4137:
4135:
4134:
4129:
4127:
4126:
4121:
4120:
4112:
4101:
4099:
4098:
4093:
4091:
4090:
4085:
4084:
4076:
4065:
4063:
4062:
4057:
4039:
4037:
4036:
4031:
4029:
4028:
4023:
4022:
4014:
3991:
3989:
3988:
3983:
3978:
3977:
3972:
3969:
3952:
3934:
3932:
3931:
3926:
3924:
3909:
3907:
3906:
3901:
3883:
3881:
3880:
3875:
3867:
3866:
3844:
3842:
3841:
3836:
3824:
3822:
3821:
3816:
3814:
3795:
3793:
3792:
3787:
3733:
3731:
3730:
3725:
3707:
3705:
3704:
3699:
3691:
3690:
3667:extra-large type
3664:
3662:
3661:
3656:
3638:
3636:
3635:
3630:
3622:
3621:
3574:
3572:
3571:
3566:
3520:
3518:
3517:
3512:
3471:
3469:
3468:
3463:
3455:
3454:
3432:
3430:
3429:
3424:
3412:
3410:
3409:
3404:
3392:
3390:
3389:
3384:
3373:if the vertices
3372:
3370:
3369:
3364:
3356:
3355:
3333:
3331:
3330:
3325:
3313:
3311:
3310:
3305:
3275:
3273:
3272:
3267:
3255:
3253:
3252:
3247:
3230:are also common.
3209:
3207:
3206:
3201:
3177:
3175:
3174:
3169:
3157:
3155:
3154:
3149:
3123:
3121:
3120:
3115:
3103:
3101:
3100:
3095:
3083:
3081:
3080:
3075:
3073:
3072:
3056:
3054:
3053:
3048:
3032:
3030:
3029:
3024:
3002:
3000:
2999:
2994:
2992:
2991:
2986:
2957:Egbert Brieskorn
2942:
2940:
2939:
2934:
2932:
2931:
2915:
2913:
2912:
2907:
2905:
2904:
2888:
2886:
2885:
2880:
2878:
2877:
2861:
2859:
2858:
2853:
2851:
2850:
2834:
2832:
2831:
2826:
2824:
2823:
2807:
2805:
2804:
2799:
2797:
2796:
2780:
2778:
2777:
2772:
2761:
2760:
2744:
2742:
2741:
2736:
2734:
2733:
2717:
2715:
2714:
2709:
2707:
2706:
2690:
2688:
2687:
2682:
2680:
2679:
2662:
2660:
2659:
2654:
2613:
2611:
2610:
2605:
2581:
2579:
2578:
2573:
2549:
2547:
2546:
2541:
2514:
2512:
2511:
2506:
2494:
2492:
2491:
2486:
2462:
2460:
2459:
2454:
2452:
2451:
2442:
2441:
2429:
2428:
2419:
2418:
2402:
2400:
2399:
2394:
2382:
2380:
2379:
2374:
2356:
2354:
2353:
2348:
2309:
2307:
2306:
2301:
2243:
2241:
2240:
2235:
2223:
2221:
2220:
2215:
2213:
2212:
2196:
2194:
2193:
2188:
2186:
2185:
2169:
2167:
2166:
2161:
2149:
2147:
2146:
2141:
2129:
2127:
2126:
2121:
2109:
2107:
2106:
2101:
2099:
2098:
2062:
2060:
2059:
2054:
2024:
2022:
2021:
2016:
2008:
2007:
2006:
2005:
1993:
1992:
1971:
1969:
1968:
1963:
1933:
1931:
1930:
1925:
1917:
1916:
1915:
1914:
1902:
1901:
1880:
1878:
1877:
1872:
1860:
1858:
1857:
1852:
1826:
1823:
1821:
1820:
1811:
1810:
1798:
1797:
1788:
1787:
1754:
1751:
1749:
1748:
1739:
1738:
1729:
1728:
1716:
1715:
1706:
1705:
1696:
1695:
1683:
1682:
1658:
1657:
1630:
1628:
1627:
1622:
1604:
1602:
1601:
1596:
1588:
1587:
1565:
1563:
1562:
1557:
1545:
1543:
1542:
1537:
1469:
1467:
1466:
1461:
1443:
1441:
1440:
1435:
1427:
1426:
1404:
1402:
1401:
1396:
1384:
1382:
1381:
1376:
1338:
1336:
1335:
1330:
1300:
1298:
1297:
1292:
1280:
1278:
1277:
1272:
1264:
1263:
1247:
1245:
1244:
1239:
1223:
1221:
1220:
1215:
1203:
1201:
1200:
1195:
1183:
1181:
1180:
1175:
1167:
1166:
1150:
1148:
1147:
1142:
1130:
1128:
1127:
1122:
1110:
1108:
1107:
1102:
1070:symmetric matrix
1067:
1065:
1064:
1059:
1057:
1056:
1031:
1029:
1028:
1023:
988:
987:
969:
968:
953:
950:
945:
944:
943:
942:
907:
906:
905:
904:
862:
860:
859:
854:
840:
839:
811:
809:
808:
803:
792:
791:
763:
761:
760:
755:
743:
741:
740:
735:
723:
721:
720:
715:
703:
701:
700:
695:
683:
681:
680:
675:
673:
672:
644:
642:
641:
636:
594:
592:
591:
586:
