677:
912:
Wise (2005) claimed that the strengthened Hanna
Neumann conjecture implies another long-standing group-theoretic conjecture which says that every one-relator group with torsion is
1157:
811:
782:
863:
Khan (2002) and, independently, Meakin and Weil (2002), showed that the conclusion of the strengthened Hanna
Neumann conjecture holds if one of the subgroups
1277:
1306:
1031:
493:
745:
Warren Dicks (1994) established the equivalence of the strengthened Hanna
Neumann conjecture and a graph-theoretic statement that he called the
960:
682:
The strengthened Hanna
Neumann conjecture was proved in 2011 by Joel Friedman. Shortly after, another proof was given by Igor Mineyev.
1272:
1199:
1068:
1190:
Combinatorial and geometric group theory (New York, 2000/Hoboken, NJ, 2001), 155–170, Contemporary
Mathematics, vol. 296,
977:"Sheaves on Graphs, Their Homological Invariants, and a Proof of the Hanna Neumann Conjecture: With an Appendix by Warren Dicks"
691:
In 1971 Burns improved Hanna
Neumann's 1957 bound and proved that under the same assumptions as in Hanna Neumann's paper one has
1301:
1191:
894:
Ivanov and Dicks and Ivanov obtained analogs and generalizations of Hanna
Neumann's results for the intersection of
1085:
1011:
308:
53:
In 2017, a third proof of the
Strengthened Hanna Neumann conjecture, based on homological arguments inspired by
1063:
Groups–Canberra 1989, pp. 161–170. Lecture Notes in
Mathematics, vol. 1456, Springer, Berlin, 1990;
1139:
1121:
1103:
921:
917:
429:
70:
32:
739:
722:(1992) established the strengthened Hanna Neumann Conjecture for the case where at least one of the subgroups
1247:
716:
In a 1990 paper, Walter
Neumann formulated the strengthened Hanna Neumann conjecture (see statement above).
193:
She also conjectured that the factor of 2 in the above inequality is not necessary and that one always has
1263:
Mathematical Proceedings of the Cambridge Philosophical Society, vol. 144 (2008), no. 3, pp. 511–534
934:
1296:
1152:
844:
established the strengthened Hanna Neumann conjecture for the case where at least one of the subgroups
738:) has rank two. As most other approaches to the Hanna Neumann conjecture, Tardos used the technique of
174:
In a 1957 addendum, Hanna Neumann further improved this bound to show that under the above assumptions
1217:
356:
1216:
Proceedings of the Conference on Geometric and Combinatorial Group Theory, Part I (Haifa, 2000).
1170:
841:
1195:
1136:
Equivalence of the strengthened Hanna Neumann conjecture and the amalgamated graph conjecture.
1064:
976:
1233:
International Journal of Algebra and Computation, vol. 11 (2001), no. 3, pp. 281–290
787:
409:
401:
328:
300:
81:
66:
28:
719:
767:
484:
1290:
764:), then, in a certain statistical meaning, for a generic finitely generated subgroup
148:
43:
992:
906:
375:
20:
1274:
The Coherence of One-Relator Groups with Torsion and the Hanna Neumann Conjecture.
1005:
54:
1188:
Positively generated subgroups of free groups and the Hanna Neumann conjecture.
1244:
On the Kurosh rank of the intersection of subgroups in free products of groups
1211:
433:
332:
324:
320:
241:
77:
39:
463: ≠ {1}. Suppose that at least one such double coset exists and let
883:, that is, generated by a finite set of words that involve only elements of
1007:
Approximation by subgroups of finite index and the Hanna Neumann conjecture
672:{\displaystyle \sum _{i=1}^{n}\leq ({\rm {rank}}(H)-1)({\rm {rank}}(K)-1).}
1213:
Subgroups of free groups: a contribution to the Hanna Neumann conjecture.
895:
382:
316:
93:
73:
35:
1134:
1116:
1098:
1080:
1082:
On the intersection of finitely generated subgroups of a free group.
1047:
Publicationes Mathematicae Debrecen, vol. 4 (1956), 186–189.
829:. Thus, the strengthened Hanna Neumann conjecture holds for every
1261:
On the intersection of free subgroups in free products of groups.
50:) was proved independently by Joel Friedman and by Igor Mineyev.
1177:
Journal of Group Theory, vol. 4 (2001), no. 2, pp. 113–151
1061:
On intersections of finitely generated subgroups of free groups.
958:
On the intersection of finitely generated free groups. Addendum.
46:
in 1957. In 2011, a strengthened version of the conjecture (see
742:
for analyzing subgroups of free groups and their intersections.
479:
be all the distinct representatives of such double cosets. The
443:) then there exist at most finitely many double coset classes
65:
The subject of the conjecture was originally motivated by a
335:
is equal to the size of any free basis of that free group.
