1289:
Note that the Adian–Rabin theorem also implies that the complement of a Markov property in the class of finitely presentable groups is algorithmically undecidable. For example, the properties of being nontrivial, infinite, nonabelian, etc., for finitely presentable groups are undecidable. However,
389:
1290:
there do exist examples of interesting undecidable properties such that neither these properties nor their complements are Markov. Thus
Collins (1969) proved that the property of being
649:
284:
451:
712:
227:
879:
611:
1046:
1013:
792:
1269:
1234:
1207:
960:
933:
906:
846:
819:
759:
202:
171:
132:
98:
1066:
980:
732:
676:
571:
471:
304:
1537:. Algorithms and classification in combinatorial group theory (Berkeley, CA, 1989), pp. 1–59, Math. Sci. Res. Inst. Publ., 23, Springer, New York, 1992,
1426:
497:, proved by analogous methods. It was also in the semigroup context that Markov introduced the above notion that that group theorists came to call the
501:
of finitely presented groups. This Markov, a prominent Soviet logician, is not to be confused with his father, the famous
Russian probabilist
1076:
The following properties of finitely presented groups are Markov and therefore are algorithmically undecidable by the Adian–Rabin theorem:
1557:
1542:
1436:
1410:
323:
1271:
does not embed into any finitely presentable simple group. Hence being a finitely presentable simple group is a Markov property.)
1562:
1449:
1459:
1567:
1303:
521:
1478:
1335:
652:
539:
616:
251:
1294:
is undecidable for finitely presentable groups, while neither being
Hopfian nor being non-Hopfian are Markov.
397:
1165:
1158:
1151:
25:
1241:
1395:
1308:
1515:
1372:
1237:
1130:
317:. More formally, the conclusion of the Adian–Rabin theorem means that set of all finite presentations
1275:
526:
543:
473:
is a finite set of relations in these generators and their inverses) defining groups with property
29:
681:
210:
1509:
1471:
1236:
to be a finitely presented group with unsolvable word problem whose existence is provided by the
1137:
851:
576:
494:
651:
be a finitely presented group with undecidable word problem, whose existence is provided by the
1538:
1455:
1432:
1406:
67:
1018:
985:
764:
1362:
1144:
314:
37:
1247:
1212:
1185:
938:
911:
884:
824:
797:
737:
180:
149:
110:
76:
1402:
1172:
1109:
1123:
1116:
1051:
965:
717:
661:
556:
538:
In modern sources, the proof of the Adian–Rabin theorem proceeds by a reduction to the
510:
456:
289:
245:
1551:
1291:
1282:
1095:
1081:
547:
506:
502:
478:
174:
1474:, "Невозможность алгорифмов распознавания некоторых свойств ассоциативных систем" .
1329:
Algorithmic unsolvability of problems of recognition of certain properties of groups
1048:, it follows that it is undecidable whether a finitely presented group has property
1387:
1179:
1088:
205:
33:
17:
530:
review of Rabin's 1958 paper containing Rabin's proof of the Adian–Rabin theorem.
489:
The statement of the Adian–Rabin theorem generalizes a similar earlier result for
244:
be a Markov property of finitely presentable groups. Then there does not exist an
1391:
1532:
1102:
490:
237:
In modern sources, the Adian–Rabin theorem is usually stated as follows:
135:
1366:
1431:
Lectures in
Mathematics ETH Zürich. Birkhäuser Verlag, Basel, 1993.
146:
For example, being a finite group is a Markov property: We can take
1491:
1348:
658:
The proof then produces a recursive procedure that, given a word
384:{\displaystyle \langle x_{1},x_{2},x_{3},\dots \mid R\rangle }
24:
is a result that states that most "reasonable" properties of
613:
be as in the definition of the Markov property above. Let
1454:, Graduate Texts in Mathematics, Springer, 4th edition;
1401:
1534:
Decision problems for groups — survey and reflections
1250:
1215:
1188:
1054:
1021:
988:
968:
941:
914:
887:
854:
827:
800:
767:
740:
720:
684:
664:
619:
579:
559:
459:
400:
326:
292:
254:
213:
183:
152:
113:
79:
1368:
Recursive unsolvability of group theoretic problems
1281:Being a group admitting a uniform embedding into a
1263:
1228:
1201:
1060:
1040:
1007:
974:
954:
927:
900:
873:
840:
813:
786:
753:
726:
706:
670:
643:
605:
565:
465:
445:
383:
313:The word 'algorithm' here is used in the sense of
298:
278:
221:
196:
165:
126:
92:
55:of finitely presentable groups is one for which:
138:in any finitely presentable group with property
1493:The Adian-Rabin theorem: an English translation
1350:The Adian-Rabin theorem: an English translation
8:
638:
626:
453:is a fixed countably infinite alphabet, and
378:
327:
273:
261:
233:Precise statement of the Adian–Rabin theorem
306:defined by this presentation has property
107:There exists a finitely presentable group
73:There exists a finitely presentable group
1504:
1502:
1255:
1249:
1220:
1214:
1193:
1187:
1053:
1029:
1020:
996:
987:
967:
946:
940:
919:
913:
892:
886:
862:
853:
832:
826:
805:
799:
775:
766:
745:
739:
719:
695:
683:
663:
644:{\displaystyle G=\langle X\mid R\rangle }
618:
597:
584:
578:
558:
458:
431:
418:
405:
399:
360:
347:
334:
325:
291:
279:{\displaystyle G=\langle X\mid R\rangle }
253:
215:
214:
212:
188:
182:
157:
151:
118:
112:
84:
78:
446:{\displaystyle x_{1},x_{2},x_{3},\dots }
1451:An Introduction to the Theory of Groups
1320:
520:, as defined above, was introduced by
1428:Topics in combinatorial group theory.