578:
577:
555:
553:
552:
547:
535:
533:
532:
527:
515:
513:
512:
507:
465:
463:
462:
457:
433:
431:
430:
425:
401:
399:
398:
393:
385:
384:
362:
360:
359:
354:
342:
340:
339:
334:
316:
314:
313:
308:
285:
283:
282:
277:
259:
257:
256:
251:
239:
237:
236:
231:
213:
211:
210:
205:
163:
161:
160:
155:
143:
141:
140:
135:
123:
121:
120:
115:
38:, also known as
21:
5518:
5517:
5513:
5512:
5511:
5509:
5508:
5507:
5493:
5492:
5448:Flores, Ramon;
5447:
5433:10.5802/wbln.17
5417:
5404:
5373:
5332:
5329:
5327:Further reading
5324:
5323:
5274:
5273:
5269:
5268:
5264:
5215:
5214:
5210:
5168:
5167:
5163:
5128:
5127:
5123:
5077:
5076:
5072:
5041:
5040:
5036:
4997:
4996:
4992:
4949:
4948:
4944:
4902:
4901:
4897:
4866:
4865:
4861:
4827:
4826:
4822:
4776:
4775:
4771:
4731:Deligne, Pierre
4729:
4728:
4724:
4681:
4680:
4676:
4633:
4632:
4628:
4579:
4578:
4574:
4538:
4537:
4533:
4510:10.2307/1969218
4490:
4489:
4485:
4480:
4452:
4412:
4411:
4377:
4372:
4371:
4341:
4336:
4335:
4305:
4300:
4299:
4269:
4264:
4263:
4233:
4228:
4227:
4202:
4201:
4171:
4166:
4165:
4140:
4139:
4109:
4104:
4103:
4073:
4068:
4067:
4042:
4041:
4011:
4006:
4005:
3970:
3962:
3945:
3937:
3936:
3917:
3912:
3911:
3886:
3885:
3852:
3847:
3846:
3827:
3826:
3807:
3802:
3801:
3766:
3765:
3758:
3710:
3709:
3676:
3671:
3670:
3641:
3640:
3607:
3602:
3601:
3593:
3542:
3541:
3497:
3496:
3440:
3435:
3434:
3415:
3414:
3395:
3394:
3375:
3374:
3341:
3336:
3335:
3316:
3315:
3278:
3277:
3258:
3257:
3238:
3237:
3180:
3179:
3160:
3159:
3140:
3139:
3131:
3106:
3105:
3086:
3085:
3064:
3059:
3058:
3039:
3038:
3037:, meaning that
3015:
3014:
3010:(Ruth Charney).
2981:
2976:
2975:
2951:is determined (
2923:
2918:
2917:
2896:
2891:
2890:
2869:
2864:
2863:
2842:
2837:
2836:
2815:
2810:
2809:
2788:
2783:
2782:
2752:
2747:
2746:
2725:
2720:
2719:
2698:
2693:
2692:
2671:
2666:
2665:
2645:
2644:
2633:
2625:
2584:
2583:
2552:
2551:
2520:
2519:
2497:
2496:
2465:
2464:
2443:
2433:
2420:
2410:
2405:
2404:
2385:
2384:
2359:
2358:
2327:
2326:
2277:
2276:
2226:
2225:
2204:
2199:
2198:
2177:
2172:
2171:
2152:
2151:
2132:
2131:
2112:
2111:
2090:
2085:
2084:
2074:Garside methods
2070:
2027:
2026:
1997:
1984:
1979:
1974:
1973:
1936:
1935:
1906:
1893:
1888:
1883:
1882:
1863:
1862:
1824: for
1812:
1802:
1789:
1779:
1752: for
1740:
1730:
1720:
1707:
1697:
1687:
1668:
1649:
1635:
1634:
1607:
1606:
1573:
1568:
1567:
1548:
1547:
1474:
1473:
1446:
1445:
1412:
1407:
1406:
1387:
1386:
1349:
1348:
1345:
1303:
1302:
1283:
1282:
1255:
1250:
1249:
1230:
1229:
1206:
1205:
1186:
1185:
1158:
1153:
1152:
1133:
1132:
1113:
1112:
1081:
1080:
1072:, known as the
1042:
1037:
1036:
973:
954:
928:
923:
890:
885:
868:
867:
831:
814:
813:
783:
766:
765:
746:
745:
726:
725:
706:
705:
686:
685:
664:
647:
646:
597:
596:
563:
558:
557:
538:
537:
518:
517:
468:
467:
436:
435:
404:
403:
370:
365:
364:
345:
344:
319:
318:
317:and, for every
299:
298:
262:
261:
242:
241:
216:
215:
166:
165:
146:
145:
126:
125:
94:
93:
87:
58:. Examples are
28:
23:
22:
15:
12:
11:
5:
5516:
5514:
5506:
5505:
5495:
5494:
5491:
5490:
5445:
5415:
5402:
5371:
5344:(1): 141–158,
5328:
5325:
5322:
5321:
5296:
5293:
5290:
5287:
5284:
5281:
5262:
5226:(1): 231–282,
5208:
5180:(2): 185–228,
5161:
5143:(3): 277–289,
5121:
5092:(3): 486–512.