57:
considerations, was published by Andrei Jaikin-Zapirain.
47:
1154:
A property of subgroups of infinite index in a free group
1231:
Intersecting free subgroups in free products of groups.
993:"Submultiplicativity and the Hanna Neumann Conjecture."
1045:
On the intersection of finitely generated free groups.
1029:
On the intersection of finitely generated free groups.
756:
is a finitely generated subgroup of infinite index in
240:) be two nontrivial finitely generated subgroups of a
790:
770:
496:
1175:
The rank three case of the Hanna Neumann conjecture.
805:
776:
671:
80:is always finitely generated, that is, has finite
1100:On the intersection of subgroups of a free group.
979:Mem. Amer. Math. Soc., 233 (2015), no. 1100.
8:
69:who proved that the intersection of any two
1278:Bulletin of the London Mathematical Society
1250:, vol. 218 (2008), no. 2, pp. 465–484
1142:, vol. 117 (1994), no. 3, pp. 373–389
151:improved this bound by showing that :
1280:, vol. 37 (2005), no. 5, pp. 697–705
1124:, vol. 71 (1983), no. 3, pp. 551–565
1106:, vol. 108 (1992), no. 1, pp. 29–36.
1032:Journal of the London Mathematical Society
995:Ann. of Math., 175 (2012), no. 1, 393-414.
789:
769:
686:Partial results and other generalizations
633:
632:
596:
595:
568:
563:
550:
522:
521:
512:
501:
495:
271:. The conjecture says that in this case
971:
969:
987:
985:
946:
952:
950:
487:(1990), states that in this situation
84:. In this paper Howson proved that if
1088:, vol. 119 (1971), pp. 121–130.
481:strengthened Hanna Neumann conjecture
339:Strengthened Hanna Neumann conjecture
7:
1055:
1053:
1034:, vol. 29 (1954), pp. 428–434
961:Publicationes Mathematicae Debrecen
212:This statement became known as the
1220:, vol. 94 (2002), pp. 33–43.
752:Arzhantseva (2000) proved that if
643:
640:
637:
634:
606:
603:
600:
597:
532:
529:
526:
523:
432:subgroups of a finitely generated
307:, that is, the smallest size of a
14:
1307:Conjectures that have been proved
1259:Warren Dicks, and S. V. Ivanov.
19:In the mathematical subject of
1018:(2017), no. 10, pp. 1955-1987
800:
794:
663:
654:
648:
629:
626:
617:
611:
592:
586:
577:
537:
518:
279:) − 1 ≤ (rank(
42:. The conjecture was posed by
1:
1192:American Mathematical Society
112: ≥ 1 then the rank
1160:128 (2000), 3205–3210.
920:subgroup in such a group is
747:amalgamated graph conjecture
283:) − 1)(rank(
31:of the intersection of two
16:Proposition in group theory
1323:
1118:Topology of finite graphs.
1086:Mathematische Zeitschrift
1012:Duke Mathematical Journal
860:) has rank at most three.
821: = {1} for all
740:Stallings subgroup graphs
27:is a statement about the
1210:J. Meakin, and P. Weil.
1194:, Providence, RI, 2002;
1140:Inventiones Mathematicae
1122:Inventiones Mathematicae
1104:Inventiones Mathematicae
1004:Andrei Jaikin-Zapirain,
483:, formulated by her son
214:Hanna Neumann conjecture
200: − 1 ≤ (
158: − 1 ≤ 2
131: − 1 ≤ 2
25:Hanna Neumann conjecture
1248:Advances in Mathematics
963:, vol. 5 (1957), p. 128
408:and thus have the same
355:are two subgroups of a
287:) − 1).
263:be the intersection of
1302:Geometric group theory
1158:Proc. Amer. Math. Soc.
935:Geometric group theory
807:
778:
673:
517:
412:. It is known that if
208: − 1).
204: − 1)(
67:1954 theorem of Howson
808:
779:
674:
497:
881:positively generated
806:{\displaystyle F(X)}
788:
768:
707: − 2
703: − 3
494:
1218:Geometriae Dedicata
1151:G. N. Arzhantseva.
1115:John R. Stallings.
576:
379:HaK = HbK
166: −
162: −
139: −
135: −
108: ≥ 1 and
1169:Warren Dicks, and
922:finitely presented
918:finitely generated
909:of several groups.
840:In 2001 Dicks and
803:
774:
669:
559:
430:finitely generated
295:the quantity rank(
104:) of finite ranks
71:finitely generated
33:finitely generated
1079:Robert G. Burns.
777:{\displaystyle K}
291:Here for a group
1314:
1281:
1270:
1264:
1257:
1251:
1240:
1234:
1227:
1221:
1208:
1202:
1184:
1178:
1167:
1161:
1149:
1143:
1131:
1125:
1113:
1107:
1095:
1089:
1077:
1071:
1059:Walter Neumann.
