251:
845:, i.e. can be expressed as a finite product of easier subgroups, such as the elementary subgroups consisting of matrices differing from the identity matrix in one given off-diagonal position.
817:
Historically property (T) was established for discrete groups Γ by embedding them as lattices in real or p-adic Lie groups with property (T). There are now several direct methods available.
730:,1) is a simple Lie group of real rank 1 that has property (T). By Kazhdan's theorem, lattices in this group have property (T). This construction is significant because these lattices are
1048:
993:, that is, graphs with the property that every subset has a uniformly large "boundary". This connection led to a number of recent studies giving an explicit estimate of
952:
and has been successfully implemented by several researchers. It is based on the algebraic characterization of property (T) in terms of an inequality in the real
734:; thus, there are groups that are hyperbolic and have property (T). Explicit examples of groups in this category are provided by arithmetic lattices in Sp(
178:
84:, property (T) can often be checked even when there is little or no explicit knowledge of the unitary dual. Property (T) has important applications to
1374:
1427:
461:
The equivalence of (4) and (5) (Property (FH)) is the
Delorme-Guichardet theorem. The fact that (5) implies (4) requires the assumption that
458:, ε)-invariant unit vector and does not have an invariant vector. Look at the direct sum of all such representation and that will negate (4).
1321:
1250:
302:
1067:
observed that the positivity of the bottom of the spectrum of a "twisted" Laplacian on a closed manifold is related to property (T) of the
1267:
de la Harpe, P.; Valette, A. (1989), "La propriété (T) de
Kazhdan pour les groupes localement compactes (with an appendix by M. Burger)",
961:
1442:
1437:
853:
1202:
Ballmann, W.; Swiatkowski, J. (1997), "L-cohomology and property (T) for automorphism groups of polyhedral cell complexes",
865:
786:), as a result of the existence of complementary series representations near the trivial representation, although SL(2,
1432:
538:
531:. Amenability and property (T) are in a rough sense opposite: they make almost invariant vectors easy or hard to find.
85:
920:
1242:
369:
81:
842:
571:
957:
50:
944:
860:. Its simplest combinatorial version is due to Zuk: let Γ be a discrete group generated by a finite subset
1333:
1211:
1058:
916:
325:
276:
141:
97:
58:
38:
592:
363:
1367:
1024:
1283:(1967), "On the connection of the dual space of a group with the structure of its closed subgroups",
614:
1216:
1076:
1015:
1004:
610:
310:
299:
1342:
1329:
1309:
928:
805:
1317:
1246:
1068:
125:
105:
27:
1233:
1050:. Sorin Popa subsequently used relative property (T) for discrete groups to produce a type II
1406:
1292:
1221:
1185:
1156:
1125:
1072:
1064:
974:
936:
892:
739:
731:
630:
602:
415:
89:
66:
1383:
Zuk, A. (1996), "La propriété (T) de
Kazhdan pour les groupes agissant sur les polyèdres",
1303:
1260:
1121:
1071:. This observation yields Brooks' result which says that the bottom of the spectrum of the
960:
problem numerically on a computer. Notably, this method has confirmed property (T) for the
1300:
1256:
1129:
1117:
990:
932:
662:
395:
109:
24:
790:) has property (τ) with respect to principal congruence subgroups, by Selberg's theorem.
1087:
1019:
949:
794:
723:
524:
483:
314:
280:
101:
93:
42:
1145:"Unitary representations of fundamental groups and the spectrum of twisted Laplacians"
1421:
1280:
1161:
1144:
528:
380:
292:
269:
70:
62:
864:, closed under taking inverses and not containing the identity, and define a finite
1000:
924:
857:
575:
284:
246:{\displaystyle \forall g\in K\ :\ \left\|\pi (g)\xi -\xi \right\|<\varepsilon .}
46:
1176:
Brooks, Robert (1981). "The fundamental group and the spectrum of the
Laplacian".
450:
be a locally compact group satisfying (3), assume by contradiction that for every
1346:
666:
20:
1410:
1108:
Watatani, Yasuo (1981). "Property T of
Kazhdan implies property FA of Serre".