5070:
5034:
5008:(2): 201–220,
4990:
4960:(2): 246–276,
4942:
4916:(3): 445–470,
4895:
4877:(3): 442–460,
4859:
4841:(4): 671–683,
4820:
4794:(4): 245–271,
4769:
4722:
4674:
4644:(3): 609–637,
4626:
4572:
4531:
4504:(1): 101–126.
4482:
4481:
4479:
4476:
4475:
4474:
4469:
4464:
4461:Artinian group
4458:
4451:
4448:
4447:
4446:
4434:
4431:
4428:
4425:
4422:
4419:
4393:
4386:
4383:
4357:
4350:
4347:
4321:
4314:
4311:
4285:
4278:
4275:
4249:
4242:
4239:
4215:
4212:
4209:
4187:
4180:
4177:
4153:
4150:
4147:
4125:
4118:
4115:
4089:
4082:
4079:
4055:
4052:
4049:
4027:
4020:
4017:
3998:of affine type
3994:
3981:
3976:
3968:
3965:
3961:
3958:
3955:
3951:
3948:
3944:
3923:
3920:
3899:
3896:
3893:
3873:
3870:
3865:
3862:
3859:
3855:
3834:
3813:
3810:
3796:is said to be
3785:
3782:
3779:
3776:
3773:
3757:
3754:
3753:
3752:
3749:
3746:
3735:
3723:
3720:
3717:
3697:
3694:
3689:
3686:
3683:
3679:
3654:
3651:
3648:
3628:
3625:
3620:
3617:
3614:
3610:
3592:
3589:
3588:
3587:
3576:
3564:
3561:
3558:
3555:
3552:
3549:
3538:
3527:direct product
3510:
3507:
3504:
3485:complete graph
3473:
3461:
3458:
3453:
3450:
3447:
3443:
3422:
3402:
3382:
3362:
3359:
3354:
3351:
3348:
3344:
3323:
3303:
3300:
3297:
3294:
3291:
3288:
3285:
3265:
3245:
3231:
3224:semifree group
3199:
3196:
3193:
3190:
3187:
3167:
3147:
3130:
3127:
3126:
3125:
3113:
3093:
3071:
3067:
3046:
3022:
3011:
3004:
2990:
2985:
2964:
2953:Pierre Deligne
2944:
2930:
2926:
2903:
2899:
2876:
2872:
2849:
2845:
2822:
2818:
2795:
2791:
2770:
2767:
2764:
2759:
2755:
2732:
2728:
2705:
2701:
2678:
2674:
2652:
2638:spherical type
2632:
2629:
2624:
2621:
2620:
2619:
2615:
2603:
2600:
2597:
2594:
2591:
2571:
2568:
2565:
2562:
2559:
2539:
2536:
2533:
2530:
2527:
2516:
2504:
2484:
2481:
2478:
2475:
2472:
2450:
2446:
2440:
2436:
2432:
2427:
2423:
2417:
2413:
2392:
2372:
2369:
2366:
2346:
2343:
2340:
2337:
2334:
2323:
2316:
2315:
2299:
2296:
2293:
2290:
2287:
2284:
2272:
2271:
2267:
2266:
2262:
2261:
2252:– solving the
2246:
2245:
2233:
2211:
2207:
2184:
2180:
2159:
2139:
2119:
2097:
2093:
2081:
2069:
2066:
2065:
2064:
2052:
2049:
2046:
2043:
2040:
2037:
2034:
2014:
2011:
2004:
2000:
1996:
1991:
1987:
1982:
1961:
1958:
1955:
1952:
1949:
1946:
1943:
1923:
1920:
1913:
1909:
1905:
1900:
1896:
1891:
1881:strands; here
1870:
1850:
1847:
1844:
1841:
1838:
1835:
1832:
1829:
1819:
1815:
1809:
1805:
1801:
1796:
1792:
1786:
1782:
1778:
1775:
1772:
1769:
1766:
1763:
1760:
1757:
1747:
1743:
1737:
1733:
1727:
1723:
1719:
1714:
1710:
1704:
1700:
1694:
1690:
1686:
1681:
1678:
1675:
1671:
1667:
1664:
1661:
1656:
1652:
1648:
1645:
1642:
1632:
1620:
1617:
1614:
1594:
1591:
1586:
1583:
1580:
1576:
1555:
1535:
1532:
1529:
1526:
1523:
1520:
1517:
1514:
1511:
1508:
1505:
1502:
1499:
1496:
1493:
1490:
1487:
1484:
1481:
1471:
1459:
1456:
1453:
1433:
1430:
1425:
1422:
1419:
1415:
1394:
1374:
1371:
1368:
1365:
1362:
1359:
1356:
1344:
1341:
1328:
1325:
1322:
1319:
1316:
1313:
1310:
1290:
1270:
1267:
1262:
1258:
1237:
1213:
1193:
1173:
1170:
1165:
1161:
1140:
1120:
1100:
1097:
1094:
1091:
1088:
1076:of the group.