734:, outputs a finitely presented group
516:According to Don Collins, the notion
7:
1383:
1381:
1421:
1419:
286:, decides whether or not the group
1015:. Since it is undecidable whether
248:that, given a finite presentation
62:is an abstract property, that is,
14:
1375:(2), vol. 67, 1958, pp. 172–194
16:In the mathematical subject of
1518:, vol. 20, 1969, pp. 235–240.
1:
1481:vol. 77, (1951), pp. 953–956.
1413:; Ch. IV, Theorem 4.1, p. 192
573:be a Markov property and let
134:that cannot be embedded as a
1209:to be the trivial group and
707:{\displaystyle X\cup X^{-1}}
222:{\displaystyle \mathbb {Z} }
1338:vol. 103, 1955, pp. 533–535
874:{\displaystyle w\neq _{G}1}
606:{\displaystyle A_{+},A_{-}}
36:(1955) and, independently,
26:finitely presentable groups
1584:
1511:On recognizing Hopf groups
1479:Doklady Akademii Nauk SSSR
1397:Combinatorial group theory
1336:Doklady Akademii Nauk SSSR
1304:Higman's embedding theorem
1558:Theorems in group theory
1274:Being a group of finite
1164:Being a group of finite
32:. The theorem is due to
1462:; Theorem 12.32, p. 469
1166:cohomological dimension
1159:residually finite group
1041:{\displaystyle w=_{G}1}
1008:{\displaystyle w=_{G}1}
787:{\displaystyle w=_{G}1}
1563:Geometric group theory
1265:
1230:
1203:
1062:
1042:
1009:
976:
956:
929:
902:
875:
842:
815:
788:
755:
728:
708:
672:
645:
607:
567:
467:
447:
385:
300:
280:
223:
198:
167:
128:
94:
1516:Archiv der Mathematik
1490:C.-F. Nyberg-Brodda,
1373:Annals of Mathematics
1347:C.-F. Nyberg-Brodda,
1266:
1264:{\displaystyle A_{-}}
1238:Novikov-Boone theorem
1231:
1229:{\displaystyle A_{-}}
1204:
1202:{\displaystyle A_{+}}
1152:solvable word problem
1150:Being a group with a
1131:word-hyperbolic group
1063:
1043:
1010:
977:
957:
955:{\displaystyle G_{w}}
930:
928:{\displaystyle A_{-}}
903:
901:{\displaystyle G_{w}}
876:
843:
841:{\displaystyle A_{+}}
816:
814:{\displaystyle G_{w}}
789:
756:
754:{\displaystyle G_{w}}
729:
709:
673:
653:Novikov–Boone theorem
646:
608:
568:
540:Novikov–Boone theorem
468:
448:
386:
301:
281:
224:
199:
197:{\displaystyle A_{-}}
168:
166:{\displaystyle A_{+}}
129:
127:{\displaystyle A_{-}}
95:
93:{\displaystyle A_{+}}
1568:Undecidable problems
1276:asymptotic dimension
1248:
1213:
1186:
1052:
1019:
986:
966:
939:
935:as a subgroup. Thus
912:
885:
852:
825:
798:
765:
738:
718:
682:
662:
617:
577:
557:
544:amalgamated products
542:via a clever use of
527:Mathematical Reviews
457:
398:
324:
290:
252:
211:
181:
150:
111:
77:
28:are algorithmically
1531:C. F. Miller, III,
1508:Donald J. Collins,
1439:; Theorem 5, p. 112
1242:Kuznetsov's theorem
204:to be the infinite
66:is preserved under
22:Adian–Rabin theorem
1261:
1226:
1199:
1138:torsion-free group
1058:
1038:
1005:
972:
952:
925:
898:
871:
838:
811:
784:
751:
724:
704:
678:in the generators
668:
641:
603:
563:
495:Andrey Markov, Jr.