1057:
1048:
1041:
1035:
1025:
1019:
1002:
996:
989:
980:
973:
964:
954:
916:(that is, every
812:
810:
809:
804:
783:
781:
780:
775:
678:
676:
675:
670:
647:
646:
610:
609:
575:
567:
555:
554:
536:
535:
516:
511:
374:define the same
220:Formal statement
147:In a 1956 paper
96:of a free group
1322:
1321:
1317:
1316:
1315:
1313:
1312:
1311:
1287:
1286:
1285:
1284:
1271:
1267:
1258:
1254:
1241:
1237:
1228:
1224:
1209:
1205:
1185:
1181:
1171:Edward Formanek
1168:
1164:
1150:
1146:
1132:
1128:
1114:
1110:
1096:
1092:
1078:
1074:
1058:
1051:
1043:Hanna Neumann.
1042:
1038:
1026:
1022:
1003:
999:
990:
983:
975:Joel Friedman,
974:
967:
956:Hanna Neumann.
955:
948:
943:
931:
786:
785:
766:
765:
711: + 4.
688:
546:
492:
491:
478:
469:
341:
327:itself and the
323:is known to be
222:
63:
17:
12:
11:
5:
1320:
1318:
1310:
1309:
1304:
1299:
1289:
1288:
1283:
1282:
1265:
1252:
1242:S. V. Ivanov.
1235:
1229:S. V. Ivanov.
1222:
1203:
1179:
1162:
1144:
1133:Warren Dicks.
1126:
1108:
1097:Gábor Tardos.
1090:
1072:
1049:
1036:
1027:A. G. Howson.
1020:
997:
991:Igor Minevev,
981:
965:
945:
944:
942:
939:
938:
937:
930:
927:
926:
925:
910:
892:
861:
838:
833:and a generic
802:
799:
796:
793:
773:
750:
743:
717:
713:
712:
693:
692:
687:
684:
680:
679:
668:
665:
662:
659:
656:
653:
650:
645:
642:
639:
636:
631:
628:
625:
622:
619:
616:
613:
608:
605:
602:
599:
594:
591:
588:
585:
582:
579:
574:
571:
566:
562:
558:
553:
549:
545:
542:
539:
534:
531:
528:
525:
520:
515:
510:
507:
504:
500:
485:Walter Neumann
474:
467:
340:
337:
309:generating set
289:
288:
221:
218:
210:
209:
191:
190:
181:− 1 ≤ 2(
172:
171:
145:
144:
62:
59:
15:
13:
10:
9:
6:
4:
3:
2:
1319:
1308:
1305:
1303:
1300:
1298:
1295:
1294:
1292:
1279:
1276:
1275:
1269:
1266:
1262:
1256:
1253:
1249:
1245:
1239:
1236:
1232:
1226:
1223:
1219:
1215:
1214:
1207:
1204:
1201:
1200:0-8218-2822-3
1197:
1193:
1189:
1183:
1180:
1176:
1172:
1166:
1163:
1159:
1156:
1155:
1148:
1145:
1141:
1138:
1137:
1130:
1127:
1123:
1120:
1119:
1112:
1109:
1105:
1102:
1101:
1094:
1091:
1087:
1084:
1083:
1076:
1073:
1070:
1069:3-540-53475-X
1066:
1062:
1056:
1054:
1050:
1046:
1040:
1037:
1033:
1030:
1024:
1021:
1017:
1013:
1009:
1008:
1001:
998:
994:
988:
986:
982:
978:
972:
970:
966:
962:
959:
953:
951:
947:
940:
936:
933:
932:
928:
923:
919:
915:
911:
908:
904:
900:
897:
893:
890:
886:
882:
878:
874:
870:
866:
862:
859:
855:
851:
847:
843:
839:
836:
832:
828:
824:
820:
817: ∩
816:
797:
791:
771:
763:
759:
755:
751:
748:
744:
741:
737:
733:
729:
725:
721:
718:
715:
714:
710:
706:
702:
698:
695:
694:
690:
689:
685:
683:
666:
660:
657:
651:
623:
620:
614:
589:
583:
580:
572:
569:
564:
560:
556:
551:
547:
543:
540:
513:
508:
505:
502:
498:
490:
489:
488:
486:
482:
477:
473:
466:
462:
459: ∩
458:
454:
450:
446:
442:
438:
435:
431:
427:
423:
419:
415:
411:
407:
403:
399:
396: ∩
395:
391:
388: ∩
387:
384:
380:
377:
373:
369:
365:
361:
358:
354:
350:
346:
338:
336:
334:
330:
326:
322:
318:
314:
310:
306:
302:
298:
294:
286:
282:
278:
274:
273:
272:
270:
266:
262:
259: ∩
258:
255: =
254:
250:
246:
243:
239:
235:
231:
227:
219:
217:
215:
207:
203:
199:
196:
195:
194:
188:
184:
180:
177:
176:
175:
169:
165:
161:
157:
154:
153:
152:
150:
149:Hanna Neumann
142:
138:
134:
130:
127:
126:
125:
123:
120: ∩
119:
115:
111:
107:
103:
99:
95:
91:
87:
83:
79:
75:
72:
68:
60:
58:
56:
51:
49:
45:
44:Hanna Neumann
41:
37:
34:
30:
26:
22:
1297:Group theory
1273:
1268:
1260:
1255:
1243:
1238:
1230:
1225:
1212:
1206:
1187:
1186:Bilal Khan.