1074:Coxeter matrix
1055:
1052:
1049:
1045:
1033:
1032:
1021:
1018:
1015:
1012:
1009:
1006:
1003:
1000:
997:
994:
991:
986:
983:
980:
976:
972:
967:
964:
961:
957:
948:
941:
938:
935:
931:
926:
922:
919:
916:
913:
910:
903:
900:
897:
893:
888:
884:
881:
878:
875:
852:
849:
846:
843:
838:
834:
830:
827:
824:
821:
801:
798:
795:
790:
786:
782:
779:
776:
773:
753:
733:
713:
693:
671:
667:
663:
660:
657:
654:
634:
631:
628:
625:
622:
619:
616:
613:
610:
607:
604:
584:
581:
576:
573:
570:
566:
545:
525:
505:
502:
499:
496:
493:
490:
487:
484:
481:
478:
475:
455:
452:
449:
446:
443:
423:
420:
417:
414:
411:
391:
388:
383:
380:
377:
373:
352:
332:
329:
326:
306:
275:
272:
269:
249:
229:
226:
223:
203:
200:
197:
194:
191:
188:
185:
182:
179:
176:
173:
153:
133:
113:
110:
107:
104:
101:
86:
83:
56:Coxeter groups
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
5515:
5504:
5501:
5500:
5498:
5487:
5483:
5479:
5475:
5470:
5465:
5461:
5457:
5456:
5451:
5446:
5443:
5439:
5434:
5429:
5425:
5421:
5416:
5413:
5409:
5405:
5399:
5395:
5391:
5386:
5381:
5377:
5372:
5369:
5365:
5361:
5357:
5352:
5347:
5343:
5339:
5335:
5334:Charney, Ruth
5331:
5330:
5326:
5317:
5312:
5308:
5291:
5288:
5285:
5279:
5272:Proof of the
5266:
5263:
5259:
5255:
5251:
5247:
5243:
5239:
5234:
5229:
5225:
5221:
5220:
5212:
5209:
5205:
5201:
5197:
5193:
5188:
5183:
5179:
5175:
5171:
5165:
5162:
5158:
5154:
5150:
5146:
5142:
5138:
5137:
5132:
5131:Charney, Ruth
5125:
5122:
5117:
5113:
5109:
5105:
5100:
5095:
5091:
5087:
5086:
5081:
5078:Holt, Derek;
5074:
5071:
5067:
5063:
5059:
5055:
5051:
5047:
5046:
5038:
5035:
5031:
5027:
5023:
5019:
5015:
5011:
5007:
5003:
5002:
4994:
4991:
4987:
4983:
4978:
4973:
4968:
4963:
4959:
4955:
4954:
4946:
4943:
4939:
4935:
4931:
4927:
4923:
4919:
4915:
4911:
4910:
4905:
4899:
4896:
4892:
4888:
4884:
4880:
4876:
4872:
4871:
4863:
4860:
4856:
4852:
4848:
4844:
4840:
4836:
4835:
4830:
4829:Charney, Ruth
4824:
4821:
4817:
4813:
4809:
4805:
4801:
4797:
4793:
4789:
4788:
4783:
4779:
4773:
4770:
4766:
4762:
4758:
4754:
4750:
4746:
4742:
4738:
4737:
4732:
4726:
4723:
4719:
4715:
4710:
4705:
4700:
4695:
4691:
4687:
4686:
4678:
4675:
4671:
4667:
4662:
4657:
4652:
4647:
4643:
4639:
4638:
4630:
4627:
4623:
4619:
4615:
4611:
4607:
4603:
4598:
4593:
4589:
4585:
4584:
4576:
4573:
4569:
4565:
4560:
4555:
4551:
4547:
4546:
4541:
4540:Tits, Jacques
4535:
4532:
4527:
4523:
4519:
4515:
4511:
4507:
4503:
4499:
4498:
4493:
4487:
4484:
4477:
4473:
4470:
4468:
4465:
4462:
4459:
4457:
4454:
4453:
4449:
4429:
4426:
4423:
4417:
4409:
4391:
4384:
4381:
4355:
4348:
4345:
4319:
4312:
4309:
4283:
4276:
4273:
4247:
4240:
4237:
4213:
4210:
4207:
4185:
4178:
4175:
4151:
4148:
4145:
4123:
4116:
4113:
4087:
4080:
4077:
4053:
4050:
4047:
4025:
4018:
4015:
4003:
3999:
3995:
3974:
3966:
3963:
3959:
3956:
3953:
3949:
3946:
3921:
3918:
3897:
3894:
3891:
3868:
3863:
3860:
3857:
3853:
3832:
3811:
3808:
3799:
3780:
3777:
3774:
3763:
3762:
3761:
3755:
3750:
3747:
3744:
3743:Kenneth Appel
3740:
3736:
3721:
3718:
3715:
3695:
3692:
3687:
3684:
3681:
3677:
3668:
3652:
3649:
3646:
3626:
3623:
3618:
3615:
3612:
3608:
3599:
3595:
3594:
3590:
3585:
3581:
3577:
3559:
3556:
3553:
3547:
3539:
3536:
3532:
3528:
3524:
3508:
3505:
3502:
3494:
3493:HNN extension
3490:
3486:
3482:
3478:
3474:
3456:
3451:
3448:
3445:
3441:
3400:
3380:
3360:
3357:
3352:
3349:
3346:
3342:
3321:
3301:
3298:
3295:
3292:
3289:
3286:
3283:
3263:
3236:
3232:
3229:
3225:
3221:
3217:
3213:
3197:
3194:
3191:
3188:
3185:
3145:
3137:
3133:
3132:
3128:
3111:
3091:
3069:
3065:
3044:
3036:
3035:Garside group
3020:
3012:
3009:
3005:
2988:
2973:
2969:
2965:
2962:
2958:
2954:
2950:
2945:
2928:
2924:
2901:
2897:
2874:
2870:
2847:
2843:
2820:
2816:
2793:
2789:
2765:
2757:
2753:
2730:
2726:
2703:
2699:
2676:
2672:
2650:
2643:
2642:Coxeter group
2639:
2635:
2634:
2630:
2628:
2622:
2616:
2598:
2595:
2592:
2566:
2563:
2560:
2534:
2531:
2528:
2517:
2502:
2482:
2479:
2476:
2473:
2470:
2448:
2444:
2438:
2434:
2430:
2425:
2421:
2415:
2411:
2390:
2370:
2367:
2364:
2341:
2338:
2335:
2324:
2321:
2320:
2319:
2313:
2294:
2291:
2288:
2282:
2274:
2273:
2269:
2268:
2264:
2263:
2259:
2255:
2251:
2250:
2249:
2231:
2209:
2205:
2182:
2178:
2157:
2137:
2117:
2095:
2091:
2082:
2079:
2078:
2077:
2075:
2067:
2050:
2047:
2041:
2038:
2035:
2012:
2009:
2002:
1998:
1994:
1989:
1985:
1980:
1959:
1956:
1950:
1947:
1944:
1921:
1918:
1911:
1907:
1903:
1898:
1894:
1889:
1868:
1845:
1842:
1836:
1833:
1830:
1817:
1813:
1807:
1803:
1799:
1794:
1790:
1784:
1780:
1776:
1773:
1770:
1764:
1761:
1758:
1745:
1741:
1735:
1731:
1725:
1721:
1717:
1712:
1708:
1702:
1698:
1692:
1688:
1684:
1679:
1676:
1673:
1669:
1665:
1662:
1659:
1654:
1650:
1643:
1640:
1633:
1618:
1615:
1612:
1592:
1589:
1584:
1581:
1578:
1574:
1553:
1527:
1524:
1521:
1518:
1515:
1512:
1509:
1506:
1503:
1500:
1497:
1491:
1488:
1482:
1479:
1472:
1457:
1454:
1451:
1428:
1423:
1420:
1417:
1413:
1392:
1366:
1363:
1357:
1354:
1347:
1346:
1342:
1340:
1323:
1320:
1317:
1314:
1311:
1288:
1268:
1265:
1260:
1256:
1235:
1227:
1226:Coxeter group
1211:
1191:
1171:
1168:
1163:
1159:
1138:
1118:
1095:
1092:
1089:
1077:
1075:
1071:
1053:
1050:
1047:
1043:
1035:The integers
1019:
1010:
1007:
1004:
1001:
998:
995:
989:
984:
981:
978:
974:
970:
965:
962:
959:
955:
946:
939:
936:
933:
929:
920:
917:
914:
908:
901:
898:
895:
891:
882:
879:
876:
866:
865:
864:
850:
847:
844:
841:
836:
828:
825:
822:
799:
796:
793:
788:
780:
777:
774:
751:
731:
711:
691:
669:
661:
658:
655:
632:
629:
626:
623:
620:
617:
614:
611:
608:
605:
602:
579:
574:
571:
568:
564:
543:
523:
503:
500:
497:
494:
491:
488:
485:
482:
479:
476:
473:
453:
450:
447:
444:
441:
421:
418:
415:
412:
409:
389:
386:
381:
378:
375:
371:
350:
330:
327:
324:
304:
295:
293:
289:
273:
270:
267:
247:
227:
224:
221:
214:for distinct
201:
198:
195:
192:
189:
186:
183:
180:
177:
174:
171:
151:
131:
108:
105:
102:
92:
84:
82:
80:
76:
71:
69:
65:
61:
57:
53:
52:presentations
49:
45:
41:
37:
33:
19:
5503:Braid groups
5459:
5453:
5423:
5419:
5375:
5351:math/0610668
5341:
5337:
5271:
5265:
5223:
5217:
5211:
5177:
5173:
5164:
5140:
5134:
5124:
5089:
5083:
5073:
5052:(1): 15–56,
5049:
5043:
5037:
5005:
4999:
4993:
4957:
4951:
4945:
4913:
4907:
4898:
4874:
4868:
4862:
4838:
4832:
4823:
4791:
4785:
4782:Saito, Kyoji
4772:
4740:
4734:
4725:
4689:
4683:
4677:
4651:math/0102002
4641:
4635:
4629:
4597:math/0003133
4590:(1): 19–36,
4587:
4581:
4575:
4549:
4543:
4534:
4501:
4495:
4486:
4407:
3997:
3935:, the group
3797:
3759:
3666:
3597:
3523:free product
3488:
3334:, for which
3227:
3223:
3219:
3215:
3211:
3210:. The names
3136:right-angled
3135:
2637:
2626:
2317:
2247:
2071:
1078:
1034:
296:
287:
91:presentation
88:
79:Jacques Tits
72:
68:braid groups
43:
39:
36:Artin groups
35:
32:group theory
29:
5462:: 111–129.
5080:Rees, Sarah
4743:: 273–302,
4692:: 739–784,
4492:Artin, Emil
3756:Other types
3521:, with the
3477:free groups
3220:trace group
3216:graph group
2961:Kyoji Saito
60:free groups
5469:1802.04870
5316:1907.11795
5187:1606.08991
4967:1512.06609
4699:1505.02058
4552:: 96–116,
4478:References
3845:such that
3798:of FC type
3598:large type
3472:otherwise.
2949:cohomology
2618:effective.
764:— so that
724:of length
466:such that
85:Definition
75:Emil Artin
5385:1105.1048
5286:π
5233:1312.7770
5099:1003.6007
4424:π
4385:~
4349:~
4313:~
4277:~
4241:~
4211:⩾
4179:~
4149:⩾
4117:~
4081:~
4051:⩾
4019:~
3980:⟩
3960:∩
3954:∣
3943:⟨
3872:∞
3869:≠
3784:⟩
3778:∣
3772:⟨
3719:≠
3693:⩾
3650:≠
3624:⩾
3554:π
3506:−
3460:∞
3421:Γ
3296:…
3244:Γ
3166:∞
2602:⟩
2596:∣
2590:⟨
2570:⟩
2564:∣
2558:⟨
2538:⟩
2532:∣
2526:⟨
2345:⟩
2339:∣
2333:⟨
2289:π
2232:σ
2138:σ
2039:−
1999:σ
1986:σ
1948:−
1908:σ
1895:σ
1849:⟩
1843:⩾
1834:−
1814:σ
1804:σ
1791:σ
1781:σ
1762:−
1742:σ
1732:σ
1722:σ
1709:σ
1699:σ
1689:σ
1685:∣
1677:−
1670:σ
1663:…
1651:σ
1647:⟨
1534:⟩
1525:∈
1513:∣
1492:∣
1486:⟨
1432:∞
1373:⟩
1370:∅
1367:∣
1361:⟨
1318:…
1184:for each
1099:⟩
1093:∣
1087:⟨
1014:∞
1008:…
990:∈
925:⟩
912:⟨
887:⟩
874:⟨
833:⟩
820:⟨
785:⟩
772:⟨
666:⟩
653:⟨
633:…
615:…
583:∞
504:…
486:…
454:…
422:…
387:⩾
202:…
184:…
112:⟩
106:∣
100:⟨
5497:Category
4526:30514042
4450:See also
3967:′
3950:′
3922:′
3884:for all
3812:′
3226:or even
2614:(Paris).