463:
443:
381:
296:
276:
219:
194:
163:
124:
90:
1309:Bass–Serre theory
1061:{\displaystyle P}
975:{\displaystyle P}
821:is isomorphic to
727:{\displaystyle G}
671:{\displaystyle w}
566:{\displaystyle P}
534:Idea of the proof
466:{\displaystyle R}
299:{\displaystyle G}
68:group isomorphism
1575:
1519:
1506:
1497:
1488:
1482:
1477:
1469:
1463:
1446:
1440:
1423:
1414:
1405:, Berlin, 2001.
1385:
1376:
1363:Michael O. Rabin
1360:
1354:
1345:
1339:
1334:
1325:
1270:
1268:
1267:
1262:
1260:
1259:
1235:
1233:
1232:
1227:
1225:
1224:
1208:
1206:
1205:
1200:
1198:
1197:
1182:. (One can take
1145:polycyclic group
1067:
1065:
1064:
1059:
1047:
1045:
1044:
1039:
1034:
1033:
1014:
1012:
1011:
1006:
1001:
1000:
981:
979:
978:
973:
961:
959:
958:
953:
951:
950:
934:
932:
931:
926:
924:
923:
907:
905:
904:
899:
897:
896:
880:
878:
877:
872:
867:
866:
847:
845:
844:
839:
837:
836:
820:
818:
817:
812:
810:
809:
793:
791:
790:
785:
780:
779:
760:
758:
757:
752:
750:
749:
733:
731:
730:
725:
713:
711:
710:
705:
703:
702:
677:
675:
674:
669:
650:
648:
647:
642:
612:
610:
609:
604:
602:
601:
589:
588:
572:
570:
569:
564:
511:Markov processes
485:Historical notes
472:
470:
469:
464:
452:
450:
449:
444:
436:
435:
423:
422:
410:
409:
390:
388:
387:
382:
365:
364:
352:
351:
339:
338:
315:recursion theory
305:
303:
302:
297:
285:
283:
282:
277:
228:
226:
225:
220:
218:
203:
201:
200:
195:
193:
192:
177:and we can take
172:
170:
169:
164:
162:
161:
133:
131:
130:
125:
123:
122:
99:
97:
96:
91:
89:
88:
38:Michael O. Rabin
1583:
1582:
1578:
1577:
1576:
1574:
1573:
1572:
1548:
1547:
1528:
1526:Further reading
1523:
1522:
1507:
1500:
1489:
1485:
1475:
1470:
1466:
1448:Joseph Rotman,
1447:
1443:
1424:
1417:
1403:Springer-Verlag
1386:
1379:
1361:
1357:
1346:
1342:
1332:
1326:
1322:
1317:
1300:
1251:
1246:
1245:
1216:
1211:
1210:
1189:
1184:
1183:
1173:automatic group
1110:nilpotent group
1074:
1050:
1049:
1025:
1017:
1016:
992:
984:
983:
982:if and only if
964:
963:
942:
937:
936:
915:
910:
909:
888:
883:
882:
858:
850:
849:
828:
823:
822:
801:
796:
795:
771:
763:
762:
741:
736:
735:
716:
715:
691:
680:
679:
660:
659:
615:
614:
593:
580:
575:
574:
555:
554:
536:
518:Markov property
499:Markov property
487:
455:
454:
427:
414:
401:
396:
395:
356:
343:
330:
322:
321:
288:
287:
250:
249:
235:
209:
208:
184:
179:
178:
153:
148:
147:
114:
109:
108:
80:
75:
74:
50:Markov property
46:
44:Markov property
12:
11:
5:
1581:
1579:
1571:
1570:
1565:
1560:
1550:
1549:
1546:
1545:
1527:
1524:
1521:
1520:
1498:
1483:
1464:
1441:
1415:
1377:
1355:
1340:
1319:
1318:
1316:
1313:
1312:
1311:
1306:
1299:
1296:
1287:
1286:
1279:
1272:
1258:
1254:
1223:
1219:
1196:
1192:
1176:
1169:
1162:
1155:
1148:
1141:
1134:
1127:
1124:amenable group
1120:
1117:solvable group
1113:
1106:
1099:
1092:
1085:
1073:
1070:
1057:
1037:
1032:
1028:
1024:
1004:
999:
995:
991:
971:
949:
945:
922:
918:
895:
891:
870:
865:
861:
857:
835:
831:
808:
804:
783:
778:
774:
770:
748:
744:
723:
701:
698:
694:
690:
687:
667:
640:
637:
634:
631:
628:
625:
622:
600:
596:
592:
587:
583:
562:
548:HNN extensions
535:
532:
486:
483:
462:
442:
439:
434:
430:
426:
421:
417:
413:
408:
404:
392:
391:
380:
377:
374:
371:
368:
363:
359:
355:
350:
346:
342:
337:
333:
329:
295:
275:
272:
269:
266:
263:
260:
257:
234:
231:
217:
191:
187:
160:
156:
144:
143:
121:
117:
105:
100:with property
87:
83:
71:
45:
42:
13:
10:
9:
6:
4:
3:
2:
1580:
1569:
1566:
1564:
1561:
1559:
1556:
1555:
1553:
1544:
1543:0-387-97685-X
1540:
1536:
1535:
1530:
1529:
1525:
1517:
1513:
1512:
1505:
1503:
1499:
1495:
1494:
1487:
1484:
1480:
1473:
1468:
1465:
1461:
1457:
1453:
1452:
1445:
1442:
1438:
1437:3-7643-2921-1
1434:
1430:
1429:
1425:G. Baumslag.