953:
801:
720:
911:, then Γ has property (T). A more general geometric version, due to Zuk and
1012:
606:
446:
Definition (4) evidently implies definition (3). To show the converse, let
1397:
Zuk, A. (2003), "Property (T) and
Kazhdan constants for discrete groups",
1225:
891:. If this graph is connected and the smallest non-zero eigenvalue of the
366:
1296:
1189:
997:, quantifying property (T) for a particular group and a generating set.
520:
Any countable discrete group with property (T) is finitely generated.
1316:, Progress in Mathematics, vol. 125, Basel: Birkhäuser Verlag,
613:
at least two have property (T). This family of groups includes the
465:
is σ-compact (and locally compact) (Bekka et al., Theorem 2.12.4).
494:
has property (T). Equivalently, if a homomorphic image of a group
272:, and any of them can be used as the definition of property (T).
434:)-invariant unit vector, then it has a non-zero vector fixed by
336:)-invariant unit vector for any ε > 0 and any compact subset
935:
at each vertex, then Γ has property (T). Many new examples of
964:
of rank at least 5. No human proof is known for this result.
1232:
Bekka, Bachir; de la Harpe, Pierre; Valette, Alain (2008),
1003:
used discrete groups with property (T) to find examples of
65:
and has "almost invariant vectors", then it has a nonzero
359:)-invariant unit vector, has a nonzero invariant vector.
1314:
Discrete groups, expanding graphs and invariant measures
989:≥ 3) has property (T) to construct explicit families of
895:
of the corresponding simple random walk is greater than
931:
with the same graph theoretic conditions placed on the
1027:
939:
with property (T) can be exhibited using this method.
181:
1376:
International
Congress of Mathematicians Madrid 2006
1082:
equals zero if and only if the fundamental group of
454:
and ε there is a unitary representation that has a (
912:
1042:
343:(4) There exists an ε > 0 and a compact subset
245:
1075:on the universal covering manifold over a closed
956:, for which a solution may be found by solving a
841:≥ 3; the method relies on the fact that Γ can be
418:if there exists an ε > 0 and a compact subset
426:such that whenever a unitary representation of
77:), gives this a precise, quantitative meaning.
474:Property (T) is preserved under quotients: if
90:lattices in algebraic groups over local fields
1241:, New Mathematical Monographs, vol. 11,
8:
852:method has its origins in ideas of Garland,
661:≥ 3. More generally, this holds for simple
599:) and all finite groups have property (T).
351:such that every unitary representation of
1215:
1160:
1034:
1030:
1029:
1026:
915:, states that if a discrete group Γ acts
180:
1285:Functional Analysis and its Applications
80:Although originally defined in terms of
1100:
545:then Γ has property (T) if and only if
74:
69:. The formal definition, introduced by
1054:factor with trivial fundamental group.
574:have property (T). In particular, the
527:which has property (T) is necessarily
825:method of Shalom applies when Γ = SL(
7:
1368:"The algebraization of property (T)"
1018:, so in particular not the whole of
962:automorphism group of the free group
1057:Groups with property (T) also have
948:method is based on a suggestion by
340:, has a non-zero invariant vector.
506:itself does not have property (T).
182:
14:
913:Ballmann & Swiatkowski (1997)
715:≥ 2, the noncompact Lie group Sp(
704:) have relative property (T) for
553:≥ 3, the special linear group SL(
53:. Informally, this means that if
1043:{\displaystyle \mathbb {R} ^{+}}
778:The special linear groups SL(2,
754:The additive groups of integers
124:be a σ-compact, locally compact
738:, 1) and certain quaternionic
317:, converges to 1 uniformly on
230:
217:
211:
204:
1:
1428:Unitary representation theory
148:on a (complex) Hilbert space
1162:10.1016/0040-9383(89)90015-3
665:of rank at least two over a
256:The following conditions on
750:have property (T) include
549:has property (T). Thus for
303:positive definite functions
86:group representation theory
82:irreducible representations
1459:
1243:Cambridge University Press
1143:Sunada, Toshikazu (1989).