1605:for all
1444:for all
1343:Examples
5486:3874519
5442:3922033
5412:3203644
5368:2322545
5258:3698343
5238:Bibcode
5204:3634782
5157:1755729
5116:2900234
5066:1390670
5030:0700768
5010:Bibcode
4986:3851323
4938:1465330
4918:Bibcode
4891:2546582
4855:1157320
4816:0323910
4796:Bibcode
4765:0422673
4745:Bibcode
4718:1839284
4670:1933791
4622:1839284
4602:Bibcode
4568:0206117
4518:1969218
3739:torsion
1566:; here
1405:; here
5484:
5440:
5410:
5400:
5366:
5256:
5202:
5155:
5114:
5064:
5028:
4984:
4936:
4889:
4853:
4814:
4763:
4716:
4668:
4620:
4566:
4524:
4516:
4370:, and
4164:, and
4002:affine
3580:CAT(0)
3433:, and
2916:, and
2495:is in
1972:, and
292:monoid
124:where
48:groups
5464:arXiv
5380:arXiv
5346:arXiv
5311:arXiv
5228:arXiv
5182:arXiv
5094:arXiv
4962:arXiv
4694:arXiv
4646:arXiv
4592:arXiv
4522:S2CID
4514:JSTOR
3235:graph
3033:is a
2170:into
1224:is a
290:is a
5398:ISBN
4200:for
4138:for
4040:for
3525:and
3393:and
2959:and
2256:and
2254:word
2048:>
2025:for
1934:for
704:and
536:and
434:and
5474:doi
5460:519
5428:doi
5390:doi
5356:doi
5342:125
5246:doi
5224:210
5192:doi
5145:doi
5104:doi
5090:104
5054:doi
5050:110
5018:doi
4972:doi
4958:117
4926:doi
4914:129
4879:doi
4843:doi
4839:292
4804:doi
4753:doi
4704:doi
4690:301
4656:doi
4610:doi
4588:145
4554:doi
4506:doi
3910:in
3825:of
3669:if
3600:if
3256:on
3158:or
2974:in
2463:if
2403:is
2383:of
2150:of
2083:If
1204:of
1079:If
343:in
240:in
42:or
5499::
5482:MR
5480:.
5472:.
5458:.
5438:MR
5436:,
5422:,
5408:MR
5406:,
5396:,
5388:,
5364:MR
5362:,
5354:,
5340:,
5309:,
5254:MR
5252:,
5244:,
5236:,
5222:,
5200:MR
5198:,
5190:,
5176:,
5153:MR
5151:,
5141:79
5139:,
5112:MR
5110:.
5102:.
5088:.
5062:MR
5060:,
5048:,
5026:MR
5024:,
5016:,
5006:72
5004:,
4982:MR
4980:,
4970:,
4956:,
4934:MR
4932:,
4924:,
4912:,
4887:MR
4885:,
4873:,
4851:MR
4849:,
4837:,
4812:MR
4810:,
4802:,
4792:17
4790:,
4780:;
4761:MR
4759:,
4751:,
4741:17
4739:,
4714:MR
4712:,
4702:,
4688:,
4666:MR
4664:,
4654:,
4642:77
4640:,
4618:MR
4616:,
4608:,
4600:,
4586:,
4564:MR
4562:,
4548:,
4520:.
4512:.
4502:48
4500:.
4334:,
4298:,
4262:,
4102:,
4066:,
3537:).
3222:,
3218:,
3214:,
2889:,
2862:,
2835:,
2808:,
2745:,
2718:,
2691:,
1339:.
812:,
66:,
62:,
34:,
5488:.
5476::
5466::
5430::
5424:4
5392::
5382::
5358::
5348::
5313::
5295:)
5292:1
5289:,
5283:(
5280:K
5248::
5240::
5230::
5194::
5184::
5178:1
5147::
5118:.
5106::
5096::
5056::
5020::
5012::
4974::
4964::
4928::
4920::
4881::
4875:2
4845::
4806::
4798::
4755::
4747::
4706::
4696::
4658::
4648::
4612::
4604::
4594::
4556::
4550:4
4528:.