1182:
1174:
1165:
1153:
1147:
1135:
1129:
1117:
1111:
1099:
1093:
1081:
1075:
1060:
1044:
1039:
1028:
1023:
1015:
1006:
1000:
957:
913:
907:free product
902:
898:
888:
884:
880:
876:
872:
868:
864:
857:
853:
849:
845:
834:
830:
826:
822:
818:
814:
761:
757:
753:
746:
735:
731:
727:
723:
708:
704:
700:
696:
681:
480:
475:
471:
464:
460:
456:
455:) such that
452:
448:
444:
440:
436:
425:
421:
417:
413:
405:
397:
393:
389:
385:
378:
376:double coset
371:
367:
363:
359:
352:
348:
344:
342:
312:
304:
296:
292:
290:
284:
280:
276:
268:
264:
260:
256:
252:
248:
244:
237:
233:
229:
225:
223:
213:
211:
205:
201:
197:
192:
186:
182:
178:
173:
167:
163:
159:
155:
146:
140:
136:
132:
128:
121:
117:
113:
109:
105:
101:
97:
89:
85:
64:
52:
24:
21:group theory
18:
891:as letters.
887:but not of
189:− 1).
185:− 1)(
124:satisfies:
55:pro-p-group
1291:Categories
941:References
813:, we have
434:free group
333:free group
321:free group
251:) and let
242:free group
78:free group
40:free group
896:subgroups
658:−
621:−
590:≤
581:−
570:−
544:∩
499:∑
402:conjugate
383:subgroups
381:then the
299:) is the
94:subgroups
74:subgroups
36:subgroups
929:See also
914:coherent
842:Formanek
317:subgroup
315:. Every
362:and if
61:History
1198:
1067:
720:Tardos
428:) are
23:, the
905:of a
879:) is
470:,...,
357:group
331:of a
319:of a
275:rank(
76:of a
48:below
38:of a
1196:ISBN
1065:ISBN
901:and
848:and
726:and
410:rank
400:are
392:and
329:rank
325:free
311:for
301:rank
267:and
224:Let
92:are
88:and
82:rank
29:rank
1016:166
871:of
852:of
825:in
819:gKg
784:in
730:of
699:≤ 2
461:aKa
447:in
445:HaK
404:in
398:bKb
390:aKa
343:If
303:of
116:of
1293::
1246:.
1173:.
1052:^
1014:,
1010:,
984:^
968:^
949:^
924:).
867:,
701:mn
420:≤
416:,
370:∈
366:,
351:≤
347:,
232:≤
228:,
216:.
164:2m
160:mn
133:mn
903:K
899:H
889:X
885:X
877:X
875:(
873:F
869:K
865:H
858:X
856:(
854:F
850:K
846:H
837:.
835:K
831:H
827:F
823:g
815:H
801:)
798:X
795:(
792:F
772:K
762:X
760:(
758:F
754:H
749:.
736:X
734:(
732:F
728:K
724:H
709:n
705:m
697:s
667:.
664:)
661:1
655:)
652:K
649:(
644:k
641:n
638:a
635:r
630:(
627:)
624:1
618:)
615:H
612:(
607:k
604:n
601:a
598:r
593:(
587:]
584:1
578:)
573:1
565:i
561:a
557:K
552:i
548:a
541:H
538:(
533:k
530:n
527:a
524:r
519:[
514:n
509:1
506:=
503:i
476:n
472:a
468:1
465:a
457:H
453:X
451:(
449:F
441:X
439:(
437:F
426:X
424:(
422:F
418:K
414:H
406:G
394:H
386:H
372:G
368:b
364:a
360:G
353:G
349:K
345:H
313:G
305:G
297:G
293:G
285:K
281:H
277:L
269:K
265:H
261:K
257:H
253:L
249:X
247:(
245:F
238:X
236:(
234:F
230:K
226:H
206:n
202:m
198:s
187:n
183:m
179:s
170:.
168:n
156:s
143:.
141:n
137:m
129:s
122:K
118:H
114:s
110:m
106:n
102:X
100:(
98:F
90:K
86:H
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