572:Compact topological groups
160:, then a unit vector ξ in
1411:10.1007/s00039-003-0425-8
1443:Computer-assisted proofs
958:semidefinite programming
917:properly discontinuously
746:Examples of groups that
719:, 1) of isometries of a
591:-adic integers, compact
502:have property (T) then
156:is a compact subset of
1438:Geometric group theory
1385:C. R. Acad. Sci. Paris
1347:"What is property (τ)"
1338:, monograph to appear.
1235:Kazhdan's property (T)
1044:
977:used the fact that SL(
593:special unitary groups
513:has property (T) then
326:unitary representation
277:trivial representation
260:are all equivalent to
247:
142:unitary representation
98:geometric group theory
39:trivial representation
1226:10.1007/s000390050022
1045:
615:special linear groups
578:, the additive group
478:has property (T) and
412:relative property (T)
362:(5) Every continuous
248:
1025:
872:and an edge between
629:≥ 3 and the special
298:(2) Any sequence of
179:
1366:Shalom, Y. (2006),
1178:Comment. Math. Helv
1077:Riemannian manifold
1059:Serre's property FA
843:boundedly generated
806:free abelian groups
561:) has property (T).
383:has a fixed point (
1433:Topological groups
1297:10.1007/BF01075866
1190:10.1007/bf02566228
1040:
929:simplicial complex
758:, of real numbers
469:General properties
410:) is said to have
243:
170:)-invariant vector
152:. If ε > 0 and
110:theory of networks
49:equipped with the
1332:, A. and A. Zuk,
1323:978-3-7643-5075-8
1252:978-0-521-88720-5
1069:fundamental group
1016:fundamental group
995:Kazhdan constants
945:computer-assisted
937:hyperbolic groups
740:reflection groups
732:hyperbolic groups
631:orthogonal groups
535:Kazhdan's theorem
202:
196:
126:topological group
106:operator algebras
28:topological group
1450:
1413:
1392:
1379:
1372:
1361:
1351:
1326:
1299:
1275:
1263:
1240:
1228:
1219:
1194:
1193:
1173:
1167:
1166:
1164:
1140:
1134:
1133:
1105:
1065:Toshikazu Sunada
1049:
1047:
1046:
1041:
1039:
1038:
1033:
991:expanding graphs
975:Grigory Margulis
910:
908:
907:
904:
901:
663:algebraic groups
355:that has an (ε,
332:that has an (ε,
309:converging to 1
252:
250:
249:
244:
233:
229:
200:
194:
67:invariant vector
16:Mathematics term
1458:
1457:
1453:
1452:
1451:
1449:
1448:
1447:
1418:
1417:
1396:
1382:
1370:
1365:
1349:
1341:
1335:On property (τ)
1324:
1308:
1279:
1266:
1253:
1238:
1231:
1201:
1198:
1197:
1175:
1174:
1170:
1142:
1141:
1137:
1107:
1106:
1102:
1097:
1053:
1028:
1023:
1022:
1008:
971:
905:
902:
899:
898:
896:
815:
813:Discrete groups
795:solvable groups
774:
586:
568:
541:in a Lie group
471:
444:
396:closed subgroup
315:compact subsets
207:
203:
177:
176:
118:
25:locally compact
17:
12:
11:
5:
1456:
1454:
1446:
1445:
1440:
1435:
1430:
1420:
1419:
1416:
1415:
1405:(3): 643–670,
1394:
1380:
1363:
1339:
1327:
1322:
1306:
1277:
1264:
1251:
1229:
1217:10.1.1.56.8641
1210:(4): 615–645,
1196:
1195:
1168:
1155:(2): 125–132.