4508::
4433:)
4430:1
4427:,
4421:(
4418:K
4392:2
4382:G
4356:4
4346:F
4320:8
4310:E
4284:7
4274:E
4248:6
4238:E
4214:3
4208:n
4186:n
4176:D
4152:2
4146:n
4124:n
4114:C
4088:n
4078:B
4054:1
4048:n
4026:n
4016:A
3975:2
3964:S
3957:R
3947:S
3919:S
3898:t
3895:,
3892:s
3864:t
3861:,
3858:s
3854:m
3833:S
3809:S
3781:R
3775:S
3734:.
3722:t
3716:s
3696:4
3688:t
3685:,
3682:s
3678:m
3653:t
3647:s
3627:3
3619:t
3616:,
3613:s
3609:m
3563:)
3560:1
3557:,
3551:(
3548:K
3509:1
3503:r
3489:r
3457:=
3452:t
3449:,
3446:s
3442:m
3401:t
3381:s
3361:2
3358:=
3353:t
3350:,
3347:s
3343:m
3322:M
3302:n
3299:,
3293:,
3290:2
3287:,
3284:1
3264:n
3198:s
3195:t
3192:=
3189:t
3186:s
3146:2
3112:W
3092:A
3070:+
3066:A
3045:A
3021:A
3003:.
2989:n
2984:C
2943:.
2929:4
2925:H
2902:3
2898:H
2875:4
2871:F
2848:8
2844:E
2821:7
2817:E
2794:6
2790:E
2769:)
2766:n
2763:(
2758:2
2754:I
2731:n
2727:D
2704:n
2700:B
2677:n
2673:A
2651:W
2599:R
2593:S
2567:R
2561:S
2535:R
2529:S
2503:R
2483:s
2480:t
2477:=
2474:t
2471:s
2449:2
2445:s
2439:2
2435:t
2431:=
2426:2
2422:t
2416:2
2412:s
2391:S
2371:t
2368:,
2365:s
2342:R
2336:S
2298:)
2295:1
2292:,
2286:(
2283:K
2210:+
2206:A
2183:+
2179:A
2158:W
2118:W
2096:+
2092:A
2063:.
2051:1
2045:|
2042:j
2036:i
2033:|
2013:2
2010:=
2003:j
1995:,
1990:i
1981:m
1960:1
1957:=
1954:|
1951:j
1945:i
1942:|
1922:3
1919:=
1912:j
1904:,
1899:i
1890:m
1869:n
1846:2
1840:|
1837:j
1831:i
1828:|
1818:i
1808:j
1800:=
1795:j
1785:i
1777:,
1774:1
1771:=
1768:|
1765:j
1759:i
1756:|
1746:j
1736:i
1726:j
1718:=
1713:i
1703:j
1693:i
1680:1
1674:n
1666:,
1660:,
1655:1
1644:=
1641:G
1631:.
1619:t
1616:,
1613:s
1593:2
1590:=
1585:t
1582:,
1579:s
1575:m
1554:S
1531:}
1528:S
1522:t
1519:,
1516:s
1510:s
1507:t
1504:=
1501:t
1498:s
1495:{
1489:S
1483:=
1480:G
1470:.
1458:t
1455:,
1452:s
1429:=
1424:t
1421:,
1418:s
1414:m
1393:S
1364:S
1358:=
1355:G
1327:}
1324:n
1321:,
1315:,
1312:1
1309:{
1289:n
1269:1
1266:=
1261:2
1257:s
1236:W
1212:R
1192:s
1172:1
1169:=
1164:2
1160:s
1139:A
1119:A
1096:R
1090:S
1054:t
1051:,
1048:s
1044:m
1020:.
1017:}
1011:,
1005:,
1002:3
999:,
996:2
993:{
985:s
982:,
979:t
975:m
971:=
966:t
963:,
960:s
956:m
947:,
940:s
937:,
934:t
930:m
921:s
918:,
915:t
909:=
902:t
899:,
896:s
892:m
883:t
880:,
877:s
851:s
848:t
845:s
842:=
837:3
829:t
826:,
823:s
800:t
797:s
794:=
789:2
781:t
778:,
775:s
752:s
732:m
712:t
692:s
670:m
662:t
659:,
656:s
630:s
627:t
624:s
621:t
618:=
612:t
609:s
606:t
603:s
580:=
575:t
572:,
569:s
565:m
544:t
524:s
501:s
498:t
495:s
492:t
489:=
483:t
480:s
477:t
474:s
451:s
448:t
445:s
442:t
419:t
416:s
413:t
410:s
390:2
382:t
379:,
376:s
372:m
351:S
331:t
328:,
325:s
305:S
274:t
271:,
268:s
248:S
228:t
225:,
222:s
199:s
196:t
193:s
190:t
187:=
181:t
178:s
175:t
172:s
152:R
132:S
109:R
103:S
20:)
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