1422:
1420:
1416:
1412:
1411:3-540-41158-5
1408:
1404:
1400:
1398:
1393:
1389:
1384:
1382:
1378:
1374:
1370:
1369:
1364:
1359:
1356:
1352:
1351:
1344:
1341:
1337:
1330:
1327:S. I. Adian,
1324:
1321:
1314:
1310:
1307:
1305:
1302:
1301:
1297:
1295:
1293:
1284:
1283:Hilbert space
1280:
1277:
1273:
1256:
1252:
1244:implies that
1243:
1239:
1221:
1217:
1194:
1190:
1181:
1177:
1174:
1170:
1167:
1163:
1160:
1156:
1153:
1149:
1146:
1142:
1139:
1135:
1132:
1128:
1125:
1121:
1118:
1114:
1111:
1107:
1104:
1100:
1097:
1096:abelian group
1093:
1090:
1086:
1083:
1082:trivial group
1079:
1078:
1077:
1071:
1069:
1055:
1035:
1030:
1026:
1022:
1002:
997:
993:
989:
969:
962:has property
947:
943:
920:
916:
893:
889:
868:
863:
859:
855:
833:
829:
806:
802:
781:
776:
772:
768:
761:such that if
746:
742:
721:
699:
696:
692:
688:
685:
665:
656:
654:
635:
632:
629:
623:
620:
598:
594:
590:
585:
581:
560:
551:
549:
545:
541:
533:
531:
529:
528:
523:
522:William Boone
519:
514:
512:
508:
507:Markov chains
504:
503:Andrey Markov
500:
496:
492:
484:
482:
480:
479:recursive set
476:
460:
440:
437:
432:
428:
424:
419:
415:
411:
406:
402:
375:
372:
369:
366:
361:
357:
353:
348:
344:
340:
335:
331:
320:
319:
318:
316:
311:
309:
293:
270:
267:
264:
258:
255:
247:
243:
238:
232:
230:
207:
189:
185:
176:
175:trivial group
158:
154:
141:
137:
119:
115:
106:
103:
85:
81:
72:
69:
65:
61:
58:
57:
56:
54:
51:
43:
41:
39:
35:
31:
27:
23:
19:
1533:
1510:
1492:
1486:
1476:(in Russian)
1467:
1450:
1444:
1427:
1396:
1388:Roger Lyndon
1367:
1358:
1349:
1343:
1333:(in Russian)
1328:
1323:
1288:
1180:simple group
1089:finite group
1075:
1072:Applications
657:
552:
537:
525:
517:
515:
498:
488:
474:
393:
312:
307:
241:
239:
236:
206:cyclic group
145:
139:
101:
63:
59:
52:
49:
47:
34:Sergei Adian
21:
18:group theory
15:
1392:Paul Schupp
513:are named.
505:after whom
477:, is not a
30:undecidable
1552:Categories
1460:0387942858
1315:References
1103:free group
1080:Being the
491:semigroups
173:to be the
1472:A. Markov
1257:−
1222:−
1171:Being an
1094:Being an
921:−
908:contains
860:≠
848:, and if
697:−
689:∪
639:⟩
633:∣
627:⟨
599:−
441:…
379:⟩
373:∣
370:⋯
328:⟨
274:⟩
268:∣
262:⟨
246:algorithm
190:−
120:−
1298:See also
1178:Being a
1157:Being a
1143:Being a
1136:Being a
1129:Being a
1122:Being a
1115:Being a
1108:Being a
1101:Being a
1087:Being a
394:(where
136:subgroup
40:(1958).
1496:. 2022.
1353:. 2022.
1292:Hopfian
1240:. Then
524:in his
1541:
1458:
1435:
1409:
20:, the
881:then
794:then
1539:ISBN
1456:ISBN
1433:ISBN
1407:ISBN
1390:and
553:Let
546:and
509:and
240:Let
714:of
493:by
1554::
1514:,
1501:^
1418:^
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