1135:
1099:
1098:
1096:
1093:
1092:
1091:
1062:
1055:
1051:
1037:
1032:
1020:positive reals
1006:
998:
970:
967:
966:
965:
950:Narutaka Ozawa
940:
927:2-dimensional
868:with vertices
846:
814:
811:
810:
809:
798:
791:
776:
770:
766:-adic numbers
744:
743:
726:of signature (
724:hermitian form
709:
670:
600:
582:
567:
564:
563:
562:
532:
525:amenable group
521:
518:
507:
484:quotient group
470:
467:
443:
440:
281:isolated point
254:
253:
242:
239:
236:
232:
228:
225:
222:
219:
216:
213:
210:
206:
199:
193:
190:
187:
184:
128:and π :
117:
114:
94:ergodic theory
43:isolated point
15:
13:
10:
9:
6:
4:
3:
2:
1455:
1444:
1441:
1439:
1436:
1434:
1431:
1429:
1426:
1425:
1423:
1412:
1408:
1404:
1400:
1395:
1390:
1386:
1381:
1378:
1377:
1369:
1364:
1359:
1355:
1348:
1345:, A. (2005),
1344:
1340:
1337:
1336:
1331:
1328:
1325:
1319:
1315:
1312:, A. (1994),
1311:
1307:
1305:
1302:
1298:
1294:
1290:
1286:
1282:
1278:
1274:
1270:
1265:
1262:
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1254:
1248:
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1237:
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1230:
1227:
1223:
1218:
1213:
1209:
1205:
1200:
1199:
1191:
1187:
1183:
1179:
1172:
1169:
1163:
1158:
1154:
1150:
1146:
1139:
1136:
1131:
1127:
1123:
1119:
1115:
1111:
1104:
1101:
1094:
1089:
1085:
1081:
1078:
1074:
1070:
1066:
1063:
1060:
1056:
1035:
1021:
1017:
1014:
1010:
1002:
999:
996:
992:
988:
984:
980:
976:
973:
972:
968:
963:
959:
955:
954:group algebra
951:
947:
946:
941:
938:
934:
930:
926:
922:
918:
914:
894:
890:
886:
883:
879:
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871:
867:
863:
859:
855:
851:
847:
844:
840:
836:
832:
828:
824:
820:
819:
818:
812:
807:
803:
799:
796:
792:
789:
785:
781:
777:
773:
769:
765:
761:
757:
753:
752:
751:
749:
741:
737:
733:
729:
725:
722:
718:
714:
710:
707:
703:
699:
695:
691:
687:
683:
679:
675:
671:
668:
664:
660:
656:
652:
648:
644:
640:
636:
632:
628:
624:
620:
616:
612:
608:
604:
601:
598:
594:
590:
585:
581:
577:
573:
570:
569:
565:
560:
556:
552:
548:
544:
540:
536:
533:
530:
526:
522:
519:
517:/ is compact.
516:
512:
508:
505:
501:
497:
493:
489:
485:
481:
477:
473:
472:
468:
466:
464:
459:
457:
453:
449:
441:
439:
437:
433:
429:
425:
421:
417:
413:
409:
405:
401:
397:
393:
388:
386:
385:property (FH)
382:
381:Hilbert space
379:
375:
371:
368:
365:
360:
358:
354:
350:
346:
341:
339:
335:
331:
327:
322:
320:
316:
312:
308:
304:
301:
296:
294:
293:Fell topology
290:
286:
282:
278:
273:
271:
267:
263:
259:
240:
237:
234:
226:
223:
220:
214:
208:
197:
191:
188:
185:
175:
174:
173:
171:
169:
164:is called an
163:
159:
155:
151:
147:
143:
139:
135:
131:
127:
123:
115:
113:
111:
107:
103:
99:
95:
91:
87:
83:
78:
76:
72:
71:David Kazhdan
68:
64:
63:Hilbert space
60:
56:
52:
51:Fell topology
48:
44:
40:
36:
32:
29:
26:
22:
1402:
1398:
1388:
1384:
1375:
1360:(6): 626–627
1357:
1353:
1334:
1313:
1291:(1): 63–65,
1288:
1284:
1272:
1268:
1234:
1207:
1203:
1181:
1177:
1171:
1152:
1148:
1138:
1113:
1109:
1103:
1083:
1079:
1001:Alain Connes
994:
986:
982:
978:
969:Applications
943:
925:contractible
888:
884:
881:
877:
873:
869:
861:
858:Pierre Pansu
849:
838:
834:
830:
826:
822:
816:
787:
783:
782:) and SL(2,
779:
771:
767:
763:
759:
755:
747:
745:
735:
727:
721:quaternionic
716:
712:
705:
701:
697:
693:
689:
685:
681:
677:
673:
658:
654:
650:
646:
642:
638:
634:
626:
622:
618:
596:
588:
583:
579:
576:circle group
558:
554:
550:
546:
542:
537:: If Γ is a
534:
514:
510:
503:
499:
495:
491:
487:
479:
475:
462:
460:
455:
451:
447:
445:
435:
431:
427:
423:
419:
411:
407:
403:
402:, the pair (
399:
391:
389:
384:
377:
373:
361:
356:
352:
348:
344:
342:
337:
333:
329:
323:
318:
306:
297:
288:
285:unitary dual
274:
266:property (T)
265:
261:
257:
255:
167:
165:
161:
157:
153:
149:
145:
137:
133:
129:
121:
119:
79:
54:
47:unitary dual
35:property (T)
34:
30:
18:
1354:AMS Notices
1281:Kazhdan, D.
1184:: 581–598.
1110:Math. Japon
921:cocompactly
837:a ring and
802:free groups
800:Nontrivial
793:Noncompact
672:The pairs (
667:local field
649:≥ 2 and SO(
430:has an (ε,
116:Definitions
21:mathematics
1422:Categories
1269:Astérisque
1130:0489.20022
1116:: 97–103.
1095:References
607:Lie groups
442:Discussion
324:(3) Every
300:continuous
1391:: 453–458
1212:CiteSeerX
1073:Laplacian
1013:countable
893:Laplacian
880:whenever
850:geometric
823:algebraic
367:isometric
311:uniformly
238:ε
227:ξ
224:−
221:ξ
209:π
189:∈
183:∀
102:expanders
59:unitarily
1343:Lubotzky
1330:Lubotzky
1310:Lubotzky
1149:Topology
1088:amenable
887:lies in
609:of real
566:Examples
416:Margulis
275:(1) The
264:having
231:‖
205:‖
108:and the
1304:0209390
1261:2415834
1122:0649023
1009:factors
1005:type II
985:) (for
909:
897:
833:) with
762:and of
688:) and (
539:lattice
529:compact
283:of the
270:Kazhdan
45:in its
37:if the
1320:
1259:
1249:
1214:
1128:
1120:
854:Gromov
748:do not
657:) for
641:) for
625:) for
603:Simple
370:action
364:affine
279:is an
201:
195:
140:) a
41:is an
1371:(PDF)
1350:(PDF)
1239:(PDF)
1011:with
923:on a
866:graph
692:⋊ SL(
676:⋊ SL(
645:>
605:real
498:does
490:then
482:is a
394:is a
376:on a
291:with
61:on a
57:acts
1399:GAFA
1318:ISBN
1247:ISBN
1204:GAFA
942:The
933:link
919:and
876:and
856:and
848:The
821:The
804:and
711:For
708:≥ 2.
611:rank
378:real
235:<
172:if
166:(ε,
144:of
120:Let
75:1967
33:has
23:, a
1407:doi
1389:323
1293:doi
1273:175
1222:doi
1186:doi
1157:doi
1126:Zbl
1086:is
700:),
684:),
633:SO(
617:SL(
595:SU(
587:of
523:An
509:If
500:not
486:of
422:of
414:of
398:of
390:If
387:).
372:of
347:of
328:of
313:on
305:on
287:of
268:of
19:In
1424::
1403:13
1401:,
1387:,
1373:,
1358:52
1356:,
1352:,
1301:MR
1287:,
1271:,
1257:MR
1255:,
1245:,
1220:,
1206:,
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